## All horses are the same colour

All horses are the same colour; we can prove this by induction on the number of horses in a given set. Here's how: "If there's just one horse then it's the same colour as itself, so the basis is trivial. For the induction hypothesis, horses $$1$$ through $$n - 1$$ are the same colour, and similarly horses $$2$$ through $$n$$ are the same colour. But the middle horses, $$2$$ through $$n - 1$$, can't change colour when they're in different groups; these are horses, not chameleons. So horses $$1$$ and $$n$$ must be the same colour as well, by transitivity. Thus all $$n$$ horses are the same colour; QED." What, if anything, is wrong with this reasoning?

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The powers $$2^n$$ and $$5^n$$ start with the same digit where $$n$$ is a positive integer. What are the possible values for this digit?

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## Efficient gossiping

Each of $$n$$ male citizens knows a different piece of gossip. They are allowed to exchange the gossip they know by phone. During a call, just one of the men speaks and tells the other all the gossip he knows. What is the minimum number of calls required to enable each man to know all the gossip?

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## Separating each integer from its double

The positive integers are to be partitioned into several sets such that two times an integer and the integer itself do not belong to the same set, and every positive integer belongs to exactly one set. What is the minimum number of such sets required?

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## Two-move chess

The two-move chess game has the same rules as the regular one, with only one exception. Each player has to make two consecutive moves at a time. Does the White player, who makes the first two moves, have a non-losing strategy?

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