This reminds me of the (not really a joke) math puzzler. Find the logical misstep in the following induction proof.
To prove:
All horses are the same color.
Base case: Consider a set of one horse. Clearly, one horse is the same color as itself, therefore a set of one horse is all the same color.
Induction step:
- Consider a set of N horses. Partition the set into a subset of 1 horse, call this horse i, and a subset of N-1 horses. By the inductive hypothesis, the N-1 horses are all the same color.
- Now re-partition the original set into a subset of 1 horse j ≠ i and a subset of the other N-1 horses. By the inductive hypothesis, the set of N-1 horses is all the same color, including, in particular, horse i.
- But we’ve previously established that j is the same color as all other horses in the set other than i. By the transitive property, horse i and horse j are thus the same color.
- Therefore, by the principle of mathematical induction, all N horses are the same color.
Now, where is the poison? The battle of wits has begun.
