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

== by : $202$ ==
;Noughts and Crosses
== 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:
$\qquad \begin {array} {|c|c|c|}
\hline 1 & 2 & 3 \\
\hline 4 & 5 & 6 \\
\hline 7 & 8 & 9 \\
\hline \end {array}$
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 $\text O$:
$\qquad \begin {array} {|c|c|c|}
\hline \ \ & \text X & \ \ \\
\hline & \text O & \\
\hline & & \\
\hline \end {array}$
as $\text O$ then plays $1$ then $4$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & \text X & \\
\hline \text O & \text O & \\
\hline & & \text X \\
\hline \end {array}$
and there is nothing $\text X$ can do to stop $\text O$ winning.
The following opening is a draw:
$\qquad \begin {array} {|c|c|c|}
\hline \text X & & \ \ \\
\hline & \text O & \\
\hline & & \\
\hline \end {array}$
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:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & \text X & \ \ \\
\hline & & \\
\hline & & \\
\hline \end {array}$
$\text O$ then plays $5$ then $4$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & \text X & \\
\hline \text O & \text O & \\
\hline & & \text X \\
\hline \end {array}$
and there is nothing $\text X$ can do to stop $\text O$ winning.
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \\
\hline & \ \ & \\
\hline & & \text X \\
\hline \end {array}$
$\text O$ then plays $7$ then $3$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \text O\\
\hline \text X & \ \ & \\
\hline \text O & & \text X \\
\hline \end {array}$
and $\text O$ wins.
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \text X \\
\hline & \ \ & \\
\hline & & \\
\hline \end {array}$
$\text O$ then plays $9$ then $7$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \text X\\
\hline & \text X & \\
\hline \text O & & \text O \\
\hline \end {array}$
and $\text O$ wins.
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \\
\hline & \ \ & \text X \\
\hline & & \\
\hline \end {array}$
$\text O$ then plays $5$ then $3$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \text O\\
\hline & \text O & \text X \\
\hline & & \text X \\
\hline \end {array}$
and $\text O$ wins.
Hence on a corner opening, $\text X$ needs to play center:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \\
\hline & \text X & \ \ \\
\hline & & \\
\hline \end {array}$
If $\text O$ plays $2$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & \text O & \\
\hline & \text X & \ \ \\
\hline & & \\
\hline \end {array}$
then all subsequent moves are forced, leading to a draw.
If $\text O$ plays $3$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & & \text O \\
\hline & \text X & \ \ \\
\hline & & \\
\hline \end {array}$
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:
$\qquad \begin {array} {|c|c|c|}
\hline & \text O & \ \ \\
\hline \text X & & \\
\hline & & \\
\hline \end {array}$
$\text O$ then plays $5$ then $1$, leaving:
$\qquad \begin {array} {|c|c|c|}
\hline \text O & \text O & \ \ \\
\hline \text X & \text O & \\
\hline & \text X & \\
\hline \end {array}$
and $\text O$ wins.
$\qquad \begin {array} {|c|c|c|}
\hline & \text O & \ \ \\
\hline & & \\
\hline \text X & & \\
\hline \end {array}$
$\text O$ then plays $5$ then $9$ (forced), leaving:
$\qquad \begin {array} {|c|c|c|}
\hline & \text O & \ \ \\
\hline & \text O & \\
\hline \text X & \text X & \text O \\
\hline \end {array}$
The subsequent moves are forced, leading to a draw.
The other three cases:
$\qquad \begin {array} {|c|c|c|}
\hline \text X & \text O & \ \ \\
\hline & & \\
\hline & & \\
\hline \end {array} \qquad 
\begin {array} {|c|c|c|}
\hline & \text O & \ \ \\
\hline \ \ & \text X & \\
\hline & & \\
\hline \end {array} \qquad 
\begin {array} {|c|c|c|}
\hline & \text O & \ \ \\
\hline \ \ & & \\
\hline & \text X & \\
\hline \end {array}$
end as a draw with best play.
== Sources ==
* : Solutions: $202$. -- Noughts and Crosses
* : Answers: $471$. Tic Tac Toe
Category:Henry Ernest Dudeney/Modern Puzzles/Solutions