What is Bigger and rectangular boards?
Dots and Boxes needs nothing but a grid of dots, so it can be played at any size. Besides the usual 2×2 to 6×6 boards you can choose 7×7 or 8×8 boxes, or a rectangle: 3×4, 3×5, 4×5, 4×6 or 5×6 boxes. Sizes count boxes, not dots, so the 8×8 board has 9 rows of 9 dots.
The rules stay the same. Players take turns drawing one line between two neighbouring dots, a player who completes the fourth side of a box scores it and draws another line, and when every line has been drawn the player with more boxes wins. Equal scores are a draw, which can only happen when the board has an even number of boxes.
A bigger board means a longer quiet phase before anyone has to give boxes away, more chains and loops, and more room for the long chain rule, the counting technique that tells a player whether to make the number of long chains odd or even. The 8×8 board has 144 lines and 64 boxes, against 40 lines and 16 boxes on the standard 4×4.
Rule changes
| Rule | Standard Dots & Boxes | Bigger and rectangular boards |
|---|---|---|
| Board | Standard Dots & Boxes4×4 boxes (5×5 dots), or any square size from 2×2 to 6×6. | Bigger and rectangular boards7×7 or 8×8 boxes, or a rectangle of 3×4, 3×5, 4×5, 4×6 or 5×6 boxes. |
| Lines and boxes | Standard Dots & Boxes40 lines and 16 boxes on 4×4. | Bigger and rectangular boards112 lines and 49 boxes on 7×7, 144 lines and 64 boxes on 8×8, 49 lines and 20 boxes on 4×5. |
| Game length | Standard Dots & BoxesOne move per line: 40 moves on 4×4. | Bigger and rectangular boardsOne move per line: up to 144 moves on 8×8. |
| Perfect play | Standard Dots & BoxesSmall square boards have been solved by computer. | Bigger and rectangular boards4×5 was solved in 2011: it is a draw with perfect play. 7×7 and 8×8 are far too large to solve. |
| Other options | Standard Dots & BoxesEvery option in the table setup can be used. | Bigger and rectangular boardsWorks in Misère, with the pre-drawn border and with three or four players. |
History
Dots and Boxes was first described in print by the French mathematician Édouard Lucas in the 19th century, who called it la pipopipette. He gave no fixed size, and the game has always been played on whatever grid the paper allows. Small boards such as 2×2 are the usual way to learn, and the 5×5 board is often recommended for experienced players.
The game's best-known analysis is by Elwyn Berlekamp, who wrote the Dots-and-Boxes chapter of Winning Ways for your Mathematical Plays with John Conway and Richard Guy in 1982, and in 2000 a whole book, The Dots-and-Boxes Game: Sophisticated Child's Play. In 2011 Joseph Barker and Richard Korf used a computer search to solve the 4×5 board, which turned out to be a draw with perfect play; they presented the result at the AAAI conference, and it was the largest board solved at the time.