Morpion Solitaire - Rules of the Game

"Voici un passe-temps qui risque de coûter plus cher aux bureaux et aux administrations que la grippe, le téléphone et les cocottes réunis."
("Dilbert would praise this pasttime as an efficient way of neutralizing an office worker far beyond flu, phone, email and origami")
Pierre Berloquin, April 1974 (and his own free translation of May 2008)

Only one player is needed, and the rules are very simple. The equipment is also very simple: a paper (cross-ruled, if possible) and a pencil.

First draw the initial pattern below, looking as a large Greek cross. Then add a new small cross where you want, but only if 5 small crosses (including your new one) can be aligned. Draw the line across the 5 small crosses, and your score becomes 1.

  
Initial pattern at 5T and 5D Morpion Solitaire, and example of a first move joining 5 crosses.

Each time you add one small cross and draw its line (one and only one line), you add 1 to your score. A line can be horizontal, vertical, or diagonal. A line can cross or touch, but cannot overlap, a previously drawn line.

     
Examples of next moves

The game ends when you become unable to add a new cross (not caused by the paper, which is supposed to be unlimited in size).
The goal is to have the maximum score, and that's it! Very easy, isn't it?

However, a remark. If you draw small crosses as above, it will later be difficult (*) to replay exactly the same game without knowing their order. If you reach 179 moves... better than the current record... it will be laborious to reconstruct your record. That's why it is recommended to number immediately each move, writing move numbers instead of crosses. The above game becomes:

It is recommended to number the moves

(*) but possible, see "Verifying a drawing", part 4.1 of the Demaine et al paper

What are 5T, 5D, 4T, 4D games?

• The above rules are for the original 5T game. The "5" means that a line needs 5 crosses, and the "T" means "Touching" model: two lines on the same alignment can touch each other.
• The 5D game is again with 5 crosses but the "D" means "Disjoint" model: two lines on the same alignment can't touch each other, they have to be disjoint.

The move 2 on the left image is possible in 5T game, but impossible in 5D.
The move 2 on the right image is possible in both games.

• In the 4T and 4D games, the "4" means that a line needs 4 crosses. And the "T" and "D" mean again "Touching" and "Disjoint". The initial pattern is this new smaller one:

Initial pattern at 4T and 4D Morpion Solitaire

Why no 3T, 3D, 6T, 6D games?

• Because with 3T or 3D games, the score can be infinite.
• Because with 6T or 6D games (or higher), the score would be limited to 12.

                 
On the left, an infinite score at 3T or 3D game. On the right, none of the possible moves at 6T or 6D will generate any new possible move.

This figure on the left proves that the score of a 3T or 3D games CAN BE infinite. But as remarked in February 2008 by Lee Morgenstern, USA, it even seems that any game IS ALWAYS infinite. Who can prove that, or who can provide a counter example (a 3T or 3D game having a finite number of moves)? See also this variant: finite 3D/3T.

Michael Quist, USA, computed in March 2008 the number of different grids after n moves, n < 10. The number of grids grows very fast.

© Christian Boyer, www.morpionsolitaire.com