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You know that old game, where a player challenges another if he can count to 11 using his fingers?

Example: Player A attempts to count to 11, starting from his very left, and moving to his rightmost finger (both hands). The max he goes up to is 10.

Player B starts counting using the left hand, and counts up to five. Then on the other side, he counts backwards, starting with 10, going down to 6 as he counts fingers backwards. He concludes that since 5+6 = 11, he has won the game.

What is the logical fallacy present in this? I can't seem to think of a specific name for this, and it's really bugging me.

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    Why doesn't he stop one sooner? Count to five, swap hands, count 10, 9, 8, 7, and stop. Then since 5 + 7 = 12, I suppose I would win.
    – davidlowryduda
    Jan 19, 2014 at 6:43
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    It isn't a fallacy; it is more simply an error.
    – Mauro ALLEGRANZA
    Jan 19, 2014 at 16:21

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This is not a specific (argumentative) fallacy. It is a humorous conflation between the ordinal and the cardinal use of numbers, e.g. "the sixth finger" vs. "five fingers" in this case.

As an aside: There was an interesting experiment with children (age 4, I think) that shows how humans have an early understanding of cardinality. (I don't remember the source.) It goes like this: the subject is shown ten cookies in a row. The subject is asked how many cookies there are. The subject proceeds to count them and answers "ten". A cookie mid-row is taken away and the question is repeated. Now, the interesting result is that a certain percentage of children count the cookies by their cardinal order, note that the "six(th)" is gone, but that the "ten(th)" is still there and conclude that there are still "ten" cookies in front of them.

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You are mixing up two things :

  • the number of element you use to code your number
  • the number of element you can code with this elements

Assuming you have ten fingers and that you begin to count at zero, using only fingers as an atomic information storage, you may count up to 210-1=1023.

Now you may use your fingers as "non-atomic" information storage. For example there's a technique which use the three visible division of each finger and use the thumb to mark one of the other digit of the same hand. This way you can easily count to 12 on one hand. There are also other techniques which enable to count bigger number on one hand.

Moreover in your example, the user must remember in his mind how many times he used each hand. So it's not a pure finger coding technique but a mix with mind memory.

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I don't know what the fallacy is called but the fallacy is that counting down from 10 to land on any number X does not imply that X is the number of fingers you counted! Thus you can't add X to the 5 (the first part) and call that the total.

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