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Devilish Difference (Posted on 2005-08-22) Difficulty: 3 of 5
Suppose you select 501 numbers from 1 to 1000, inclusive. Show that if no two of the selected numbers differ by 334, then some two of the selected numbers differ by 666.

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Solution Solution | Comment 3 of 6 |
For each number N you pick, reject N-666 or N+334 -- one of them will be between 1 and 1000, and the other won't.

For the first 500 numbers you pick, you'll reject other 500.

But when you pick the 501st, you'll necessarily pick a rejected one.

  Posted by Federico Kereki on 2005-08-23 02:25:12
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