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

## Puzzles and Curious Problems by Henry Ernest Dudeney: $319$

The Ten Cards
Place any ten playing cards in a row face up.
There are two players.
The first player may turn face down any single card he chooses.
Then the second player can turn face down any single card or any $2$ adjacent cards.
And so on.
Thus the first player must turn face down a single, but afterwards either player may turn down either a single or two adjacent cards.
The player who turns down the last card wins.
Should the first or second player win?

## 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.

$000 \text X 000000$

## Historical Note

The analysis of the solution where the third card is turned over came from Dudeney.

The other two cases were the work of Victor Meally.