In What Has Become No Horse of a Different Color , Polya Shows the Importance of Following

In what has become “no horse of a different color”, Polya shows the importance of following the rules of induction strictly by misapplying them and using a bit of “handwaving” where he uses a logic that works for n > 1 to backtrack and falsely say that he has proven it for n = 1. In his day, by those who understood, Polya’s presentation was seen as an amusing example of how easily one can fool oneself, or one’s audience.

The flaw in the proof was its basis on the following logic, that leaps off of a single example.

Notice that if he have 5 horses and the rule that any 4 horses must be the same color.

1. Remove one horse from the group and you have 4 horses. They must all be the same color. Now replace the horse and remove a different horse. Again you have four horses, so they too must all be the same color. The new horse then must be the same horse as the rest of the horse. Therefore all five of them have the same color.

2. Continue this logic with 4, 3, and 2 horses. Clearly, it works for any n > 1. Thus if the rule is true for n, then it is true for n + 1.

3. But since we know that for 1 horse, it is the same color as itself. Therefore, it must be true for all horses.

The sleight of hand is that the rule fails for 2 horses. If one horse is removed, there is only one horse left. If the removed horse is replaced and the other horse is removed, there is still only one horse. So there is no other horse in the group of size 1 for that color to match in color. Thus, the induction hypothesis is not met.

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Goldbach's conjecture states: Any even integer greater than 2 can be written as the sum of two primes.

Part of what makes this hard to prove is that there is no formula for the generation of primes. Without such a formula, then it is hard to relate any number in general to specific primes. In the case of Goldbach, the conjecture has been proven for specific numbers that can be shown to equally the sum of specific primes (8 = 5+3, for example). Computers have extended the proof, but the theorem applies to an infinite number of integers, therefore generalized theorems usually require formulas of derivation for their key elements.

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Unproven conjecture in math:

The “twin prime conjecture” suggests that there are infinitely many primes p such that p + 2 is prime. For example 3 and 5 are twin primes. So are 5 and 7. But not 7 and 9 (because 9 is not prime), nor 7 and 11 (because 7 + 2 is not equal to 11).