# The Parentheses Riddle

Because I hate you, I included this question on the SSC survey:

It’s a weird trick question, but I would say B is right. Imagine converting “(” to X and “)” to Y. Then the first answer is XYXY, and the second answer is YXXY. I suppose you could group the parentheses in pairs, in which case the answer would be “both”, but in practice few people wanted to say that. Of the 6,000 answers I received, most were either A or B. And one factor had a dramatic effect: age.

This is a big effect. People in their 20s were more than twice as likely to choose B as people their 60s. There’s a slight improvement after 70, but I think that’s just noise caused by a low sample size in that group.

My first thought was that the younger population on this blog is disproportionately techies, and techies have to work with very finicky parentheses all day. There was indeed a slight tendency for techies to do better on this, but it was a very small part of the effect. Even controlling for that, or limiting the analysis to only non-techies, most of the effect remained.

My second thought was that maybe this was an effect of older people gradually getting less sharp. But IQ itself was correlated with the parentheses question much less than age was. SAT score also didn’t correlate very well. And I would expect the most dramatic age-related declines to be after fifty, but the parentheses effect seems to – if anything – slow down then.

My third thought is that older people had less time to waste staring at a dumb survey and trying to solve everything exactly right. But the Squares And Circles Illusion also measures how long you’re willing to stare at a dumb question, and age was negatively correlated with this one. Big Five conscientiousness also did not affect parentheses answers.

I am pretty stumped by this. Right now my guess is that it is caused by age-related cognitive decline, but the connections go deeper than IQ – there’s something about age that affects the parentheses-reading faculty in particular regardless of how smart you are. I know there’s a medium-sized dementia-screening industry. I don’t know how good they are at detecting the normal level of cognitive decline in relatively young and intelligent people – but I would be interested to hear their opinion on this question and whether it has interesting properties beyond those they already know about.

EDIT: Commenters propose that young people might have seen the riddle before on Facebook or Reddit or some other young-person-Internet-place.

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### 248 Responses to The Parentheses Riddle

1. Rasputin says:

Flynn effect.

2. jimmygandhi says:

1) We enjoy palindromes for the “symmetry” they provide. I suspect many older people mistook symmetry for palindrome.

2) Older people may approach this question with the assumption that this is a straightforward question designed to test your knowledge of what a palindrome is. Younger people used to these sort of questions expect there to be a twist that challenges your perceptions and assumptions. I saw the question and immediately assumed the obvious answer won’t be the correct answer.

3. L. Iguana says:

My apologies is this has already been proposed. I looked at this, and my thought was that people who pick A are “thinking in wholes” while people who pick B are “thinking in parts”. I picked A, I look at it and see “good parentheses” “good parentheses”, which is as symmetrical and palindromic as “gg”. I’m reading the pairs as units. B is then “bad parentheses” “good parentheses” – “bg”, clearly not symmetrical, and not a palindrome. Arriving at the “correct” solution of the riddle requires ignoring the pairing of parentheses and focusing on a smaller level of detail.

I know I have a more intuitive, seeing-the-wholes thinking style, and answering this question in the way I did is completely typical of the way I think. It’s a great approach for some problems and leads me astray on others. I would agree with some other commenters who are pointing out its not so much that some people are wrong and others right, they’re just reading the options in a different way.

I think there are questions in the survey that get at whether people are “big picture” thinkers or more detail oriented, I’d be really interested to see if there was any correlation between answers on this riddle and those traits.

4. Ted Levy MD says:

I wonder if there is a greater tendency for older people to confuse a mirror image with a palindrome?

5. formid0 says:

Young people are less intellectually confident, with good reason, so are more likely to double-check their instincts.

I’m 42 and have always been a very slow but high-scoring* test taker. My first instinct on this one was of course A, but I always validate my instinctual errors where possible, so I corrected it to B.

* When time didn’t count against me too much!

6. sansdomino says:

If we’re doing palindromes glyph-by-glyph, )(() is obviously a palindrome. On the other hand, by this definition, also e.g. gág is not a palindrome, while gág is.

No, that’s not a typo: the first is made of a and a combining accent, while the second is a precomposed letter as provided in Unicode. (The first string would reverse, glyph-by-glyph, into ǵag.)

The conclusion to draw from this is that palindromes should not be defined as glyph-by-glyph, they should be defined grapheme-by-grapheme. Glyphs are merely a component of a technological realization of a string, not the string in itself; reversing glyphs is as perverse as the example above about the strings nv or B! being palindromes.

This does not solve the question though, it just passes it on as whether ( and ) are independent graphemes, or if ( … ) is a single circumfixed grapheme. Both can be surely argued for. I would side with the latter and hence pick A.

— I also would not consider →←→← a palindrome, since arrows are semantically independent characters. Or, indeed, consider the strings {}{} and {}}{; if these are regular curly braces, then the former is a palindrome and the latter is not; but if these are X-SAMPA transcription, then the latter is a palindrome and the former is not.

(Age: thirties.)

7. Witness says:

The Categories Were Made For Man, Not Man For The Categories.

8. herculesorion says:

This is a lot of words to restate the joke about William Safire ordering two Whoppers Junior.

9. Joyously says:

I’m in my twenties, and I’d answer A, because A is a palindrome in the deeper spiritual sense of what a palindrome is.

10. fazalmajid says:

I think it’s not as much cognitive decline as the fact pattern-matching from experience becomes a more important part of older people’s cognition, and thus they are more vulnerable to false positives.

11. jhertzlinger says:

If you think of characters as sets of 8 bits, then “nv” is a palindrome, If you think of them as 7 bits, then “B!” is a palindrome.

12. Simulated Knave says:

Unless you define palindrome in the question, all you’re testing is people’s definition of palindrome and how they apply that to the options given. I honestly can’t remember what I answered. But whatever answer I gave, I was right. And so was pretty much everyone else who answered.

Alternately, if one takes the view that not everyone can be right, I think a ‘correct’ definition of a palindrome is a sentence or word that, when reversed and with the punctuation applied in the same places, reads the same.

By that definition, the most correct answer is A or D, depending on how much weight you put on there needing to be letters in a palindrome. C is probably wrong. As far as I can see, B is DEFINITELY wrong, because there is no reasonable definition of a palindrome that you can come up with that works for various common palindromes (madam, I’m Adam or what have you) and which also works for B but not for A.

13. SEE says:

The younger you are the more likely you are to be in the habit of guessing the teacher’s password to prove your knowledge; the older you are, the more likely you are to be in the habit of answering questions to actually inform others who are seeking knowledge.

People approaching the question in the first manner will sit there and try to parse what you mean by “palindrome” in this particular context. People approaching it in the second manner will instead tell you what “palindrome” should mean in this context. Telling the latter they’re wrong is something you should only try after you’ve carefully determined there’s no reasonable way to interpret “palindrome” to mean what they informed you it means.

14. carvenvisage says:

wikitionary alleges that “palindrome” comes from ancient greek palin: “back, again, back again”, and dromos: “running, race, racecourse” i.e. something that reads the same backwards as forwards. ()() literally reads the same backwards as forwards, run your eyes one way, then run them back, it’s the same. ABBA reads the same backwards because letters are stand-ins for sounds and thus unmirrorable. — The latter seems like a subsiduary special case of the former to me.

15. Rachael says:

I chose A, and unlike all the commenters arguing about definitions, I consider myself to have got it wrong, just as much as if I’d given the obvious-sounding answer in the bat and ball question.

As an insight into the thought process by which I got it wrong, I approached it too visually. I read it from right to left and mistakenly subvocalised “open, close, open, close” and recognised that that matched the original string. Misreading B from the right in the same way gives “open, close, close, open”, which doesn’t match the original string (although it is itself symmetrical, and I think I failed to notice the significance of that). I think I made this error because, although I’m fairly used to reading text backwards (and thus wouldn’t mistake a b for a d), I’m not used to reading parentheses backwards, and was misled by the visual look of them.

It surprises me that I approached it that way, because I’m normally a very non-visual thinker. I’m prosopagnosic and aphantasic. (Age 36, for reference.)

• dacimpielitat says:

Palindrom is “a word, phrase, or sequence that ____reads____ the same backwards as forwards, e.g. madam or nurses run.”
reading is visual, or at least has a big visual component.

• acrimonymous says:

This comports with my previous conjecture that people who tried to “read” the parentheses would be more likely to make the visual symmetry mistake than those trying to scan the characters. When you were introduced to palindromes, how were they explained/demonstrated to you? Do you remember?

• Rachael says:

It was very early on, probably pre-school, and in spoken conversation, so it was probably explained in terms of spelling out letter names, like “pee eye pee” or possibly even “puh ih puh” .

16. Doctor Mist says:

This strikes me as a good time to remind folks about GAMES magazine’s invention of the “plaindrome”, which preserves the tortured syntax of palindromes, but sheds the requirement of reversibility, as in

Money man I, an Adam, not even a doom.
Stella, Edna and Otis deified Satan.

• HeelBearCub says:

Huh?

If you just had a few links in here, some italics, and and a reference to the Amish or the USMC, you could be … a certain someone.

• Doctor Mist says:

Is there a prize?

• The Nybbler says:

L.A. Cigar, leaf magical.

17. saintonge235 says:

You may be overthinking this.

I looked at A & B, and thought, “It appears to be obvious that A is correct, but he’s already signaled he’s being devious, so it must be B.”

So I’d have gotten it right, if I’d clicked on a choice, without really thinking it through.

• cuke says:

Agreed, which is why I wonder what this question is designed to test for. Scott seems to be using it to signal some aspect of cognitive functioning, but that seems spurious given that a “B” answer could mean “I reasoned this through literally” or “I knew it was a trick question and didn’t reason at all.” And an “A” answer could mean “I didn’t think too much about this” or “I thought about various levels of meaning of ‘palindrome’ and reasoned through both options and decided that linguistically paired parentheses make a better analog to a letter than one paren does.”

18. Jakub Łopuszański says:

In my case, I answered A, because that was automatic for me. If I had to guess how I evolved this particular kind of automatic response as a programmer with 10+ years experience, then I’d say it’s because very often I have to parse brackets backwards: each time I have to find an opening if, for else branch, the begining of template arguments, the position of conjunction start in condition, I automatically scan the string backwards. And this leads to failure of choosing A, as I don’t really see it as a sequence of XYXY but more like two segments, _ _.
In some sense I have a similar failure mode when I try to push the door instead of pulling, because I see a transparent sticker with label “push ” on door window outside. Yes, the sticker is backwards and mirrored, but this does not prevent me from knowing immediately that it means I should perform a particular gesture with hand. A failure.
Once I realized the question is tricky, because you devoted a whole article to it, I’ve switched to system 2 and noticed that B is also correct. Yes, “also”, was my thought. Only a minute later I was able to accept that A is wrong

19. Jack V says:

I still think A is a better answer. If you write right-to-left text in a language you write the brackets that way round. I guess it’s, do you think palindrome is more like “write the same thing right-to-left, following the spirit as much as possible” or “write the same letters in a reverse order”.

And I’m not sure which it should be. If I was trying to do the second, I’d expect I’d be better-than-average at finding answer B, because programming etc means a lot of methodically working out small steps without getting distracted by what it feels the answer would be. But I’m not sure how I’d write a question to clearly suggest that, without making it so obvious people wouldn’t all just answer B. I guess Scott felt he DID write the question clearly by saying “palindrome”, but I had a broader idea what might count as a palindrome.

Of course, part of it might be, how happy you are with edge cases. My logic means you clearly DO reverse brackets (and less than signs, etc), and DON’T reverse regular letters. But there’s bound to be some symbols where it’s ambiguous, so I might avoid that interpretation for that reason, because I usually prefer an interpretation that always works, not only works with this particular input string. But OTOH, there’s SOME ambiguities in the simple definition of palindrome: e.g. languages with “final sigma”, when you reverse the word, do you get final sigma at the beginning of the word (can you even draw that?) or do you get regular sigma (but isn’t that different?)

…maybe it correlates with how many different languages you speak?? 🙂

• Toby Bartels says:

languages with “final sigma”

Yes, that's a good thought! Ignoring the differences between beginning, middle, final, and isolated forms would seem reasonable, just like ignoring capitalization. At https://en.wiktionary.org/wiki/Appendix:Greek_palindromes there are only a few examples, but a good portion of them begin and end with Sigma, (as you might expect, given how many Greek words end with that letter). There you see that the distinction between the two forms of the letter is indeed ignored, along with capitalization, and the accent as well. (The simplest example, for those who don't want to look, is ‘σοφός’.) This is all before getting to the multi-word examples, where (as usual) spacing and punctuation are also ignored.

So one could argue that there is really only one parenthesis punctuation mark, but with beginning and final forms. Then A is a palindrome, even analyzed character by character; meanwhile, B is the very same text, but written incorrectly. This justifies either answer A or answer C (both), depending on how much tolerance you have for incorrect formatting.

• A1987dM says:

But there’s bound to be some symbols where it’s ambiguous,

There is an official list.

20. acrimonymous says:

I’d like to propose an explanation different from IQ or cognitive decline for older cohorts’ choice of A. That is, I think the meaning, or at least the common explanation, of “palindrome” may have changed over time.

I was exposed to palindromes in high school with the sentence “Madam, I’m Adam.” I understood this to be a palindrome because it reads the same both backwards and forwards, not because the character string is the same backwards and forwards. (Indeed, the character string is not the same, even discounting punctuation.) The same is true of “race car.” I understood it to be a palindrome because it reads “race car” both ways, not because it is “rac e car” both ways.

The way palindromes were explained to me matches the dictionary definition of palindromes. It also generally matches the explanation of palindromes given in Wikipedia, which includes the subset of “phonetic palindromes” (same meaning without same text string both directions), but does not include nonsense strings such as “YXXY”.

In short, the traditional meaning of “palindrome” is connected to strings that have meaning. From this perspective, the correct answer to Scott’s question is D.

What I’d like to propose is that older cohorts are more inclined to fall for the visual symmetry fallacy and answer A not because they are old or dumb but because they are looking for meaning in the text string. While ()() doesn’t really have meaning, applying visual symmetry allows the text string to be “read” both directions, whereas )(() can’t be “read” at all.

Of course, this theory implies that older cohorts are trying to “read” when assessing palindromicity, while younger cohorts are just scanning text strings. This might happen if the meaning of “palindrome” or typical explanations or examples of palindromes have changed over time. Is there any reason to think this? I guess not really, except that Google Ngrams shows a dramatic change in the use of the word “palindrome” that kind of opposes the change in the cohorts answering B. Perhaps, with the dramatic increase in the popularity of palindromes, their explanation became increasing sloppy or changed in a way that leads younger people not to associate palindromicity with readability or meaning.

• acrimonymous says:

When I comment at SSC, I feel like the Bruce Willis character in The Sixth Sense.

• Speaker To Animals says:

• Toby Bartels says:

the subset of “phonetic palindromes” (same meaning without same text string both directions)

You mean pronunciation here, not ‘meaning’.

Although a semantic palindrome would be an interesting idea. In fact, that's how some of the people who answered A have justified their answer in comments here.

I can’t remember which one of the four I eventually chose, but the way I look at it, B is technically correct, but A is practically correct. It seemed like a test of whether you cared more about literal meanings or actual usage, and it seems like younger people being more idealistic is a fairly typical result.

Only “technically correct” for a definition that doesn’t match real-world palindromes. So I prefer the term “not correct”.

22. Fj says:

Did you control for 40 and older people needing reading glasses?

5th graders can revel in riddle books for hours. Most of us grow out of that.

Old people are wise.

Palindromes are symmetric visually as a phrase with meaning however simple minded. There’s a missing verbal aspect to your puzzle that’s missing, there’s only visual response here. (B( doesn’t visually present that way. (A) does and if interpreted as a pair or short phrase, is a palindrome.

The autocorrect feature in this dialog box is annoying to old people who know how to spell and type. Young people don’t know there was once another process. Do they realize quotation marks used to lean left and right too? Perhaps that correlates.

• Toby Bartels says:

Do they realize quotation marks used to lean left and right too?

They do if they comment here, which automatically formats them ‘correctly’. It's making them straight that I (a 43-year-old with some uncommon opinions about typography) have to work at.

23. deciusbrutus says:

The proper graph to display the results of that question isn’t “percent that chose B” on one axis.

It’s one bar that is “percent that chose B – percent that chose A”, possibly next to a bar or bars showing the size of that cohort that chose the other responses.

I’m old and have some trouble reading text on screens without reading glasses. Just now, sans glasses, I didn’t see “()”, I saw “0”. I think that might be your answer.

25. Jotto999 says:

I’m 27 and fell for the wrong answer — hook, line and sinker. I remember thinking of it as mirroring, instead of an actual palindrome. It frightens me that I didn’t think about it clearly or critically enough, especially in a context where it is likely to be a trick question. It’s even worse because I totally know what palindromes are; I know that “racecar” is one, for instance. Something about the parentheses made my brain forget what a palindrome was, and insert something vaguely similar but quite different, and then not even worry about double-checking.

26. bassicallyboss says:

I got this “right”, but only because I thought “How on earth could anyone confuse option 2 for a palindrome?”, squinted at it for a full minute, and decided it was probably the intended answer.

I agree with some commenters upthread that this is likely associated with computer use. I would expect moderate-to-strong correlation between option 2 and [words typed on a keyboard or phone per day], because entering the characters yourself is what makes it feel discrete. I’d also expect weak-to-moderate correlation between option 2 and [ratio of time spent reading from formal, edited publications (books, magazines, newspapers) to time spent reading informal texts (social media, blogs, text messages)], since informal writings are more likely to use unmatched parentheses in writing or emoticons.

I can’t think of a great proxy for either of those that will be reliable w.r.t. self-estimates, though.

Does anyone know if there’s a program or language that treats parentheticals (or their equivalent) more strongly as a unit? E.g., maybe instead of typing the characters separately, maybe you press a button, “()” appears with the cursor in the middle, and you hit esc to get outside the parentheses again? I predict that heavy users any program, or speakers of any language, that handles parentheticals like this would heavily favor ()() as the palindrome.

• colomon says:

Probably at least half of the text editors designed for computer programming will do just what you suggest in your last paragraph.

• bassicallyboss says:

I was thinking more in the sense of treating them as a fixed pair, along the lines of, say, Microsoft Word math expressions/equations. Visual Studio auto-creates the ) and centers my cursor when I type (, but it still lets you edit, delete, and sometimes add them as individual characters.

On the other hand, maybe I just spend enough time on Reddit or here to cancel out the ()-grouping effect from VS.

• Toby Bartels says:

Excel, another Microsoft product, will insert both parentheses and set the cursor between them when you select a function (although again you can't get out with escape and you can edit them individually). Similarly for its open-source clone, OpenOffice Calc.

• Toby Bartels says:

It's hardly a programming language, but the textbook company Pearson has a lot of math books with online assignments, and their web site puts buttons for common mathematical symbols on every page with a problem with free text entry in the answer section. Even though it's hardly necessary (unlike, say, the ‘π’ button), this includes a button for matched parentheses (actually several variations on this theme). If you click this button, then you not only get the matched parentheses and a cursor inside them, but also (if you don't first move the cursor with the mouse or arrow keys) hitting ESC moves the cursor outside of them (immediately after them).

Nevertheless, you can still edit the parentheses as individual characters, and you can input them from the keyboard without getting any of these features. By having the button at all, however, Pearson does seem to be trying to get students to think of the matched parentheses as a single mathematical symbol.

I honestly don’t remember what I answered on the survey. When I looked at it this time, my first instinct was A, but then I actually tried substituting the parentheses for letters (LRLR vs. RLLR) and realized it was B. I could easily see myself doing the same thing when I took the survey, but I could also see myself saying “ugh whatever, it’s 3 am and I’ve been doing these questions for the past half-hour, I’m too tired to think about this shit now” and going with my initial answer of A.

Given that older people tend to get physically and mentally exhausted more quickly, this could explain the age-related discrepancy. As for why the same effect didn’t apply to the squares and circles illusion, maybe it’s just because that one is more visually engaging and tends to wake people up more when they see it, so they’ll actually snap out of autopilot mode and focus on it. Whereas the parentheses one just seems tedious and dull, so people will be more prone to half-assing it, especially if they’re already old or tired or otherwise low-energy.

28. PDP10 says:

Everything you’ve ever written about rationalist examination of purely arbitrary axioms in language, morals, and societies… and then you went and wrote about the “correct” answer to a twerpy buzzfeed riddle.

Yeah, I wonder if there’s going to be some kind of double reveal …

29. Speaker To Animals says:

Is there a correlation between age and autism in your readership? A younger cohort of autistics would be more likely to interpret the symbols as individual symbols rather than as pairs.

30. meh says:

I probably posted this the last time palindromes were discussed, but I can’t help it

• Edward Scizorhands says:

Half-expected that, half-expected this.

31. danjelski says:

A palindrome implies mirror symmetry, and answer A has that. B does not. Substituting in X and Y destroys the symmetry, and therefore changes the problem. So I picked A, and I’m sticking with it.

Nah, that’s not really it, nobody calls bid or qip palindromes, but boob and radar are legit palindromes.

The problem is that ())( is a palindrome only if you use an absurd but ultra-precise technical definition of “palindrome” that sucks all the fun out of it to make it a concept only useful in algorithmic class and logic puzzles.

The strictly character-based definition of palindrome not only destroys all that is good and pure and interesting about palindromes, it also even breaks common rules around parentheses. Humans and programmers both should reject it!

• danjelski says:

Then there are two different kinds of palindromes. The ones of which you speak are about phonemes rather than geometric figures. But parentheses have no phonetic value, and therefore they can’t form palindromes in the way letters can. So only the geometric definition actually works, i.e., mirror symmetry.

Similarly, 321123 is a palindrome in the sense that you get the same integer in either direction. But parentheses have no numerical value, so they can’t be a palindrome in that way, either.

32. EGI says:

Maybe this measures different cognitive styles, something like algebraic / symbolic vs geometric style? (If you parse “palindrome” as “mirror symmetry” a is correct and b wrong.) I had to stare quite some time at this to see why b is right and I have a very geometric style. Why this would change with age? Maybe modern education or interacting with computers makes you more symbolic?

• EGI says:

Let my spouse have a look at this and she got it right on first try as I predicted. Unfortunately she is both more symbolic and more conscientious than I am, so it might be evidence either way.

33. Ben says:

C and D obviously cannot be correct, as they are neither combinations of parantheses as the question suggests nor are they palindromic (htob and rehtien).

For the question of what the question-asker’s intent and definition is, you can actually work it out. The premise of the question is simple and option a is instinctually correct, so therefore it cannot be option a. This is social media trick question 101. While option a, c, or d could be correct with a lot of qualifiers or gymnastics, that’s a complexity and drudge beyond what a social media post warrants, so also disqualified. Option b has a perfect amount of ‘aha, oh right, now ill move on to cat pics’ to it that it has to be what was meant.

So yes, I believe I might have seen this before on social media or reddit and that’s why it skews young for answer b.

Syntactically, since C and D are not ‘combinations of parantheses’, the question does not ask if they are in and of themselves palindromes, thus making them available again for combinatorial responses.

And I thought the simplistic answer was to just look at the characters and see if they run the same backwards, and that the ‘aha’ moment comes when you evaluate the meaning of “()” and “)(” and realize they are not the same (therefore not palindromic).

• Leonard says:

I agree: the question comes down to a matter of author’s intent.

But notice that just because you can work out his intent, that still doesn’t mean you have to answer b. Answering b is what you do if you want to be “right” according to the author. So, you should choose it if you are concerned with pleasing the imaginary Scott in your head. But really, who cares what imaginary Scott thinks? I care what I think. Well, younger people have been taking computer-graded standardized tests their whole life featuring questions just like this. They have been made to care. Old people, not as much. Screw you, stupid test question! a!

34. mdhughes says:

I would answer A because I’ve programmed a fairly extensive amount in Scheme, and to me () is a word unit, while )( is gibberish, an invalid token. I can see that LRRL is a palindrome, but “Madam I’m SYNTAX ERROR” is not.

35. A1987dM says:

IIRC I answered “())(” because I understood what you meant to ask, but technically 1. the parentheses are supposed to turn into each other when reading right to left, and 2. the usual definition of “palindrome” applies only to letters (disregarding case) and not spaces and punctuation, otherwise “A man, a plan, a canal: Panama” wouldn’t be a palindrome because it isn’t “amanaP :lanac a ,nalp a ,nam A”, therefore “Both” is the technically correct answer.

36. Mouth says:

My 63-year-old mother thought it was odd to have zeros in a question about parentheses. Unsurprisingly, she picked A.

37. Harmon Dow says:

I chose A. I’m 70. My mental process was that I viewed each set of parens as a single unit. Therefore, ()() is the equivalent of XX.

People my age (with the possible exception of those people who became secretaries and spent a lot of time typing) have spent most of their lives encountering parens passively, in the form of print – magazines, books, newspapers – where parens operate as punctuation, and where they always occur in sets, except: 1) in lists. Note that in this context, there is no use for the “(” other than as part of the set.

Younger people, on the other hand, routinely encounter parens on the keyboard, where each paren is a separate character with its own key. So that it seems likely that the mindset of a younger person is going to be one that sees a set of parens as two separate characters. 🙁

• Harmon Dow says:

I don’t know how it shows up on your computer, but on mine, the : ( which I typed at the end of my contribution showed up as a frowny face icon, because I didn’t leave a space between the colon and the bracket.

Which led me to wonder, do young people see “colon-bracket” as “colon” “bracket”? How would they read “colon-bracket bracket-colon”? It’s a word (but not letter) palindrome when typed in words, but not a palindrome when typed 🙂 🙂 (colon-bracket space colon-bracket) – or is it? And since what I just typed as the symbol version of “colon-bracket colon-bracket” lacked spaces where the dashes are, I expect it might show up as two smiley faces when I post it, meaning that the computer sees “colon-bracket colon-bracket” (or rather “:) :)”) as an iconic palindrome…when it’s clear that visually as well as symbolically, it’s not…except on a computer screen….eye of the beholder…eye…aye aye…

38. Lawrence D'Anna says:

I don’t think you’re right to think about this as a purely cognitive problem, rather than an aesthetic choice. In their natural habitat, palindromes are made of letters. I think the answer you get for this question has a lot more to do with how one thinks the “palindrome” concept should be applied to parenthesis than it does to the cognitive ability to apply a rule to an example.

• sharper13 says:

I agree. I’m in the 40-50 range and have used keyboards and programming languages for about 30 years, but I answered “Both” because I looked at the question as essentially, “Which of these answers fits philosophically into the idea of a palindrome?” and decided that both examples did, just in different ways (literal and paired).

In the most literal palindrome definitions, either neither or both would, because as others have noted, you’d remove the punctuation first.

• joncb says:

Actually you bring up an interesting point.

If the question is trying to test “can you apply a rule to an example” perhaps if the question primed the definition of a palindrome first? So something like “Given ‘ABBA’ and ‘DGHHGD’ are both palindromes but ‘ABC ABC’ is not a palindrome, which of the following …”.

39. Trofim_Lysenko says:

In the spirit of needlessly slapping on more ad hoc hypotheses to this discussion, my .02 Cents (2 whole cents is more than I have to spare ATM):

I wonder whether the difference in response by age might have something to do with heuristics and mental shortcuts. We all develop them, and the longer someone has been using one, the harder it usually is to get them to try abandoning it.

40. Freddie deBoer says:

Go hang a salami! I’m a lasagna hog.

• Freddie deBoer says:

Mac snubs Bob’s bun scam.

• Freddie deBoer says:

Amy, must I ju jitsu my ma?

• Freddie deBoer says:

41. moridinamael says:

Younger people should be a bit more likely to be programmers, and programmers are more likely to be able to think carefully about parentheses? Any correlation between the answer on this test and occupation?

42. arlie says:

Is this in fact a matter of definition? I’ve seen plenty of palindromes that included non-symmetrical punctuation. e.g. “Madam, I’m Adam”. Perhaps both (a) and (b) is the correct answer – if you think an empty string counts as a palindrome. Or neither, if you think an empty string doesn’t count.

In any case, my “fast system” answer is ()() – it looks like mirror images. My “slow system” answer is )((), since at an individual symbol answer it follows the abba pattern. But my real answer is that you’re asking people to extend the meaning of a term into a space where there’s no current definitions, so there is no “right answer”.

Related to that, consider the following string: “””” – in “traditional ascii”, and on your keyboard, that’s 4 copies of the same character. But in Windows/Mac orthography, there are two versions of the double quote character – opening and closing. The ‘traditional’ on-computer form could be replaced by any combination of opening and closing quotes, depending on the order in which you typed/copied the characters, and whether you tried to correct whatever your word processor first decided. (As it happens, I see a lot of closing quotes at the beginning of sentences, etc.)

Next perceptual question – are ( and ) logically two ways of displaying the same character, like those “smart quotes”, or are they two seperate characters. 😉

Dipthongs would also be relevant. Suppose you use the ae and oe forms that join the two letters into one. Is aeea a plaindrome, or is aeae. (I’m afraid I can’t easily produce those characters, especially in a form that will be seen as intended on readers’ browsers. So use your imagination.)

Maybe I’m just trying to prove that my 60-year-old instincts are right, and engaging in special pleading. I don’t recall answering this survey, but ()() _looked_ right to me (‘fast’ system), and I wasn’t motivated to engage the slow system until you got to the explanation. But my answer has to be “tilt – this question is underspecified and/or meaningless.”

Maybe next time, adding “Have you seen this riddle before?” would clarify a lot.

44. Lambert says:

I wonder what somebody used to a text editor which automatically completes brackets (as you can set up vim to do) would say.

45. Thegnskald says:

Eh. Still going with A.

I don’t regard ‘(‘ as a complete syntactic character; the fact that it is a complete syntactic character with respect to keyboards is more a limitation of how we store strings in computers than a meaningful statement about whether or not half a parenthetical is valid. Thus, ()() is is not XYXY, it is XX. Likewise, ())( is not XYYX, it is an error.

If the “solution” to the way we think about the problem is to convert to another character, then that implies that there is information about the original character we need to suppress in order to arrive at the “correct” answer, which begs the question of what kind of correctness we are optimizing for, and why the information we are suppressing needs to be suppressed to arrive at it.

I suspect the age gap is demonstrating an internal model based on computer utilization, as opposed to literary utilization, more than anything else. Maybe testable if we compare two cohorts of older people, one in the printing industry, one outside it? Depending on when those mental models form, and how amenable they are to modification.

I am in my 30s and don’t particularly consider B to be palindromic despite being a computer programmer vocationally (I studied economics in school and programming was originally a hobby, so I skipped palindrome detection excercises that might have trained me to regard raw ascii reversal as the definition of a palindrome).

The mirrored shape of the characters is not at issue, and I would not consider, for example, “bdbd” palindromic. Simply put, a palindrome has to *read* the same forwads and backwards, and I don’t read “()” and “)(” with the same meaning. I’m open to arguments about character level analysis making B also, and therefore C correct, but C is not correct on the grouds that all punctuation is removed and null strings are palindromes (since null strings neither read forwards nor backwards, they might even be the opposite of palindromes).

On the question of meta-strategy, I assumed simply reversing the characters was the ‘obvious’ answer and Scott was looking for a deeper level of analysis, which may have influenced me to choose A rather than C.

As a pretty new reader of ssc, the fact that Scott is asking why people got this question “wrong”, rather than exploring why they have alternative interpretations of the question is disheartening, and may change my opinion of ssc as a whole.

47. C. Y. Hollander says:

I can’t remember what I answered to this question, but what I remember without a doubt is that it wasn’t a matter of seeing or missing the “right” answer based on my intuitive feeling about whether it ‘looked’ like a palindrome. Maybe I’d react like that to seeing strings like these naturally, but encountering the question on a survey, I naturally stopped to see what the question was about, saw that one looked symmetrical and the other had character-by-character symmetry and that all four possibilities were offered.

If I remember correctly what I thought at the time, I didn’t think it was supposed to be a riddle about whether one noticed that the asymmetrical-looking one was symmetrical by character—how was one supposed to miss that when the question prompted him to look it and consider whether it was a palindrome?—but some sort of psychological question-without-a-wrong-answer about how one applied the concept of “palindrome” to non-literal strings, and answered on that basis.

48. Carl Milsted says:

I answered incorrectly, visually flipping the strings instead of reordering them. Definitely lazy.

Cognitive decline, or professional habit? It could be the latter.

Professionals in their peak earning years make more per hour because they spend fewer hours solving problems based on thinking they have done in their younger years. They have chunked their domain of expertise. What might take a junior thinker hours or days can take an experienced professional minutes. The middle aged professional is paid to look at problems with experienced eyes, not fresh eyes.

It would be interesting to compare the answers of the older takers based on those who have gone back to school for a second career, retired early, etc.

49. HeirOfDivineThings says:

One of my homework assignments in my first compsci classes was to write a program that tells whether a word/sentence is a palindrome. For that, palindrome had to be clearly defined as “first character equals last character, second character equals penultimate character” until you check the entire string.

Using that logic, something like “faded af” is a palindrome, whereas “(faded af)” is not. So that’s why I chose B.

• colomon says:

But note that your homework assignment would reject almost every standard example of a sentence which is a palindrome. The standard algorithm is something along the lines of: remove all punctuation and whitespace from the string; change all letters to the same case; and only THEN do the check you suggest.

• Toby Bartels says:

Following on what colomon just said, ‘faded af’ is not a palindrome under the rule that you stated, because ‘d’ ≠ ‘ ’. So you can see how subtle a rule that accepts ‘faded af’ but rejects ‘(faded af)’ must be.

50. njnnja says:

The interpretation allowing either (or both) to be correct seems best. Younger people are more likely to be instruction takers, and therefore more likely to respond with the most literal interpretation. Older people are more likely to be instruction makers and therefore more likely to redefine “palindrome” to whatever seems right at the time.

And A seems like the more managerial decision, with both positive and negative connotations associated with that. I wonder if younger people in more senior level positions, or older people in more junior positions, answer more contrary to their age group.

51. joncb says:

I can’t remember what my answer was to that but if it was A then my honest reason at the time of the survey would have been “because answer B doesn’t compile”.

In hindsight i think you could honestly justify any answer.

A) If you accept the palindrome in “I, Palindrome I” (Son i am able she said though you scare me watch said i beloved i said watch me scare you though said she able am i son) then accepting the “higher than single character” meta doesn’t seem to me to be a large distance from “sets of brackets are like words”.

B) Single character level palindrome. Since this is the the “official” right answer i figure not too much justification is needed here.

C) Given you accept A) and B) then this one is obvious.

D) The thing that all sample palindromes i can find (on the wikipedia page) is that the end result has some form of meaning. Arguing that “DGHHGD” is a palindrome feels like sophistry to me. Like getting a math question in the form “What is the value of X?” and answering “Boring”. or “Unknown to me”. Yes your answer is probably literally correct but i’m not going to give you any marks for it.

• Speaker To Animals says:

My attitude is if you can’t read it aloud it ain’t a palindrome.

52. regularjoeski says:

The problem and answer lies with the author of the question. The “answer” depends on what the author meant, not what objective others think. As people have stated above multiple answers can be deemed “correct”. This has become endemic in the academic world. If one looks at books for practicing board review questions this problem is seen. The books which explain why the answer is correct and why the incorrect ones are wrong have 5-10% of the questions where the explanation is inconsistent. Another 25% of the questions rely on unstated assumptions which can be best described as pettifoggery. This is the sort of questions and answers which give “intellectuals” a bad name. Younger test takers are still used to giving examiners the answer they want not the answer they need. Older test takers who bother to take these tests bring experiance. As Yoggi Berra said “In theory there is no difference between theory and practice. In practice there is.”

53. Chiffewar says:

When I (21, math student) saw that question, I grumbled and mentally translated the parentheses into ABAB and ABBA. While I’d never seen this exact problem before, I knew its kind well enough to intuit that A was not the palindrome I was looking for.

54. Rin'dzin says:

I’m 50 this year and picked answer B (reading this post – I didn’t take the survey). So I asked myself in what ways am I more like most 20-30 somethings and less like my own generation of SSC readers?

There are some clear answers to that question:

– I have high testosterone levels, roughly equivalent to 20 to 25 male age group
– My lifestyle is less predictable than usual for my age group, I live in shared houses on short-term contracts, have fewer responsibilities and keep student hours
– I have spent much of my productive time for the last decade learning STEM stuff

The last seems like the most meaningful correlate. I get up, drink coffee, get my brain in gear to learn and remember stuff with Anki, spend the day learning new things, trying to implement them, getting frustrated when I don’t understand and elated when some new piece of the puzzle clicks into place, crash at some point in the evening and stare vacantly at the ceiling, then go to the gym and lift heavy objects.

I’m not suggesting that all your 20-30 something readers spend their days this way, but it seems likely a significantly high proportion will have spent much of the last 10 to 15 years of their time in learning mode.

Alternately, there’s a hypothesis that lower testosterone levels correlate with age-related cognitive decline in men. I’m cis female. I’m not familiar with the literature and I’d be interested to hear from readers who know more about that.

I could also hypothesize that unpredictable circumstances make me more likely to look twice at something apparently obvious, but n = 1…

Does my experience match that of others in my age-group who also answered B is right?

55. Nancy Lebovitz says:

I’m in the older contingent here, and I answered C. I (probably) hadn’t see the question before, I did work b out, and they seemed like different sorts of palindromes to me.

In regards to that “(probably)”….my memory has never been extremely good (unlike like a lot of people here), which is why I tend to prefer general principles over detail. I don’t know whether I’m the only one, but sometimes I’ll read an article online, it will seem good, I’ll think of a comment, and then I’ll find that I’d read the article before years ago and made the comment back then that I was intending to make now.

56. Robert Jones says:

It is a big effect, but I still wouldn’t read too much into just this one question. While I answered B, “palindrome” isn’t sufficiently well defined to say that’s definitely correct. “A man, a plan, a canal – Panama!” is famously a palindrome, even though reversing the characters gives, “!amananP – lanac a ,nalp a ,nam A”. Perhaps we’ve just changed the way that we teach people to think about palindromes? Or brackets.

I’m not convinced the effect could be explained by greater prior exposure to the puzzle. My guess would be that SSC readers of all ages spend a decent amount of time online, and Facebook is really a middle-aged person place these days. Also, the same effect should occur for the other puzzles. Similarly explanations relying on the way we’ve been taught to approach these problems fail to explain why the age-effect occurs specifically for this puzzle.

• n8chz says:

Facebook is very much a middle-aged place, and puzzles of this “90% won’t get this” clickbait type are particularly, almost exclusively, popular with the middle-aged and elderly.

• Garrett says:

Where are the 18-25 y/o hanging out now, then?

• Nootropic cormorant says:

Instagram.

57. brmic says:

What several users already pointed out, but in a more general form: It’s highly likely that the survey actually asked several questions here. Some people understand ‘palindrome’ one way, some people another, other people don’t really know what a palindrome is, others again assume it’s a trick question from the start. Unless you have reason to assume the proportion of people in each of these groups is the same across age (and I’d argue you don’t) the age effect doesn’t really tell you anything.
[For a concrete example, consider a questionaire item on socializing which goes something like ‘it’s important to meet friends regularly’, which might have worked fine in the pre internet age except for a rare few pen/phone friendships but these days is no longer soundly correlated to the underlying construct because for a lot of people the importance of socializing and the importance of face to face meets are independent constructs.]

Also

People in their 20s were more than twice as likely to choose B as people their 60s.

should read ‘People in their 20s who read SSC and participated in the SSCS were more than twice as likely to choose B as people their 60s who read SSC and participated in the SSCS. It’s too easy to forget these are not a random sample of 20 and 60 year olds. Any just-so story (aka ‘explanation’) in terms of general trends in the population is only relevant to the extent it applies to these particular subsets of the age cohorts.

FWIW, I would have answered ‘both’

58. David Speyer says:

It would be interesting to ask a version using ordinary letters, such as comparing “bib” to “bid”.

59. Rusty says:

Or you could read it another way. A palindrome has to read the same back and forward. With letters we don’t expect them to read the same shapewise, we expect them to read the same soundwise. But with parenthesis signs there is not sound. So to be a palindrome they should read the same shapewise. So:

A, reading from left to right, convex, concave, convex, concave
A, reading from right to left convex, concave, convex, concave

That looks like a palindrome to me. If you look at the shapes produced if you read directionally from left to right and right to left they are the same.

B, reading left to right, convex, concave, concave, convex
B, reading from right to left concave, convex, convex, concave

So not a palindrome as not the same left to right versus right to left.

And what else can you look at except the shapes? Unlike letters you can’t use sounds.

• cuke says:

I like this interpretation very much.

60. scmccarthy says:

Put me down as another person who knew that B was the “correct” (test-writer-intended) answer if there was one, but still voted A after significant reflection because I think it’s more correct.

B is the conclusion you come to if your experience with palindromes is programming interview questions and you don’t think too hard about what being a palindrome might mean when it comes to non-letter objects. If you do think hard about it, I can see you coming out either way.

61. Deiseach says:

I suppose I’m on the “old” side of the scale here, and I did have to think about it. My instinctive answer was “A” but I had to check myself with “No, it’s a palindrome, write it out” and then yes, B.

So perhaps older people are more inclined to go with “gut instinct” answers. Or yes, our senile brains are decaying within our skulls and the youth have an advantage over us 🙂

dacimpielitat is definitely onto something; when you eyeball it, you (or I at least) tend to go ()() forward is the same as ()() backwards. But when you write it out, ()() becomes )()( which is not at all the same thing. So older people rely on eyeball judgement, younger people write it out? Have no idea!

62. dacimpielitat says:

I’d say the group that chosen A was because the definition of a palindrome says you read the same word from left to right as you read it from right to left; they do not take it literally letter by letter, char by char; when human eye looks at one parenthesis ( it doesn’t mean anything to it, but when it looks at () it has a meaning, in science and in programming languages this means something. so ()() has meaning whereas B doesn’t and when you read A from both sides it makes sense, when you look at B from both sides it doesn’t.
so I guess it is not age related nor IQ related, it is how you look at it and how your brain is prior trained to look at it because its encounter with parenthesis in daily life.

63. r321 says:

I answered both, based I think on contextual parsing.

I construed (a) as being pairs of parentheses, and (b) as being isolated symbols.

Doesn’t seem either wrong or right, it’s just interesting to observe the shift. Basic IT practice forces a mechanical consistency across the two, but it is perhaps more humane to make the distinction that I did.

To get a machine to capture this shift, it might begin by seeking out the multiple patterns available, and then preferring the interpretations that do yield satisfying coherence.

Age 56

64. phisheep says:

I can’t remember how I answered this question originally, but looking at it again:

“(” is not a parenthesis, “(…)” is a parenthesis and is read as a unit. So I read “()()” as palindromic.

By way of comparison, consider the four consecutive “l”s in Llanfairpwllgwyngyllgogerychwyndrobwllllantisiliogogogoch. English speakers just can’t get their heads around how to pronounce four “l”s in a line. But it is not four “l”s, it is two “ll”s as “ll” is a single letter.

That’s a really bad example for palindromes, so consider instead the Dutch letter “IJ” – which really is one letter. I suspect that in Dutch:

“IJIJ” would read as palindromic, while
“IJJI” would not

I’m not Dutch myself, so I’m not completely sure of that – we got any Dutch readers?

I suspect older readers like me are more likely to read parentheses as a single unit, because that is the way they were taught, and that younger readers are more likely to see individual parenthesis-characters because of their use singly in making smiley faces 🙂

EDIT: Rereading the post, I find I rather resent the suggestion that older people getting a different answer to younger people is – as posted by a younger person – indicative of cognitive decline in the elderly. I might as well say that it indicates parenthetical disrespect among modern youth. I am pretty sure I would have answered this the same way in my twenties.

• James says:

I’m not Dutch myself, so I’m not completely sure of that – we got any Dutch readers?

paging @Aapje….

• silver_swift says:

“IJIJ” would read as palindromic, while
“IJJI” would not

I’m not Dutch myself, so I’m not completely sure of that – we got any Dutch readers?

I would consider ijji to be a palindrome but not ijij, but I also don’t consider ()() to be a palindrome.

Worth noting that i and j are also used as separate letters in Dutch, so ijji can be parsed as i-j-j-i (IPA: I j i) or ij-j-i (IPA: ɛi j i), though both of those would be highly unusual.

• cuke says:

Oh your comment about the single use of parentheses in smiley faces is a good one! That makes sense to me. I also see parentheses as a single unit together and am not quite ready to admit cognitive decline.

65. Ketil says:

I think the list of answers is wrong. And the question. The question should be how do you define a palindrome for strings of parentheses? With answers like:

– as a symmetric list of characters/code points
– as a symmetric list of lexemes
– as a string that parses into a symmetrically nested tree
– as a string that looks the same in a mirror
– as any string, since for traditional palindromes punctuation doesn’t count
– no string, since palindromes are defined for letters that make up words

I’m pretty sure that if the question had explicitly defined “palindrome”, e.g. as number one here, you would have gotten a different set of answers. I (late forties) don’t remember what I answered, but I’m pretty sure I thought the question was trying to measure my intuition, rather than my ability to mentally execute a mechanistic model.

Anyway, the interesting thing here isn’t what answer is right, but why age appears so significant. More questions to elucidate this next time, please!

And an aside: for DNA, a palindrome is a sequence of nucleotides that is the same as its reverse complement. So AACATGTT is a palindrome, for instance. I think it is perfectly reasonable to define palindromes for parentheses different from palindromes for text sentences.

66. eterevsky says:

My guess is that this should be correlated with one of two things:

a) How likely you are to already have seen it somewhere om the internet. Younger audience is more likely to have seen it already. It is different from visual illusions, because this is a question with a correct and incorrect answers, while visual illusions only test perception.

b) How strongly are you expecting to be tricked. You see a puzzle with seamingly obvious answer. What is the probability that you will say to yourself “there must be a trick here”, and check the formal definition of polyndrome instead of just clicking on the first answer that you feel is correct.

• Johannes D says:

a) How likely you are to already have seen it somewhere om the internet. Younger audience is more likely to have seen it already.

Yes! This hypothesis seems obvious to me (but perhaps because I’m one of those who had indeed encountered this riddle earlier). There should probably have been an extra choice for “I’ve seen this before and know what’s going on”.

• cuke says:

One of the questions I had about the question is what is it testing for?

One thought I had is that it was testing for basic reasoning, of the kind you’d see on an IQ test, in which case “B” is probably the intended “right” answer and can be gotten to either by being very literal or assuming tricks.

Another thought I had is that the question designers were not testing for reasoning at all, and had no “right” answer in mind, but were testing for cognitive style (literal vs. less literal or whatever), in the same way a personality test has no “right” answers.

Ultimately that I chose “B” says more about my interpretation of the question designers’ intentions than anything about my cognitive style or my reasoning about the content of the question — ie, I concluded that the test designers were more literal-minded, believed there was a right answer, and thought they were testing for reasoning skills.

What does this question actually test for?

67. Moorlock says:

I suppose you could adjust for some of the objections to your question (about whether punctuation should be ignored in palindromes, whether parentheses’ natural pairing changes things, whether there are generational or occupational differences in the way people confront parentheses) by asking about the following two strings, in some unobtrusive font:

“bdbd”
“bddb”

• Boyd Silken says:

Good suggestion!

• cube says:

This is quite interesting. I went to the effort of logging in just to reply to this concept:

If there had only been two options, “()()” and “)(()”, I would have chosen the second answer. But because there was an option for “both”, I chose “both”.

However, now that I see your alternative typographic example, I believe that parentheses may be special to me in a way that “b” and “d” are not, since I probably would have happily chosen the second answer, but not both in your example.

• Robert L says:

You wouldn’t get any interesting or counterintuitive answers though, would you?

• Nancy Lebovitz says:

That’s what I was going to suggest.

And then see whether you get different answers from people who joined after this post.

68. drossbucket says:

Not sure about the age thing, but that question’s got a bit of a flavour of the Cognitive Reflection Test questions – looks like could be measuring something like ability to block out intuitively compelling answers and just apply the rules in a case where they ‘don’t feel right’.

Would be interesting to know how it correlated, though that test is probably too well known now to be any use on SSC readers.

Might be better than the second two of the CRT questions as well, which have a bit of a ‘brainteaser’ quality where seeing a similar puzzle in a book would help a lot.

• drossbucket says:

Thinking about it a bit more, I’d also be interested to know whether people still answer this question wrong if a very unambiguous first sentence is included saying exactly what a palindrome means in this case.

The bat and ball question in the Cognitive Reflection Test is extremely robust to adding clarifications – there’s this funny paper where they try to make the wrong answer less appealing with increasingly heavy handed interventions, including a big yellow warning sign and comments like

Be careful! Many people miss the following problem because they do not take the time to check whether their answer satisfies both the red and blue statements

This only increased the percentage who got the answer correct from 45% to 50%.

(Finally they just gave up and tried adding:

HINT: It’s not 10 cents…

and this mostly worked, though 13% still picked 10 cents…)

• Deiseach says:

I think the Bat and Ball question trips people up because every since we first learned about addition, we’re used to questions phrased as “if X and Y = Z, and X = A, how much is Y?” where the answer is “Y = Z-X”. It’s really hard to overcome this since it’s been drilled into us from the time we were kids, and the question is easily read as “bat and ball cost \$1.10, the bat costs \$1″ so instinctively we answer “\$1.10 – \$1 = the ball costs 10 cents”. We conflate “the bat costs a dollar” with “the bat costs a dollar more” when reading the question, because we’re stuck on solving the sum.

You have to carefully read and get it through your head that the question is not phrased like that. You have to test your answer by seeing “if I say the ball costs 10 cents and the bat costs \$1.00, 100 cents – 10 cents = 90 cents which is not right”. I really had to cudgel my brains to get the right answer! It’s so difficult to see past “1 dollar and 10 cents add up to \$1.10 which is a nice, neat, round answer and that fits with the bat costing a dollar more” – no, it doesn’t but it’s hard to see unless you have the kind of mind which strips out everything but the bare figures and approaches those on a fresh basis every time.

• dacimpielitat says:

I did not know the bat and the ball question, I looked it up and I solved it correctly fast and without any hesitation. However, I answered A.

• drossbucket says:

Yep, agree with this. I also don’t have that kind of mind, and the wrong answer is very ‘noisy’ and always distracts me.

I also went for ()() in the parentheses question (this morning when I first saw it, I missed the survey). I’d like to say that that was because of a careful analysis of ambiguities in the meaning of the parenthesis symbol, but actually it was because ()() ‘looked right’ and I didn’t want to think too hard.

I’m pretty sure IQ is age-normed, so if you’re looking for an age related cognitive decline, it’d be tough to find it just by looking at IQ. Also, most people are reporting IQ results from before age-related cognitive decline sets in.

70. Null Hypothesis says:

My assumption would just be that younger people are more generally primed on answering text-based riddles online. While an older person might just look to see if you’re trying to tell if we know what a palindrome is, a younger person might be more… suspicious? Expecting the question to be a trick because otherwise it’s too seemingly easy to be worth asking.

Internal Monologue of old person:
“A palindrome? That’s one of those forward and backward things. hmm. the first one looks right. mirrored image, forward and back. A is the answer. Moving on.”

Internal Monologue of a young person?
“A palindrome? That’s a forward and backward thing. Pretty easy – so he wouldn’t be asking us to identify them unless there was a trick to it. And that second one is obviously wrong, so why would he give us the option for both or neither? That first answer looks right. Is that a trap? And what’s with that weird second answer? That’s not symmetric at all. Let me think about this a bit, hmm. Yeah, actually the second one is right, painful as it is to believe. And I bet he slipped an extra ‘the’ in the text somewhere just to screw with me more. Moving on.”

I think it’s not a question of intelligence or dementia, so much as simply taking the time to consider if the obvious answer is a trick. Not out of impatience, but just out of expecting that sort of thing. No idea how to measure or validate this claim though.

Don’t get so high and mighty though, young ones. Let’s see how long it takes you to enter the Mines of Moria.

• Deiseach says:

That’s not symmetric at all.

I think that’s part of it, we seem to have a very strong bias towards preferring symmetry. So why does this show up more in older people on the survey? Blessed if I know! It’s interesting and frustrating because it could just be one of those odd ‘it happens but there’s no particular reason, hey the universe is weird, moving on’ things 🙂

71. davidweber2 says:

Palindrome is ambiguous. If we’re treating it as a visual palindrome, it’s the first, because of mirroring flipping the direction, if we’re treating it strictly as symbols, we get the second. I’m curious about the number of answers that chose both.

72. Toby Bartels says:

The problem, Scott, is that all of your analysis is flawed because you don’t realize that you are wrong about which answer is correct. Not that the answer isn't B —I answered B myself when I took the survey—, but the question is ill-posed.

(ETA: While I was writing this, it seems that half a dozen other people wrote the same thing. But my comment is the most comprehensive (so far), so let it stand.)

We've been over this before (when the survey first came out), but for the record: Palindromes can be done word-by-word or letter-by-letter. Letter-by-letter is the most common, specifically letter-by-letter ignoring punctuation (and spaces). By that standard, these are both palindromes, because they both consist of nothing but punctuation, which is ignored; and so the correct answer is C. That said, people might well refuse to accept an empty string of letters (just as the ancients refused to accept the number zero), so it's not unreasonable to think that neither is a palindrome, and the correct answer is D. Ignoring the letter of the law for its spirit[^1], we can look at palindromes character-by-character instead of letter-by-letter, and in that case the correct answer is B. But character-by-character is not the only way to do palindromes, and not even quite the normal way, and we could also do palindromes pixel-by-pixel, in which case the correct answer is A. (While this is certainly an unusual way to do palindromes, the discussion in the survey post showed that it's exactly what some people did.)

[^1] or ‘character’, if you will

ETA: And there were also people in the discussion who took the spirit of the word-by-word type of palindrome by interpreting the pair of parentheses as a functional unit, again making the answer A.

So every answer is justifiable on some reasonable interpretation of the question. There may be an interesting reason why younger people interpret the question more often in such a way as to make the correct answer B, but you're not going to figure out that reason if you think that what's going on is that older people are getting it wrong.

There may be an interesting reason why younger people interpret the question more often in such a way as to make the correct answer B, but you’re not going to figure out that reason if you think that what’s going on is that older people are getting it wrong.

👏👏👏

• Edward Scizorhands says:

“82% of poll-givers gave the wrong question to this answer.”

• Protagoras says:

That’s a better percentage than a lot of political polling efforts manage.

73. meh says:

In Spanish, how are ‘ch’ and ‘ll’ considered for palindromes?

• Vitor says:

In spanish, “ch” is a letter in its own right. I was surprised by this when I learned it as a child, because in my native tongue (german), “ch” is not a letter but plausibly could be. Really hammered home the intrinsic arbitrariness of language.

“ll” is unproblematic either way, unless you’re planning putting it next to an “l” in the middle of the palindrome.

• ana53294 says:

Note: although “ch” is a letter of the alphabet on its own right, the letters in “ch” get treated separately in dictionaries, etc. So, in a dictionary, “ce”* would precede “ch” and “ci” follows “ch”. If “ch” was its own letter, “ch” would follow “cz”.

So while “ñ” is its own letter, “ch” is not. I would not consider “chch” a palindrome (although that could be up for debate).

*The combinations cf, cg do not exist in Spanish (I cannot imagine any word that uses them).

• Vitor says:

That’s not true. I’ve owned dictionaries where words are sorted by initial letter in the sequence (a, b, c, ch, d, e, …) so “Chile” would occurr after “cielo”. I think your observation is correct for a “ch” occurring in the middle of a word though, which just makes everything confusing.

• silver_swift says:

Interestingly, you do still appear to capitalize it as separate characters though, given that you wrote Chile and not CHile. By contrast, the Dutch word for Iceland is IJsland, not Ijsland, but we don’t, as far as I know, ever alphabetize it separately.

silver_swift: no, the letter’s lower-case form is “ch” and it’s upper-case form is “Ch”. Still works fine!

• Toby Bartels says:

But the all-cap form is ‘CH’, so it's not as straightforward as you suggest.

(I knew of that WHAT IF YOU’RE SCREAMING? counter-example too but hoped you would miss it 🙂 )

• Toby Bartels says:

It doesn't invalidate what you said; there's just three forms, that's all.

• A1987dM says:

IIRC the officially correct collation was switched from treating CH as its own letter between C and D to treating it as C plus H a couple decades ago.

Also, in Hungarian digraphs are considered single letters, even taking one square in crosswords, so I guess “Szasz” would be considered a palindrome.

• A1987dM says:

@silver_swift: I seem to recall that traditionally IJ was treated as Y for collation purposes.

• meh says:

This may depend on when the dictionary was published

https://www.nytimes.com/2010/11/26/world/europe/26spanish.html

74. awalrus says:

Maybe the amount of handwriting you grew up with is a factor,
I’d expect more non-techies and older people to think of parentheses as two halves of a unit or the same thing but flipped, rather than different “keys” or “characters”.

Also, it’s a bit of a nonstandard definition of palindrome, the usual definition of palindromes isn’t a pure unicode reversal and string comparison, it ignores punctuation entirely.

• b_jonas says:

This would be my guess too. Younger people are more likely to have some keyboard typing skills, or, these days, mobile phone touch screen typing skills, and so they know that to type “)(()” you press a symmetric sequence of keys, but to type “()()” you don’t.

• Vitor says:

There’s a general frustration I have sometimes, when everyday words (e.g. evolution) get a scientific/mathematical definition so strongly associated to them, that people will stop accepting the original meaning, reinterpreting every use as a (bad) metaphor of the formal definition.

I’m pretty sure the word palindrome existed before computer scientists came along with their coding puzzles.

• Joseph Greenwood says:

I totally relate to this frustration!

• silver_swift says:

Interesting, I have the exact opposite frustration. It annoys me when words with a nice, clean definition turn into mush by everyday usage (I recognize that this isn’t necessarily a very reasonable annoyance, but this recognition does not remove the annoyance).

• Toby Bartels says:

I have both frustrations; whoever uses the word first should win.

• imoimo says:

I have a possibly orthogonal frustration. I’m pretty pro-A Human’s Guide to Words, which I interpret to mean prescriptivism is silly and whatever best matches real-world use is winner by default.

With that in mind, and after reading this comment section, I’m sympathetic to both parties in the parentheses debate. There’s no established norm on how to treat parens in a palindrome sense, so the various approaches all seem reasonable (for now).

+1

75. Said Achmiz says:

As Vitor comments here, the first answer—“()()”—is, at the very least, also correct (and possibly it’s the only correct answer, and the other one is wrong).

That casts serious doubt on the “cognitive decline” hypothesis.

• Robert L says:

Vitor is right – i missed this the first time i read that post because he starts of with some less relevant stuff about the sizing of brackets – in saying that ( and ) must be interpreted dynamically according to direction of reading. ( is opening bracket if you are reading L to R, closing bracket if you are reading R to L, and ()() is therefore a palindrome.

• Vitor says:

Yeah, sorry if my comment was a bit rambly. But the point about sizing is relevant in showing that the “usual” rules for letters don’t obviously apply to parentheses. Same goes for punctuation and spacing, as illustrated upthread (“!amanaP ,lanac a ,nalp a ,nam A”).

• Speaker To Animals says:

My doubts about the cognitive decline hypothesis are based on the fact that so many young people can convert ( and ) to X and Y to answer a trick palindrome question but are unable to use X and Y when determining which bathroom to use.

• Toby Bartels says:

If you're referring to people with androgen insensitivity or a misplaced SRY gene, those have been around forever.

76. oongawa says:

I thought the point of the survey question was something about how one extends ambiguous rules to new situations. Palindromes usually use just the letters and ignore punctuation and case, so “A man, a plan, a canal, Panama!” is a classic palindrome even though the reversed ASCII (and unicode) string is “!amanaP ,lanac a ,nalp a ,nam A”. But how do you define “palindrome” when *all* the letters are punctuation, especially punctuation that comes in mirrored pairs which do not occur in the usual alphabet?

So of course this was some sort of meta-survey and I’m classifying myself with this post. Oh well, so be it.

• Deiseach says:

And if you typed out the reversed version, as with the parentheses in the question, you’d get “!amanaP, lanac a, nalp a, nam A” which is not at all the same thing as “A man, a plan, a canal, Panama!” 😀

The answer to the classic palindrome relies on some trickery: dividing the letters up in reverse as “(remove exclamation mark)(ignore small letter) a (insert space) man (insert space) (insert comma) a (insert space) (ignore capital letter) P (remove comma) (run on into next word) lan (insert space)(insert comma), a (insert space) (run on into next word) c (remove space) a (remove space) (run on into next word)(remove comma) nal (insert space) (insert comma) (ignore small letter) p (remove comma) (remove space) (run on into next word) a (remove space) (run on into next word) nam (remove space) (ignore capital letter) A (insert exclamation mark)”.

Having done all that it now strikes me that human brains are effin’ strange and why/how would anyone come up with that as a fun word game? ‘Hey let’s take a thing, make it into a different thing, then turn it back into something like the first thing by chopping and changing!’

• bullseye says:

Since palindromes ignore punctuation, both A and B are blank. They are still blank when reversed, so they are palindromes.

• HeelBearCub says:

I didn’t take the survey, but my first thought upon seeing it today, was:

“Madam I’m Adam” is essentially the canonical palindrome and it ignores spaces and punctuation marks. Basically you are asking me to make up some new rule on the spot. Yeah the “right” answer is B, but at A has the pleasing ability to be the same in mirror script. Compare this to asking if “(amma)” is a palindrome.

I wonder how many of the older people who did not answer B answered C instead. Was that significant?

• cuke says:

I’m an older person who would have answered “C” if I didn’t care about getting the “right” answer and so answered “B” even though I felt “A” was the truer right answer. Maybe because I’m not old enough yet to not care at all about getting answers “right.” I look forward to that point.

77. IdleKing says:

I bet the effect is less about how parenthesis-parsing varies with age, and more about how rigorous-palindrome-defining varies with age.

It’s easy to conflate palindromy with reflective symmetry. (E.g. “AVA” has reflective symmetry, “ABBA” does not.) It takes a bit of doing to retrieve the more precise definition of a palindrome.

But also — I find it hard to believe ALL these respondents just didn’t notice the sense in which B is a palindrome. Even if your first instinct is A, you’re going to at least take a moment to wonder why the other options are there and why the question exists at all. I remember thinking this question might be trying to ask about some sort of subjective sensibility, since the objective question was obvious and was out of keeping with the rest of the survey.

• philipkd says:

Interesting, a lot of comments on here describe being confused as to whether the definition of palindrome was subjective or objective.

• Vitor says:

Well, questions don’t just randomly appear in the SSC survey, so why was this one included? To me, it seemed obvious that it was about seeing which definition of palindrome people would apply. I never even considered that there was supposed to be one single “correct” answer.

• Not A Random Name says:

I never even considered that there was supposed to be one single “correct” answer.

Well there definitely is, the question is more if there ought?

Let me explain. During my first semester of computer science we learned what a finite state machine (FSM) is and how it works (basically it’s the most simple computer model). Deciding if a given input word is a palindrome is something like the central example I’ve seen people use when explaining what a FSM can do. And also quite commonly they’d show you with letters like A and B and then make you figure it out yourself with ( and ).

So, clearly there is a single, unambiguous definition of palindromes for parentheses, otherwise you couldn’t build a FSM (or computer for that matter) that decides whether a string made entirely out of ( and ) is one. And it’ll tell you that ( and ) work like every other letter, so that ABBA and )(() are both palindromes while ()() and ABAB aren’t.

But just because a definition exists and some people use it doesn’t mean it’s good or right. If there was a survey tomorrow, finding that virtually everybody outside a couple of weird comp sci people believe ()() to be a palindrome… Does that mean the (vast) majority is wrong or does it mean the word “palindrome” includes ()() and academics just haven’t caught on yet on the change of meaning?

• colomon says:

Funny thing is, after reading your comment here, it occurs to me that (as I understand it) the standard definition of palindrome only considers letters and digits. It leaves out spaces, punctuation, and capitalization. For instance, consider the classic palindrome “Madam, I’m Adam”. If you blindly reverse the string as a series of character glyphs, what you get is “madA m’I ,madaM”. Everyone understands that you’re actually comparing the letter sequence “madamimadam”.

Given that, I would expect that in this (again, standard) palindrome definition, parentheses are not considered at all! In which case it would be reasonable to boil the survey question down to “Is the empty string a palindrome?”

Edited to add, if I’d just scrolled down the tiniest bit further I would have seen oongawa raising the exact same issue.

• Hyman Rosen says:

Actually, detecting balanced parentheses is the central example of what a FSM can not do. A FSM cannot count. Even the next complexity level, pushdown automata, cannot recignize palindromes.

• Vitor says:

As I’ve stated in another comment, there is no change of meaning away from the formal definition of palindrome, since the word “palindrome” (greek origin) has existed way before computer science itself!

There is only an assumption by computer scientists that the formal definition must take precedence over previous definitions on account of being more “rigorous” and therefore “correct”.

As a computer scientist myself, I strongly disagree with this kind of thinking. I might as well claim that this entire comment is a word, and I would be formally correct.

• Not A Random Name says:

@Hyman Rosen

You’re right of course. FSM can’t do it for arbitrarily large words, only for words up to a given maximum length. Cellular automatons can – which is what I was thinking of first – but when I simplified to the better known concept I didn’t quite think it through.

@Vitor

Yea that’s why I said the real question is what the word palindrome ought to mean. I personally like the way we do it in CS because it takes the concept that works and makes sense on regular letters and generalizes it to strings of any characters. But I admit that’s nothing more than personal preference.

• Michael Watts says:

@Hyman Rosen —

Why would a pushdown automaton be unable to recognize palindromes? I would think a palindrome would be a classic introductory example, because it reflects the automaton’s use of a stack so cleanly. You push things on in the first half of the string and pop them back off in the second half.

It’s easy to define a context-free grammar that recognizes palindromes, as long as the number of literal symbols in the alphabet is finite. Here is one that recognizes palindromes in the alphabet {a,b,c}:

P -> a P a
P -> b P b
P -> c P c
P -> a
P -> b
P -> c
P ->

I don’t think the requirement for a finite set of symbols is a particularly meaningful constraint, because it is already part of the definition of a pushdown automaton.

What did I miss?

@Not a Random Name —

Yea that’s why I said the real question is what the word palindrome ought to mean. I personally like the way we do it in CS because it takes the concept that works and makes sense on regular letters and generalizes it to strings of any characters. But I admit that’s nothing more than personal preference.

This is pretty hard to square with what you actually said upthread:

Well there definitely is [one single “correct” answer], the question is more if there ought?

[…]

So, clearly there is a single, unambiguous definition of palindromes for parentheses, otherwise you couldn’t build a FSM (or computer for that matter) that decides whether a string made entirely out of ( and ) is one.

• Toby Bartels says:

I personally like the way we do it in CS because it takes the concept that works and makes sense on regular letters and generalizes it to strings of any characters.

By ‘the concept that works and makes sense on regular letters’, presumably you mean the colloquial sense that has worked for centuries, and any generalization of that must also treat ‘Madam, I'm Adam.’ as a palindrome. The straightforward computer-science generalization of that to strings of parenthesis symbols says that the answer is C (both palindromes), contrary to what you said earlier.

• Not A Random Name says:

I think wordpress just ate my reply. Sadness. Anyway, the tl;dr version:

@Michael Watts

Basically I what I meant can be rephrased as “Well there definitely is [a single answer that’s supposed to be “correct”], the question is more if [it] ought [to be considered the end all be all answer to the question]?”

That should hopefully clear up the confusion.

@Toby Bartels

The definition of palindromes I mentioned is a definition for words. It both works on natural language and extends to any alphabet we might desire – like the alphabet {(, )}. By figuring out the rules on natural language we then have the rules for how to decide on all the other alphabets. Which is where ABBA is a palindrome => )(() is a palindrome comes from.

Extending the concept to sentences is not obvious and I don’t know of a good solution.
Just extending the alphabet to include punctuation doesn’t even work on regular language (virtually any sentence ends on a dot but doesn’t start with one).
Differentiating between upper- and lowercase as well as punctuation symbols doesn’t really apply to a lot of alphabets (like, again the alphabet {(, )} that was presumably used in the initial riddle). But it’s what we humans do (trim all punctuation, convert to lowercase, use the single-word rules).

In any case the initial riddle – which is what I was referring to – doesn’t require any sentence based palindrome deciding, all it requires is a word based palindrome decider that can handle the alphabet {(, )}. Which is why I didn’t go into any of this initially.

• Toby Bartels says:

OK, so the simple rule that you want to generalize is the traditional rule for palindromic words. You generalize this to CS words, which may consist of any characters, including punctuation and even whitespace. This conflicts with a different generalization, the traditional rule for palindromic sentences ignoring punctuation and whitespace, but that's no concern of yours. Now I understand.

Although the traditional rule for words also ignored capitalization (and probably apostrophes, although there are no clear examples in English). Not that that affects this example.

• Toby Bartels says:

Me too. I actually thought the question was just asking in a concise way what my definition of a palindrome was, to tie into the questions about ambiguity tolerance.

A) mirror image
B) character by character
C) whatever you want it to be

I chose “c”

78. Squirrel of Doom says:

I’m older, usually quite smart, and got the wrong answer.

I think my process was just to short circuit “palindrome” to mean “mirror image”, and pick the obvious answer (a). I didn’t bother letting the meta level of “hey, that’s too simple an answer to even be in the clever survey” creep into my thinking.

This could be interpreted as “less time to waste on a survey”, or really “less prestige to put into a survey”, but I honestly think that’s how cognitive decline often manifests: You more often think “I don’t really care about figuring out that detail” but that might just be window dressing for “that’s a little too hard for me these days”.

I don’t remember precisely what answer I gave, but I think I interpreted the meaning of ‘palindrome’ creatively when I did.

80. Boyd Silken says:

Maybe if you included which wrong answers were given in the data set, that might help shed some light.

Older people generally have poorer vision so the ‘A’ combination would look a lot more like two O’s to an older set of eyes thus making miscategorization more likely. If this was the case, I would expect when looking at just the wrong answers for all age groups, that older wrong answers might be disproportionately higher on A’s than younger wrong answers.

Try rewriting the question with more distinctly separated symbols to rule out eyesight and image clarity factors. Also along these lines, perhaps record the monitor sizes of the survey participants to see if smaller screens play any role.

81. RavenclawPrefect says:

So as not to bias the sample of people who take this poll, please click here if you have about 3.5 seconds to spare. If you read on, you can’t ever take the poll again on pain of violating random sampling and angering the amateur stats gods. (I was worried about a bias towards people who take random internet surveys from a link on SSC, but then I remembered who I was sampling.)

I’d like to claim my thought process at this question was something like:

1. Obviously A
2. This is an SSC survey, obviously B
3. This is Scott here, maybe it’s just faking us out and it’s actually A
4. Okay, no, it is just B if I ignore my System 1

But actually that’s a lie because my first thought (which seems very likely to be age-coded) was “Oh, I saw this on reddit.”

That one post might not be enough exposure, but this feels like exactly the sort of 140-character gem that would get spread to people who use various newfangled forms of social media.

(Results of the poll here; if less than 25% or so of people have seen it on the internet before, then probably this effect is primarily caused by something else. More like 30-40% and it seems plausible that this is the main factor behind things.)

• Shkaal says:

My thoughts precisely! I would guess younger people would generally spend more time on the internet, so their probability of having seen that question or a similar question before is higher.
I’m not sure if something like daily internet usage was on the survey, but it would be interesting to see how well it correlates with the answer. Also, it might make sense to add follow-up questions to the riddles along the lines of “is this question familiar (yes, I’ve seen it; no, I haven’t seen it; vaguely familiar).

• silver_swift says:

I haven’t taken the scc survey poll, but since this post is the first time I encountered this particular question I answered no.

My thought process was something like
1. Obviously A
2. This is SSC, obviously B
3. “Ok, stop guessing start thinking.” (system 2 kicks in)

And then I grabbed pen and paper to figure out what happens if you write each string backwards.

• Deiseach says:

I like your poll. Any chance of pairing it up with one for the Surgeon Riddle, to compare and contrast? I have a feeling a lot more people are aware of the Surgeon Riddle because they have seen it before/heard about it, but I have no hard data. So the parentheses one is a genuine test but the surgeon one is pretty much no use (until social mores swing around so much that having two dads is a perfectly cromulent answer, and I think we might be getting to that point).

Actually, that does amuse me: the Surgeon Riddle was designed to test sexism/sexist attitudes (because “But Only Men Can Be Surgeons, and Lil’ Timmy can only have one dad, then who is the surgeon if his dad was in the accident????”), then Time Marched On and it became less useful, but it could now be repurposed for homophobia/anti-gay attitudes (“What do you mean it never occurred to you ‘Lil’ Timmy could have two dads’? Bigot!”) 🙂

• Toby Bartels says:

Also note that there are plenty of ways that Timmy could have two dads that don't even touch on homophobia. From adoptive stepfathers to the far-fetched but thoroughly heterosexual premise of the 1980s sitcom My Two Dads, this is something that could happen for a variety of reasons.

• jg29a says:

I think using the surgeon story for anti-same-sex bias is profoundly different from using it for sexism. A big part of the original phenomenon is that people hear the noun “surgeon”, *imagine* a surgeon, and invariably flesh the referent out as a man. A later realization that “he” is a second father doesn’t require revising the imagined referent like realizing “he” is a mother.

I mean, “A horse walks into a bar. The bartender says, ‘Why the long face?’ ”

Did you imagine the bartender as Japanese? Probably not, unless you’re Japanese and/or living in Japan. I probably would have imagined some dark-haired Caucasian with a beard. The existence of that prototype doesn’t mean that I think Japanese people can’t be bartenders (or would be remotely surprised to see one).

• Toby Bartels says:

‘Hey, this is my face, and I resent the implication that there's something incorrect about it.’

• Edward Scizorhands says:

If you asked me this question a second time, I would still say A.

82. BlindKungFuMaster says:

It could just be the Flynn effect. High IQ scores for older people means they score worse on actual tests than younger people with the same score. Maybe next time you need to include an IQ test.

What I noticed when giving lectures about my weird ideas to people from my research institute, is that the more experience someone has, the more prone he is to jumping to conclusions. It’s almost comical how the older guys try to frame everything in terms of their own expertise. They seem to be much more reluctant to actually use their brain to understand new stuff.

• notpeerreviewed says:

This is what I think. And note that Flynn himself doesn’t think the effect that bears his name is due to gains in broad cognitive ability, but rather, cohort-specific changes in very specialized cognitive abilities, such as manipulation of pure symbols. In other words, Flynn thinks IQ scores are not comparable across age cohorts, so even including an IQ test wouldn’t necessarily make the age effect go away.

83. smwls says:

For what it’s worth, when I (a 25 year old) saw this question in the survey, my immediate thought was “of course b) is technically the correct answer, but a) really ought to be the correct answer. Parentheses enjoy a reflective symmetry that the alphabet doesn’t have, and the definition of a parenthetical palindrome ought to take that into account!”

• IdleKing says:

Yep me too. Age 28.

• thevoiceofthevoid says:

Haha, same here! College-age, I think I put “both”.

Same here, and I’m in my late thirties.

Of course b) is the technically correct answer for a strict, “computerish” definition of a palindrome, but by that definition “(radar)” is not a palindrome and “(radar(” is, which is just ugly.

There is a pretty universal rule that parentheses and brackets must be matched and that you have to start with an opening one. This applies to English, French, Japanese, Chinese, C++, Python.

But by the ugly palindrome rule, a sentence with parenthesis that respects the parenthesis rule could *never* be a palindrome, and the “mirrored” version of a valid sentence becomes invalid (when it comes to the parenthesis rule).

Therefore, *proper* definitions of palindromes should be:

– either any parenthesis, bracket, quote mark etc. gets automatically mirrored

– or only alphanumeric symbols (and not parentheses, spaces, periods etc.) count when it comes to evaluating the palindrome-nature of sentences.

But I stand by my choice of a), the only one compatible with humane and productive definitions of palindromes. And I am slightly miffed at Scott considering this a flaw in my reasoning skills.

(now I have an image in my head of an elderly psychologist in a leather chair nodding along and checking the “goes on long rant about the definition of palindrome” box on his clipboard)

• thevoiceofthevoid says:

I think we were all too clever by half for this survey.

• cuke says:

Yes, me too. Age 53.

• anonymousskimmer says:

I don’t get it why everyone is analyzing “()()” and “)(()” as if they cannot possible have a larger context. I assume that they are theoretically embedded in a larger group of characters and try to determine which group of characters within that larger group are palindromic.

E.g. this palindromic sentence itself contains a palindromic word: “Sir, I soon saw Bob was no Osiris.”

as does this non-palindromic sentence: “The tot accidentally activated the radar this eve.”

so E.g. (53+x)(()x+y) contains the palindrome of option B with an extraneous empty bracket pair which may be necessary based on the ad-hoc programming rules which evaluate this expression (and which is far less bad than the two empty bracket pairs of option A). All the parenthesis are symmetrically paired, it’s just that only a subset of them are palindromic.

84. Joseph Greenwood says:

Could the effect be something more innocuous, like a shifting of the definition of palindrome over time? In my experience, people tend to retain the word usages they had when they were younger unless faced with significant social conditioning to do otherwise, and nobody uses the word “palindrome” often enough to create this sort of conditioning.

According to the Websters 1828 dictionary (my lazy source for what old people might use for their definition of this word–if someone wants to respond with a link to an early-ish 1900s dictionary, we can go with its definition instead), a palindrome is “A word, verse or sentence that is the same when read backwards or forwards; as madam, or ‘Roma tibi subito motibus ibit amor.'” (http://webstersdictionary1828.com/Dictionary/palindrome) This parses coherently as ‘a sequence of characters of the form ABA’ but also as ‘something which visually looks the same backwards as forwards.’ The obvious problem is that the palindromes that this definition gives in its example are palindromes in the former sense but not in the latter.

Basically, I am stumped too.

• Joseph Greenwood says:

Robert L seems to have the same kind of idea as me, but has provided a more plausible mechanism.

Sounds plausible. I chose “both” with the reasoning “depends how you define ‘palindrome.'”

85. John V says:

Total conjecture, but this question feels unusually susceptible to meta strategy. If you put a gun to my head and asked me to answer this question in 10 seconds, I’d probably give the wrong answer. If it was put in the middle of a 50 question test of mundane palindromes, then I would probably also give the wrong answer.

But in a test like this, it is almost impossible (for me personally at least) to ignore the meta strategy. I mean, there’s no way the survey maker actually cares whether one knows the definition of a palindrome or can apply it to an “easy” example. It’s more likely to be a trick question.

I can’t be certain about this, but I have a feeling that such meta testing strategy skills are more emphasized in recent years in education (tests have become higher stakes, behavior economics like Thinking Fast and Slow have encouraged various people to adopt CRT like questions or tests, etc.).

• philipkd says:

This rings true. As I’ve gotten older, and as I’ve noticed in smart adults who’ve aged, they don’t seem to get trick questions as well. I find clever jokes take me longer to register.

It could also have to do with the typographic literacy of young people. Digital natives probably know more readily what a character is thanks to their fluency with emojis.

• pansnarrans says:

This my guess too. It screams ‘trick question’; maybe trick questions are just more in vogue more than they used to be so younger people are alert for them.

86. Plumber says:

Unfortunately I’m not supposed at all.

At 50 years old I can definitely notice how much harder it is for me to learn new things than when I was younger, and how I’m just “slower”.

On the plus side I feel anger and sadness less and being bored seems like a good thing now (I’ve had my exciting times, I want safety now).

• zrkrlc says:

Is there an underlying mechanism for this, aside from general bodily decay? On one hand, it seems like there’s just less things to do when you’re older because of longstanding responsibilities, people valuing your experience vs what you know, etc. and we know that disuse can accelerate neurodegenerative diseases. One the other hand, almost all objective measures of brainpower I’ve seen (e.g., reaction time, working memory) tend to peak at around 15-30-ish.

• arlie says:

At 60, what I note is that I tend to resent/not bother learning many classes of new things. I can still learn just fine, except that I’m slower when the new material directly contradicts something already learned (e.g. new names for old meanings and new meanings for old names, particularly both at the same time).

I also have to intend to learn some classes of things – I won’t sop things up just be being around them, particularly in the case where I already have that meaning/method niche filled. So e.g. the latest entries in the business “buzz word bingo” game, or the current employer’s one-true-way of proposing a new project. (In the most recent case, I got a guy who’d interned with my new employer to critique/revise my slide deck, and got much better reactions to the new one than to my draft. And only then did I figure out what principles to apply to the next one.)

• Edward Scizorhands says:

As you get older you notice a lot of “new” things are repackaged old things, but they changed all the names. I am regularly frustrated at the stupidity I am expected to learn, and that frustration is self-inflicted and my biggest obstacle.

87. amaranth says:

(a) is more humane

anyone who tells you that (b) is correct is trying to get you to be a good little slave, usually because they’re trying to be good for their Daddy and He wants you

RUN

88. Robert L says:

If you parse () as pairofbrackets (which is a legitimate thing to do) the wrong answer becomes the right one. Perhaps this is about the computer age altering our perception of typographic characters? To the elderly ( only makes sense when set against ), to the younger it’s just a character like any other and could have an independent existence as a standalone in a password for instance.

I am 57 and got the right answer, probably only because I expected a paradoxical outcome and approached it very cautiously.

• mseebach says:

Even without parsing in pairs, brackets aren’t like regular letters: instead of reading an abstract, discrete character, you read “bracket open”, “bracket close”. But brackets are contextual, they enclose things in pairs. When reading backwards, as required for a palindrome, the context is reversed, and “bracket close” becomes “bracket open”.
In this interpretation, A is clearly a palindrome, B is ambiguous (it’s invalid, as the brackets aren’t balanced).
I don’t think “palindrome” is unambiguously defined for this case, and what is being tested for is some sort of ability to deduce the unstated rules of the game, not “sharpness”.
Thr question should have been, which of these would a programming language string “palindrome” method identify as a palindrome? In which case, I doubt there would have been variation along other axes than programming ability, because it’s clear that it’s the evaluation of abstract characters that’s being requested.

• Thegnskald says:

Unless your instructions are less ambiguous in Hungarian than in English, you do realize that “q” read backwards is “p”, right?

• sty_silver says:

You’ve left out the ‘character-by-character’ part. that implies that each character itself remains as-is. I have a hard time seeing how this can be genuinely misunderstood.

• Thegnskald says:

“Character-by-character” isn’t as unambiguous as you think it is. To demonstrate, which of the following set-of-sets is a palindrome, where a palindrome is defined for these purposes as reading the same forwards and backwards, set item by set item:

[[ABA],[ABA]]
[[ABB],[BBA]]
[[AFQ],[ZRBGUK],[AFQ]]
(And so on – each of these is valid according to some conceivably reasonable interpretation of the instructions)

Now, when I say set item by set item, do I mean that set items are to be treated as atomic units of consideration, or do I mean that each set item needs to follow the criteria specified?

And how do I handle cases, given the ambiguities, which would be correct under one interpretation, but incorrect under the other? And how does the order affect answers, by shifting people’s perception of what they think you are asking through example cases?

The precise mirroring case – pxq – would certainly bias me, after a few similar questions, towards an interpretation of the question in which that is a palindrome, because it is exactly the case where the ambiguity matters. It is tap-dancing on all of my guessing-the-teacher’s-password alarms.

• sty_silver says:

I mean, we’re really only having a factual disagreement about how confused people will be. If you tell me that you would be legitimately confused by the above instructions, then that’s a valid data point, but you certainly shouldn’t leave out the most important part of the instructions while you’re making that point. That was the thing I was complaining about.

All I say to the factual question is that I would not at all be tempted to understand them in any different way, and I’ve always believed that I am significantly below average in dealing with this kind of ambiguity, based on being frustrated by unclear instructions pretty often (unless other people just never admit it). This would not be such a case; letter-by-letter feels clear to me. The case of pxq would not bias me towards doubting that, because I would expect words that look like palindromes but aren’t.

But it’s totally possible that I’m just confused by other kinds of ambiguity than normal people, so I’m not claiming that it generalizes. My guess is that it’s not confusing to most but I don’t know.

I think your instructions are more ambiguous, though I’d still be pretty confident that sets have to be treated as atomic units.

• Toby Bartels says:

Thegnskald, when dealing with a set of sets, ‘set-by-set’ is ambiguous because there are two levels of sets. I'm not even sure that a character of characters is a sensible concept (maybe in Chinese?), but in any case, zqed's example doesn’t involve those. So I don't see how ‘character-by-character’ can be ambiguous. (If you want the other interpretation to be that each character is symmetric in and of itself, zqed's responses don't fit that either.)

In fact, when I explain how Scott's original question is ambiguous, ‘character-by-character’ is exactly the terminology that I use to remove the ambiguity. Do you find my top-level comment on this post to be ambiguous?

• Thegnskald says:

zqed –

That presumes an incorrect model of your test-takers, in which they do not update their interpretation of your instructions as they proceed through your test.

Toby –

If you are only seeing two layers of ambiguity, you are missing one. Also, the fact that you recommended it as a modification doesn’t mean anything to me, except to suggest that you’re being defensive about your own suggestion. It does not, in fact, remove the ambiguity; in fact it makes it worse, by introducing a new level of ambiguity, on top of the existing levels.

I am giving you the following instructions:

Here is a picture of an apple. Notice how it is broken into four quadrants by two lines running through the middle, horizontally and vertically. I want you to redraw this picture, upside down and backwards, quadrant by quadrant.

Now then. What did I just ask you to do? Did I ask you to draw the apple upside down and backwards, one quadrant at a time? Or did I ask you to draw each quadrant of the picture of the apple upside down and backwards, a radically different image?

“Character-by-character” is the equivalent of “quadrant-by-quadrant”, and fascinatingly, is most misleading to those people who understand the directions as you intended without the clarification, precisely because they won’t be reading the instructions as a clarification, but rather as a modification on the instructions as they understand them. It adds a new potential interpretation to the instructions.

The problem is that X-by-X does not convey the information that you think it does. Indeed, in most contexts, it conveys exactly the opposite information: “Sort these train cars, car by car”, suggests you should be sorting within train cars, not sorting the set of the contents of all train cars (although again, it is ambiguous).

“Where a palindrome is defined to be a phrase such that it is the same read in forward or reverse character order, which to clarify, means that the phrase remains the same if you rewrite this phrase in reverse character order, such that the last character becomes the first, the second to the last becomes the second, and so on” is relatively unambiguous. “Backwards, character-by-character” is not. One valid interpretation of “backwards, character-by-character” turns the string “ixMpd” into “ixMqb”.

• Izaak says:

Either apa or qxp is a palindrome. They can’t both be, and yet 40 people classified them both as palindromes.

• Toby Bartels says:

the hypothesized mechanism of ambiguity does not work for Hungarian, due to grammatical reasons

As a linguaphile, I'd love if you could explain why not. The definition at https://en.wiktionary.org/wiki/-enként seems to me to, if anything, favour the unintended interpretation (setting aside whether anyone actually interpreted it that way).

• A1987dM says:

I’m not even sure that a character of characters is a sensible concept

Would you call p̃p̃ (U+0070 LATIN SMALL LETTER P, U+0303 COMBINING TILDE, U+0070 LATIN SMALL LETTER P, U+0303 COMBINING TILDE) a palindrome or not?

• Toby Bartels says:

@ zqed :

Wow, explaining grammar is more difficult than I expected. Did I manage to answer your question?

Yes, that seems clear, thank you!

• Toby Bartels says:

@ A1987dM :

p̃p̃

This is a palindrome, because it consists of two characters, each of which is a lowercase Latin Pee with tilde. Unicode may not have a code point for that character, but it's still a single character, and Unicode actually has a mechanism for creating new characters of this sort, using combining diacritical marks. If you change Pee to O, then the resulting pair of Unicode code points is officially equivalent to the single Unicode code point U+00D5. That's how Unicode is supposed to work; and besides, if it wasn't, the fault would be Unicode's.

But that's just my answer. You're right that you've found a potential ambiguity in what a character is, which could ruin zqed's experiment if any of the people treating a pair of parentheses as a unit would view the pair as a single character. I haven't seen anybody claim that here, but who knows?

• Sniffnoy says:

I believe the Unicode terminology for this sort of thing is “grapheme cluster”.

• Toby Bartels says:

I believe the Unicode terminology for this sort of thing is “grapheme cluster”.

It seems so, and yet in traditional linguistics, ‘p̃’ (or ‹p̃› as they would put it) would traditionally be considered a single grapheme. (At least maybe, depending on the author. Definitely Spanish ‹ñ› would be a single grapheme, but maybe Portuguese ‹õ› would be analysed as two. Although the examples at https://en.wikipedia.org/wiki/Grapheme suggest that that would be one as well.)

• Toby Bartels says:

That's a good test! Scott may have thought that such clear instructions would ruin the test, but perhaps not!

• nyc says:

> In this interpretation, A is clearly a palindrome, B is ambiguous (it’s invalid, as the brackets aren’t balanced).

They aren’t actually unbalanced, there are the same number of each, the issue is that the block depth goes negative:

0(1)0(1)0
vs.
0)-1(0(1)0

There is a simple context where you could insert either string and avoid this — inside an existing block:

```( ()() ) ( )(() )```

But if you’re going to treat “()”, i.e. “open block, close block” as a token then you could also treat “)(“, i.e. “close block, open block” as a token, as you might see in an “else” statement or similar.

Then if “()” is X and “)(” is Y, “()()” is XX, a palindrome, and “)(()” is YX, not a palindrome.

The confusion is also that parentheses behave very much unlike letters. For example, if “()” is X then (()) has no word-construction-equivalent thing. It’s obviously not XX, but there is a sense in which it almost is and the word-equivalent thing doesn’t exist.

• Doug says:

I’m seconding this. If you think of parentheses as shapes or spatial delimiters then of course you should “flip” ( to ) in a palindrome. But if you think of them as lexicographic characters, from the universe as letters, then ( should stay (.

My guess is that people who first started formal writing on a keyboard are much more biased towards the latter.

• FeepingCreature says:

Personally, I knew that the second one was syntactically more of a Palindrome but I answered A because to me it more fulfilled the “spirit” of palindromeness. “)” backwards is “(“.

• C. Y. Hollander says:

I answered A because to me it more fulfilled the “spirit” of palindromeness. “)” backwards is “(“

Yet there are letter pairs like that—for instance “b” backwards is “d”—which are not conventionally treated as pairs in palindromes. How would you answer a question like this: which of the following words (if any) are palindromes?

bob
bod

• wysinwygymmv says:

“b” is not the semantic reverse of “d”, but “(” is absolutely the semantic reverse of “)”.

• C. Y. Hollander says:

“b” is not the semantic reverse of “d”, but “(” is absolutely the semantic reverse of “)”

The person I responded to wrote “)” backwards is “(“. I think it’s a real stretch to take “backwards” in that sentence to mean “semantic reverse”—enough so that if that is what Feeping Creature meant, he should be the one to say it.

• FeepingCreature says:

That is the sense which I meant, yes. If for some reason I had to parse a sentence backwards, I’d still parse “b” as “b”, but I’d have to parse “)” as “(“.

To make up words, “b” is a lexical token and “(“, “)” are structuring tokens.

If you parse “()()” backwards as “()()”, you get the same tree structure. If you parse “)(()”, either forwards or backwards, you get a syntax error.

• Edward Scizorhands says:

“)(()” is just wrong. I don’t mean ‘not a palindrome’ I mean just wrong. If SETI found that message from space they should burn down the telescope.

• Izaak says:

is “beginning end” a palindrome?

• A1987dM says:

b and d don’t have the Unicode Bidi_Mirrored=Yes property, whereas ( and ) are Bidi_Mirroring_Glyph of each other.

• Toby Bartels says:

Thanks, that makes it official! Along with Unicode whitespace and capitalization properties, this makes it easy to define and program any type of palindrome.

• Toby Bartels says:

It's not mirrored, it's rotated. (You can tell because of ∀.) But what do you mean by the same bidirectional class? Never mind, I see that you mean the Unicode property. That seems like a mistake to me, but I can't find the data of what its mirror image is claimed to be. Found it, and it says that there is no suitable mirroring glyph. Well, the reason for that is that it's a mistake! I suppose that I should find some mathematical texts in Hebrew or something and try to check that.

• Toby Bartels says:

The arrow symbols are not mirrored!? Well, that's just ridiculous. Maybe that Unicode property doesn't really mean what I think it means.

• Doug says:

The shape and orientation of “b” and “d” are not intrinsic to their meaning. You could invent a different alphabet just as easily switch them or replace them with other phonemes entirely and it would make just as much sense.

In contrast the shape and orientation of “(” and “)” are highly relevant. One indicates *everything to the left* and one indicates *everything to the right*. Imagining an alternative alphabet where they’re reversed would be very silly )see how stupid inverted parentheses look(

Inside my head, whenever I put (something), I always imagine that I’m drawing a rounded box around the parenthesized text. The ( and ) are just the ends of the box. I never imagine anything like that for letters, they’re just purely abstract symbols.

Going back to my previous contrast I think this is why people who older people have a different intuitive sense than younger people. Writing with pen and paper is fundamentally *drawing*, so therefore it’s normal to think of ( and ) as the ends of a box circling some text. In contrast typing, is fundamentally sequencing a series of pre-formatted symbols. You think of ( and ) as abstract symbols on the same level as other ASCII characters.

Most people’s first experience using parentheses comes when they’re starting to write more formal papers in middle school and and high school. Older people probably did that with pen and paper, so they’re in the “drawing a box” mindset. Whereas younger people probably used a computer, so they’re in the “symbol sequence” mindset.

• HeelBearCub says:

)see how stupid inverted parentheses look(

Sure, but/and:
>this looks a lot less stupid, unless you use XML or HTML<

Which basically shows that the shape is both important, but also at least a little bit arbitrary.

• Lambert says:

Also, see German vs Spanish quotation marks (Guillemets).
The Spanish write «quote», whereas the Germans write »quote«.

• Toby Bartels says:

Lambert, I think that you mean the Poles. The Germans write „quote“ (which is still reversed, although it's hard to tell in some fonts).

• Toby Bartels says:

)see how stupid inverted parentheses look(

French mathematicians use them for open intervals. Where an anglophone would write (0, 1) for the open internal from 0 to 1, a francophone would write ]0, 1[ instead.

The beauty of this system is precisely that it takes advantage of the ambiguity about whether bracketing symbols should be reversed in a palindrome! You could say that ]0, 1[ is semantically a palindrome (ignoring the numbers), because both ends are open. Or you could say that ]0, 1] (the interval from 0 to 1 that's open at 0 but closed at 1) is semantically a palindrome, because at both ends, the endpoint belongs with the numbers less than it. Either way make sense!

• Lambert says:

I’ve seen both forms.
The angle brackets I’ve come across have all been in literary works.

• Toby Bartels says:

Fair enough then. (Although I've also seen the round ones in literary works.) Come to think of it, the first literary work that I ever read in German (an old copy of Im Westen nichts Neues) used guillemets, although I can no longer remember which direction they went.

• Bugmaster says:

Exactly. I parse A as “expression, expression”, and B as “syntax error”. So sue me.

• wysinwygymmv says:

Agree. I’m in my 30s and a tech worker, so I don’t think getting the wrong answer should be attributed to “cognitive decline”.

In some sense, programmers are more likely to get this wrong unless they’re looking for the trick — because they will instinctively want to balance the parentheses rather than interpret the parentheses as distinct characters.

• Edward Scizorhands says:

You don’t need to defend giving the wrong answer. Know that you gave the right answer and be happy.

• belvarine says:

You and most Lisp programmers I’d imagine. Suffering in cognitive decline, all of you.

• jensfiederer says:

My reading exactly. It almost HURTS to have to look at B, I did not spend much time looking at it at all. But my reading of A was actually mirror-reversed….open close open close read EITHER way.

Did Scott check whether there was a correlation between tech/non-tech as well as age?

• mtl1882 says:

I am 57 and got the right answer, probably only because I expected a paradoxical outcome and approached it very cautiously.

This is the key, I think. I initially wanted to say A, but then I realized that might be too easy, and realized that if you reverse them in a way that doesn’t just “flip” them, B is the answer. I do standardized exam tutoring for kids ages 10-18. A lot of them would pick B because it just looks complicated to them, and A seems too easy. Adults are more likely to try to reason, and A looks like it has more logic to it, and seems to fit the palindrome definition if you don’t think about it too hard. B looks off at first glance, and I don’t think most people are cautious.