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

Asked Jul 06 '15 at 20:50

Active Jul 06 '15 at 20:50

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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?

asked Jul 06 '15 at 20:50

3

the second part of reasoning doesn't work for $n=2$ – Anurag A Jul 06 '15 at 20:53
https://en.wikipedia.org/wiki/All_horses_are_the_same_color – Robert Israel Jul 06 '15 at 20:53
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Why should there be any pencils at all? The second step assumes you have $n+1$ crayons, not pencils. – Alex R. Jul 06 '15 at 20:53
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Duplicate of basic induction probs, and see also this question. – Brian M. Scott Jul 06 '15 at 20:54
You may find this thread about Fake Induction Proofs to be quite interesting. – Daniel W. Farlow Jul 06 '15 at 20:54
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that "hence" ... – orangeskid Jul 06 '15 at 21:12

0 Answers0