Quickie

Is 94,271,013 the sum of 12 consecutive integers?

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Keeping Distance

For which values of n can n points be placed on a sphere so that all of them are equidistant from each other?

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Some Two

Given 52 integers, prove that it’s always possible to find some two of them whose sum or difference is evenly divisible by 100.

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Black and White

van dehn chess puzzle

A remarkable thematic chess puzzle by Bodo Van Dehn, 1951. White to move and win.

The solution is 10 moves long, but all Black’s moves are forced. (That’s a very valuable hint.)

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A Flea’s Journey

https://commons.wikimedia.org/wiki/File:Euclid_Tetrahedron_4.svg
Image: Wikimedia Commons

A flea sits on one vertex of a regular tetrahedron. He hops continually from one vertex to another, resting for a minute between hops and choosing vertices without bias. Prove that, counting the first hop, we’d expect him to return to his starting point after four hops.

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