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
Let $A$ denote the person who plays first, and $B$ denote the person who plays second.

Let $O$ denote a card turned up, and $\text X$ denote a card turned down.

There are $3$ ways $A$ can win.


 * Third Card

$A$ turns down the $3$rd from either end.

This leaves:
 * $00 \text X 0000000$

Whatever happens next, $A$ can always leave one of the following:
 * $000 \text X 000$
 * $00 \text X 00 \text X 0 \text X 0$
 * $0 \text X 00 \text X 000$

The order does not matter.

In the first case, $A$ copies in one triplet what $B$ does in the other triplet, until he gets the last card.

In the second case, $A$ similarly copies $B$ until he gets the last card.

In the third case, whatever $B$ does, $A$ can leave:
 * $0 \text X 0$
 * $0 \text X 0 \text X 0 \text X 0$
 * $00 \text X 00$

and again the win is apparent.


 * Second Card

$A$ turns down the $2$nd from either end.

This leaves:
 * $0 \text X 00000000$


 * Fourth Card

$A$ turns down the $4$th card from either end.

This leaves the cards with $3$ turned up, one turned down, and $6$ turned up.
 * $000 \text X 000000$