# The riddle of the squares

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Here is a well-known game of the East that is played with rules very similar to those of the famous game of "Ta-Te-Ti" (or game of the squares). One of the young Chinese writes sixteen letters in four rows on a blackboard, as seen in the drawing. After marking a straight line between A and B, he passes the board to his opponent, who connects E with A.

If the first player now connected E and F, the other would connect B with F and get "a little square", which would entitle her to play once again. But both have played so well that neither has won a small square, although each has played six times.

The game is reaching a critical point where one of them must win, since the game offers no other possibilities. The girl who is sitting has to play now, and if she connects M and N her opponent would make four squares in a single play, with the right to another play, in which she would connect H and L and win all the rest.

What play would you recommend, and how many squares would you win by comparing this play with the best possible play of the second player?

Remember that when a player closes a square he returns to play.

Suppose, for example, that a player joins D with H. Then the second player joins H and L and, regardless of the play of the first player, the second wins the nine squares continuously.

It is a game that requires considerable skill, as you will discover after playing some games.

#### Solution

This puzzle provides many opportunities to be surprised and develop a subtle game.

The first player should make 7 squares starting with a line that goes from G to H. If the second mark then from J to K, the first one can make 2 squares marking from K to O and from P to L, and then make a waiting movement, from L to H, instead of closing 2 more squares. The other player then makes the 2 squares, marking from G to K, and then is forced to another play that will give the first player the opportunity to close 5 more.

If after the first player marks from G to H, the second player marks D-H, B-F, E-F, and then makes the waiting play M-N, he is sure to make another 4 squares.

This clever technique of abandoning the possibility of making 2 squares in order to get more is the most interesting aspect of the game.

(Known among American schoolchildren as "Points and Squares", this is probably the simplest and most widespread example of a topological game. It can be played on rectangular boards of different shapes and sizes. The 9-point square board is easily analyzed, but the The 16-point board used by Loyd is complex enough to be a real challenge. I don't know of any published analysis of winning strategy for the first or second player. The game cannot end in a draw because of the odd number of squares.

In 1951, Richard Haynes, from 1215 E. 20th. Street, Tulsa, Oklahoma, invented an interesting three-dimensional version of this game, which he called "Q-bicles." A booklet of printed sheets can be obtained to play Q-bicles by sending a dollar to Mr. Haynes.

(It can also be played with dot patterns that form two-dimensional triangular or hexagonal cells. M. G.)

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