Henry Ernest Dudeney/Modern Puzzles/208 - The Nine Squares Game/Solution

Modern Puzzles by Henry Ernest Dudeney: $208$

The Nine Squares Game

Make the simple square diagram show and provide a box of matches.

The side of the large square is three matches in length.

The game is, playing one match at a time alternately, to enclose more of those small squares than your opponent.

For every small square that you enclose, you not only score one point, but you play again.

This illustration shows an illustrative game in progress.

Twelve matches are placed, my opponent and myself having made six plays each, and, as I had first play, it is now my turn to play a match.

What is my best line of play in order to win most squares?

If I play $FG$ my opponent will play $BF$ and score one point.

Then, as he has the right to play again, he will score another with $EF$ and again with $IJ$, and still again with $IJ$, and still again with $GK$.

If he now plays $CD$, I have nothing better than $DH$ (scoring one), but, as I have to play again, I am compelled, whatever I do, to give him all the rest.

So he will win by $8$ to $1$ -- a bad defeat for me.

Now, what should I have played instead of that disastrous $FG$?

There is room for a lot of skilful play in the game, and it can never end in a draw.

Solution

There are $2$ best plays:

$MN$, which is that suggested by Dudeney

$HL$

Proof

Neither of $MN$ or $HL$ allow any boxes to be created immediately.

Playing:

$(A): \quad MN$

leaves the plays $HL$, $CD$ or $DH$ for the opponent.

Any other reply is a blunder which gives you $7$ boxes immediately, and the win.

After:

$(B): \quad HL$

you play $CD$ or $DH$, allowing his play of either $DH$ or $CD$, claiming $1$, and forcing a further play which gives you the remaining $8$ boxes.

After:

$(B): \quad CD$ or $DH$

you play $DH$ or $CD$, claiming $1$, then $HL$.

$B$'s next move gives you the rest.

Playing:

$(A): \quad HL$

leaves the plays $MN$, $CD$ or $DH$ for the opponent, and the same analysis is used, exchanging $HL$ and $MN$.

$\blacksquare$

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $208$. -- The Nine Squares Game
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $477$. The Nine Squares Game