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I retrieved an old math book and I'm delighted to share following exercise.

The pencils in a box of crayons always have the same color.

Proof by induction on the number $n$ of pencils in the box:

  • This is true when $n=1$. All the pencils, only one, havs the same color.
  • Suppose the result true for a box having $n+1$ crayons. Remove one. Remains $n$ crayons. By induction hypothesis they all have the same color. Put back the crayon you removed and remove another one. The $n$ remaining crayons in the box have again the same color by induction hypothesis. Hence the two crayons that were removed have the same color. Therefore all the $n+1$ crayons have the same color.

Can you find what's wrong here?

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