Punk math philosophy and podcast

I've just signed up for Tom Henderson's Math for Primates podcast on the strength of this interview he conducted with Technoccult about his theory of punk mathematics. My dad's a mathematician and I love math, but stopped taking it after first year university calculus and stats and feel like I'm losing it by the year. I like Henderson's approach to the subject! Bonus: Tom helped Jane McGonigal and pals make the awesome Superstruct game.
So, the concept I pitched to Nick was, "Let's talk about math from the platform of 'Math that humans are likely to want to know, because it's about other humans.'" Social conflict. Sex. Beauty.

It gives us an excuse to talk extensively about game theory. And, game theory is a key place to teach humans mathematics, because we seem to have some optimized "cheat detection" in our brains.

Let me give you an example, it's something like, uh...

There are four face-down cards on a table. There is a rule: "If the number showing is even, then the back of the card MUST have a vowel." Now, given an E, 3, 8, D, what is the smallest number of cards you need to flip over to verify that the rule is being followed? Maybe I fucked up the puzzle. But, anyway, the answer as I've phrased it is NOT E and 3.

You need to make sure that 8 has a vowel on the back, and you need to make sure that D does NOT have an even number on the back.

Everyone gets this wrong, basically. Well, non-mathematicians always do, and I'm pretty sure I got it wrong because I get every answer wrong on the first try. Punk as fuck. Now, if you ask the same people a logically equivalent question: "You see four people. Two are drinking beer and two are drinking coke. Whose IDs do you have to check?" No one says you have to check the ID of the coke drinker. Because who cares how old they are? If it's the same puzzle, but phrased as a problem of possible social cheating, we nail it.

The Philosophy of Punk Rock Mathematics - Technoccult interviews Tom Henderson (via Beyond the Beyond)



  1. This is silly. He got the question completely wrong! If you say “if the number showing is even then the back must be a vowel” then you only need to check one card: the one that shows an even number. That’s because you made the rule talk about the *showing* side.

    If instead the rule said “if one side of the card is an even number, then the other side must be a vowel”, then how many cards do you need to check this time? *THREE* of them. That’s because, you see, there could be a “4” on the other side of that “3” and so it would be violating the rule. Any of those could be violating the rule, except the first one.

    And if instead the rule said “there is a number on one side and a letter on the other side of each card. If the number is even, then the letter must be a vowel. Which cards must you flip to make sure the rule is followed?” – Then you only need to check two cards. 8 and D.

    Hope that helps.

  2. I believe he did indeed “fuck up the puzzle”. At the very least it needs to be defined more concisely. I believe that we need to add the condition that the cards have numbers on the front and letters on the back. Further, we need to clarify that we are shown the front of two cards (the 3 & 8) and the back of two other cards (the E & D). If this is the case, then yes, we need to check only the 8 & D cards to ensure the rule holds.

  3. I think many people fail at this, because the description is deliberately vague.

    And the solution – 2 cards – is not 100% correct.

    You need to make sure that 8 has a vowel on the back, and you need to make sure that D does NOT have an even number on the back. (End Quote)

    You also need to make sure that the “3” card doesn’t have a even number on the other side. This means you have to flip 3 cards.

    The given solution 2 cards assumes that all cards have a number on one side and a letter on the other. The rules don’t actually claim this. So the “3” card need checking.

    1. You do not need to check the 3. The rule only states that evens must have a vowel. You can’t necessarily imply that the rule works the other way (all vowels must have an even number). Nothing in the rules controls what is on the back of odd number cards.

      Similar to “Anyone named Debbie is a female” does not imply that all females are named Debbie.

  4. I think you only need to flip the 8, since the puzzle states “If the number showing is even…” Only one card has an even number showing.

  5. To make the drinking age test equivalent you should phrase it:
    You see four persons, one drinking beer, one drinking coke, one very young and one very old but you can’t see what the two last ones are drinking. Who to check? Obviously the beer drinker and also the drink contents of the young one.

  6. If the question is “what is the smallest number of cards…”, then the answer wouldn’t be “E and 3″.
    Also, agree with Jere7my.
    Looks like the guy really did fuck up the puzzle to the point of un-relevance.

  7. Really the only vagary in his question is the definition of an “if…then” statement, which may not be known without a lesson in discrete mathematics. Here’s an explanation:
    E need not be flipped because it is vacuously true; it does not satisfy “If the number showing is even” and as such an be a vowel or not yet the statement holds.
    3 need not be flipped for the same reason.
    8 needs to be flipped to ensure the truth of the second part of the statement.
    D needs to be flipped to ensure the contrapositive holds, that is “If the back does not have a vowel, then the front is not even” which is logically equivalent to the original statement.

    Basically an “if…then” statement is only false when the “if” is true and the “then” is false.

  8. The drinking example would need a bit of work to become logically equivalent too, with some people concealing their drinks and some obviously being well over 21.

    I’m not sure it’s the social aspect that makes it easier: just rewriting the rule in the card game to “If the number showing is even, then the back of the card MUST NOT have a consonant” makes it much easier. People just understand ‘forbidden’ better than ‘mandatory’.

  9. primes of passion

    you got game….theory

    lovely n-dimensional curves

    20 sided fuzzy dice in the mustang of love

    there’s a fraction too much friction

  10. Even if you forgive the sloppily-phrased card puzzle, the drinking example isn’t equivalent; it’s far simpler.

    A better analogy would be “You see four people. One is drinking beer, one is drinking coke, one is 22 years old and has an unknown beverage, and one is 19 years old and has an unknown beverage. Whose IDs and/or drinks do you have to inspect?”

  11. For a guy supposedly devoted to discussing math that people will care about, he completely failed to make that card problem something I would want to care about.

    Calling it “punk” will make it alluring? The windblown comb-over, maybe?

  12. Wineman, you nailed it.
    The puzzle is very old and the results are consistent. When the same logic underlay puzzles in social situations, the subjects tend to ace them.

  13. In re: the puzzle source: @jdyer on twitter linked me to the original Four Card Task, invented by Peter Wason and Philip N. Johnson-Laird: http://www.socialpsychology.org/teach/wason.htm

    In re: the social phrasing: Thanks to Dan Wineman for your correction. That looks logically equivalent, and the cheat-detecting check is “Ignore the cola drinker, ignore the 22-year-old; check only the beer-drinker’s ID and the underage drinker’s beverage.”

    In re: the hair: Nothing seems to help~

    -Tom Henderson

  14. Ah yes, the Wason selection task. There’s a huge literature on the topic and several competing interpretations of what’s actually going on (including many complaints about the dissimilarities between the card case and the social case).

    Despite these complaints it’s been taken by cognitive scientists as evidence that the structure of the mind is modular. The pioneers here are Cosmides and Tooby, though they push evolutionary psychology in frankly adaptionistic terms that make many (including myself) somewhat uncomfortable.

  15. Have to agree to some extent that the question could be better worded. Current wording:
    If the number showing is even, then the back of the card MUST have a vowel.

    Better wording:
    If the card has an even number on one side, then the other side of the card MUST have a vowel.

    Otherwise the equation can arguably be said to only go in one direction. If the equation only goes in one direction, it doesn’t matter what is on the back of the D, and only the 8 needs to be checked. If the equation has symmetry, both the 8 and the D need to be checked. The original wording above, which, incidentally, is normally how the problem is presented, has no assertion of symmetry.

    Assuming there is symmetry in the above case is arguably a logic error.

    1. If the number showing is even, then the back of the card MUST have a vowel…given an E, 3, 8, D…

      Yeah, my first answer was “One Card, the 8″.

      Only two cards are showing numbers, the 3 and the 8. The two letter cards aren’t showing numbers, so they don’t need to be checked, no?

      Or did the person asking the question really mean The rule is: If the card has an even number on one face, the reverse face must have a vowel.

      Then, given four cards showing E, 3, 8, D, what is the minimum number of card that need to be turned over to verify that all four cards follow this rule? In this case the answer would be two.

      Note: there’s no rule that non-even mumbers can’t have a vowel on the reverse. So:
      E = Don’t care
      3 = Don’t care
      8 = Care, the other face must be a vowel
      D = care, the other face must not be a vowel

      Think about it, you could get the right answer (“TWO”) and still get the reasoning wrong (the wrong “E and 3″).

      I hate these questions which use terms and/or phrasing that are ambiguous – thus giving you multiple ‘correct’ answers – ESPECIALLY when the questions are for logical puzzles .
      This habit/trait/quality of reading multiple possible meanings of an imprecise question is the bane of people with Asperger Syndrome, borderline AS, and computer hackers.
      See http://catb.org/jargon/html/speech-style.html

    2. #8 – Actually, if the equation has symmetry, you also need to check the “3” in addition to the “8” and “D”. (Unless you further stipulate that all cards have a number on one side and a letter on the other.)

  16. He’s holding ONAG (On Numbers And Games) – one of the coolest math books EVER. A bit technical, but still: the book builds up a theory that encompasses both full-information two-person games AND surreal numbers. Yum.

  17. the puzzle is trivial. otoh, the description of the puzzle shows rly why technical types should never write end-user documentation, srsly my camera manual is easier to follow.

  18. I love this kind of math, especially when it is called by the name most of us who study it have decided to use, which is: psychology. I’ll admit that “punk math” has a bit more panache to it. But this problem (the Wason card task) has a long history of study in cognitive and social psychology, only some of which has to do with game theory.
    Psychology often gets a bad rep, and one of the reasons is that its cool and applicable findings are often called by other names.

  19. The rule is “”If the number showing is even, then the back of the card MUST have a vowel.”

    So, flip over 8. Make sure it has a vowel.
    Flip over D. Make sure that the side is odd.

    You don’t care about 3, because it isn’t even.
    You don’t care about E, because an odd card could have a vowel or a consonant.

    The two statements “if P, then Q” and “if not Q, not P” are logically equivalent.

    But “If not P, then not Q” and “If Q, then P” are not.

    1. The two statements “if P, then Q” and “if not Q, not P” are logically equivalent.
      But “If not P, then not Q” and “If Q, then P” are not.
      Whaahh? A typo?

      Replace the P’s and Q’s for the second statement and you get the first statement.
      “If not P, then not Q” and “If Q, then P” (logical equivalents) becomes:
      “If not Q, then not P” and “If P, then Q” (logical equivalents and your first statement).

      This one was the least intuitive one from my Discrete Math class, until I used an everyday example: If it’s Thursday, then it’s payday. If it’s not payday, then it’s not Thursday.

  20. This importance of representation in making decisions has been well researched. Check out Gerd Gigerenzer’s Calculated Risks, where he describes the problem of probability versus natural frequencies. If you think that this is simply a topic for puzzles and brain teasers, think again. Gigerenzer makes a convincing argument that some contemporary medical practices are as reliable as medieval brain surgery.

  21. The logic puzzle was something Tom gave as an off the cuff example, not something he’d prepared in advance. He’d wanted to find the original, but couldn’t find it before I wanted to run the interview. So it’s as much my fault for rushing the interview out the door.

  22. E3D8
    if the number showing is even, then the back must be a vowel
    we assume that the backs of the cards, if seen, would follow the rule.

    if p then q.
    p therefore q
    not q therefore not p

    this is the logic.
    hence, not[vowel(side)]==>not[even(side)] or even(side)==>vowel(side)

    both conclusions must hold. this is why you test D as well as 8.

    it seems that you would want to test E, but that is incorrect for this “toy problem” since vowel(side) DOES NOT ==> even(side) and odd(side) DOES NOT ==> consonant(side)

    the rule is lame however, since it refers only to what’s face up not to what’s potentially face up. this is important because otherwise, you’d have to check every card.

  23. What everyone is overlooking is that you are comparing a set of rules (even numbers must match vowels) to which the reader has only just been introduced to a set of rules that the reader encounters frequently, and has probably had personal experience of (legal drinking age). Of course the second ruleset is easier to process, they’ve already learned it!

  24. hmm well the question is clearly for “If the number showing is even, ….”, so that is the precinct..There are only two cards with number showing and only one is even..So its just one option..Its definitely not the best phrased one though :(

  25. Wow, I always liked this puzzle to demonstrate the “confirmation bias” (although see the wikipedia page for that term to see an interesting critique by Klayman and Ha, indicating that in many “real-world” cases that strategy actually might make sense).
    I would love to not only hear from this intelligent crowd their reactions to that critique, but also a response to a question I have about this task:

    It seems like turning over each card offers either a chance for confirmation or disconfirmation (or both or neither) of the rule (correctly stated as indicated in comments above):
    “E” – offers only confirmation
    “3” – offers neither
    “8” – offers both
    “D” – offers only disconfirmation

    My question is: Most people indicate that the subject should turn over “8” and “D” – is the value of turning over “8” purely in the chance for disconfirmation? Because, if there is value in confirmation, then couldn’t that be possibly achieved by turning over “E” as well?

    Thanks! This was an awesome comment thread to read – I need to come over to boingboing more often!

  26. I think the only thing which might be slightly off with this question is the “verify” part, which usually implies the idea of confirmation to a person, although this is not always necessary, as disproving the theory is a form of verification on whether the statement is true.

    BASICALLY, we just have to search every way to disprove something (find a situation where the rule doesn’t work), which means

    3: is automatically ruled out, cause the statement wasn’t concerned with odd numbers, therefore any value which they have is irrelevant.

    E: in the question it said an even number has to have a vowel. It doesn’t say an odd number can’t have a vowel as well, sharing is caring after all. So it is irrelevant as well.

    (with these two answers, some might say ‘but you have to check if its right. really? does it matter? if its right, who cares? if its wrong, we’ve said its irrelevant, so that doesn’t matter either)

    8: you gotta check cause its even, and if its got an odd number, bam! the statement is wrong.

    D: yeah, you need this. referring to the statement “an even number MUST have a vowel”, no vowel, no valid statement.

    TOK FTW!!

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