Henry Ernest Dudeney/Modern Puzzles/202 - Noughts and Crosses/Solution

by : $202$

 * Noughts and Crosses
 * Every child knows how to play this ancient game.
 * You make a square of nine cells, and each of the two players, playing alternately, puts his mark
 * (a nought or a cross, as the case may be) in a cell with the object of getting three in a line.
 * Whichever player gets three in a line wins.


 * In this game, cross has won:


 * $\begin {array} {|c|c|c|}

\hline \text X & \text O & \text O \\ \hline \text X & \text X & \text O \\ \hline \text O &        & \text X \\ \hline \end {array}$


 * I have said in my book, ,
 * that between two players who thoroughly understand the play every game should be drawn,
 * for neither party could ever win except through the blundering of his opponent.
 * Can you prove this?
 * Can you be sure of not losing a game against an expert opponent?

Solution
Against an expert player, the best that can be accomplished is a draw, whether playing first or second.


 * If playing first, you can play any square on the board.


 * If playing second, play a corner if the first player opens with the center, and the center if anything else is played.

In the words of himself:


 * The fact remains that it is a capital little game for children, and even for adults who have never analysed it,
 * ''but two experts would merely be wasting their time in playing it.
 * To them it is not a game, but a mere puzzle that they have completely solved.

Proof
Let the squares of the board be numbered:


 * $\begin {array} {|c|c|c|}

\hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end {array}$

In the following, it is assumed that each player is sufficiently rational as to always block a row of $2$ and hence prevent an immediate win for the other player.

Hence a win cannot be prevented a play leaves more than one row of $2$, which means at least one row of $2$ cannot be blocked.

, let $\text O$ play first.


 * Center Opening

The following opening is a win for $\text O$:


 * $\begin {array} {|c|c|c|}

\hline \ \ & \text X & \ \ \\ \hline & \text O & \\ \hline &  &  \\ \hline \end {array}$

as $\text O$ then plays $1$ then $4$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & \text X & \\ \hline \text O & \text O & \\ \hline &  & \text X \\ \hline \end {array}$

and there is nothing $\text X$ can do to stop $\text O$ winning.

The following opening is a draw:


 * $\begin {array} {|c|c|c|}

\hline \text X & & \ \ \\ \hline & \text O & \\ \hline &  &  \\ \hline \end {array}$

Whatever $\text O$ then does, $\text X$ can block.

This leaves a line of two $\text X$'s which $\text O$ then blocks.

This leaves a line of two $\text O$'s which $\text X$ then blocks.

And so on, till the end of the game.


 * Corner Opening

If $\text O$ starts with a corner, the following sequences are possible:


 * $\begin {array} {|c|c|c|}

\hline \text O & \text X & \ \ \\ \hline &  & \\ \hline &  &  \\ \hline \end {array}$

$\text O$ then plays $5$ then $4$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & \text X & \\ \hline \text O & \text O & \\ \hline &  & \text X \\ \hline \end {array}$

and there is nothing $\text X$ can do to stop $\text O$ winning.


 * $\begin {array} {|c|c|c|}

\hline \text O & & \\ \hline &  \ \ & \\ \hline &  & \text X \\ \hline \end {array}$

$\text O$ then plays $7$ then $3$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & & \text O\\ \hline \text X & \ \ & \\ \hline \text O & & \text X \\ \hline \end {array}$

and $\text O$ wins.


 * $\begin {array} {|c|c|c|}

\hline \text O & & \text X \\ \hline &  \ \ &  \\ \hline &  &  \\ \hline \end {array}$

$\text O$ then plays $9$ then $7$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & & \text X\\ \hline & \text X  &  \\ \hline \text O & & \text O \\ \hline \end {array}$

and $\text O$ wins.


 * $\begin {array} {|c|c|c|}

\hline \text O & & \\ \hline &  \ \ & \text X \\ \hline &  &  \\ \hline \end {array}$

$\text O$ then plays $5$ then $3$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & & \text O\\ \hline & \text O & \text X \\ \hline &  & \text X \\ \hline \end {array}$

and $\text O$ wins.

Hence on a corner opening, $\text X$ needs to play center:


 * $\begin {array} {|c|c|c|}

\hline \text O & & \\ \hline & \text X & \ \ \\ \hline &  &  \\ \hline \end {array}$

If $\text O$ plays $2$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & \text O & \\ \hline & \text X & \ \ \\ \hline &  &  \\ \hline \end {array}$

then all subsequent moves are forced, leading to a draw.

If $\text O$ plays $3$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & & \text O \\ \hline & \text X & \ \ \\ \hline &  &  \\ \hline \end {array}$

then again all subsequent moves are forced, leading to a draw.


 * Side Opening

If $\text O$ starts with a side, the following sequences are possible:


 * $\begin {array} {|c|c|c|}

\hline & \text O & \ \ \\ \hline \text X & & \\ \hline &  &  \\ \hline \end {array}$

$\text O$ then plays $5$ then $1$, leaving:


 * $\begin {array} {|c|c|c|}

\hline \text O & \text O & \ \ \\ \hline \text X & \text O & \\ \hline & \text X &  \\ \hline \end {array}$

and $\text O$ wins.


 * $\begin {array} {|c|c|c|}

\hline & \text O & \ \ \\ \hline &  & \\ \hline \text X & &  \\ \hline \end {array}$

$\text O$ then plays $5$ then $9$ (forced), leaving:


 * $\begin {array} {|c|c|c|}

\hline & \text O & \ \ \\ \hline & \text O & \\ \hline \text X & \text X & \text O \\ \hline \end {array}$

The subsequent moves are forced, leading to a draw.

The other three cases:


 * $\begin {array} {|c|c|c|}

\hline \text X & \text O & \ \ \\ \hline &  & \\ \hline &  &  \\ \hline \end {array} \qquad \begin {array} {|c|c|c|} \hline & \text O & \ \ \\ \hline \ \ & \text X & \\ \hline &  &  \\ \hline \end {array} \qquad \begin {array} {|c|c|c|} \hline & \text O & \ \ \\ \hline \ \ & & \\ \hline & \text X &  \\ \hline \end {array}$

end as a draw with best play.