What is Dots and Triangles?
Dots and Triangles is Dots and Boxes on a triangular lattice. The dots form one large triangle, and the lines between neighbouring dots, along three directions, cut it into small triangles, half of them pointing up and half pointing down. Each small triangle is a cell.
Players take turns drawing one line between two neighbouring dots. A player who completes the third side of a triangle scores it and draws another line. A line inside the board borders two triangles, so one line can close two at once. When every line has been drawn, the player with more triangles wins, and equal scores are a draw.
Because a cell has only three sides, a cell with two sides drawn is ready to be taken, and chains form and grow much faster than with boxes. The board size is the number of small triangles along each side: size 4 has 15 dots, 30 lines and 16 triangles.
Rule changes
| Rule | Standard Dots & Boxes | Dots and Triangles |
|---|---|---|
| Board | Standard Dots & BoxesA square grid of dots. | Dots and TrianglesA large triangle of dots on a triangular lattice. |
| Cells | Standard Dots & BoxesSquare boxes with four sides. | Dots and TrianglesSmall triangles with three sides, alternately pointing up and down. |
| Lines | Standard Dots & BoxesHorizontal and vertical lines between neighbouring dots. | Dots and TrianglesLines in three directions between neighbouring dots: across and along both slopes. |
| Scoring | Standard Dots & BoxesComplete the fourth side of a box to score it and draw another line. | Dots and TrianglesComplete the third side of a triangle to score it and draw another line. One line can close two triangles. |
| Board sizes | Standard Dots & Boxes2×2 to 6×6 boxes, or bigger and rectangular boards. | Dots and Triangles2 to 8 triangles a side, from 4 to 64 triangles. Rectangular sizes are not used. |
| Other options | Standard Dots & BoxesEvery option in the table setup can be used. | Dots and TrianglesWorks in Misère, with the pre-drawn border (the three sides of the large triangle) and with three or four players. |
History
Descriptions of Dots and Boxes have long noted that the game does not need a square grid and can be played on a triangular one. Dots and Triangles is the name used for that version in mathematical work on the game.
In a 2015 paper, Variations on Narrow Dots-and-Boxes and Dots-and-Triangles, Adam Jobson, Levi Sledd, Susan White and D. Jacob Wildstrom proved a conjecture of Nowakowski and Ottaway about a narrow triangle board: on the closed 1×n strip the first player wins whenever n is not 2. In 2020 Jessica Dickson and Rachel Perrier studied Dots-and-Triangles on a triangular lattice, alongside a variation called Dots-and-Polygons, and analysed both with the strategies developed for Dots and Boxes by Elwyn Berlekamp, John Conway, Richard Guy and others.