Henry Ernest Dudeney/Puzzles and Curious Problems/319 - The Ten Cards/Solution

by : $319$

 * The Ten Cards

Solution
The first player can always win.

Proof
The first player turns down the $3$rd from either end.

This leaves the cards with $2$ turned up, one turned down, and $7$ turned up.

Whatever happens next, the first player can always leave one of the following:
 * two groups of three up
 * two groups of two up, and two instances of one up
 * one group of three up, one group of two up, one instance of one up.

In the first case, the first player copies in one triplet what the second player does in the other triplet, until he gets the last card.

In the second case, the first player similarly copies the second player until he gets the last card.

In the third case, whatever the second player does, the first player can leave:
 * two instances of one up
 * four instances of one up
 * two instances of two up

and again the win is apparent.