What is Northcott's game?
Northcott's game is played on a board of rows. Each row holds two counters, one for each player: the first player's counter is always the left one, the second player's the right one. On your turn you slide one of your own counters any number of squares along its row, in either direction, but you may not jump over the other counter or land on it. The player who cannot move loses.
The board has 6 rows of 6 squares (small), 8 rows of 8 (medium, the default) or 10 rows of 10 (large), chosen as the size in the table setup. The counters start in a random layout, dealt for each game and saved with its record, so a replay starts from the same place. Although each player moves only their own counters, the game is Nim in disguise: the empty squares between the two counters of a row act as a heap.
Rule changes
| Rule | Standard Nim | Northcott's game |
|---|---|---|
| Board | Standard NimSeveral heaps of objects: 1, 3, 5 and 7 by default, or another set chosen in the table setup. | Northcott's game6 × 6, 8 × 8 (default) or 10 × 10 squares, chosen as the size in the table setup. Each row holds one counter of each player, placed at random. |
| Moves | Standard NimTake one or more objects from one heap. | Northcott's gameSlide one of your own counters any number of squares along its row, toward the other counter or away from it, without jumping over it or landing on it. |
| Goal | Standard NimMisère play by default: whoever takes the last object loses. The setup can switch to normal play, where taking the last object wins. | Northcott's gameWhoever cannot move loses. There is no misère switch. |
| Length of the game | Standard NimEvery move removes at least one object, so the game ends within as many moves as there are objects. | Northcott's gameMoving a counter back makes its gap bigger again, so there is no fixed limit and no draw rule. The player who is winning can always bring the game to an end by moving forward only. |
| Winning play | Standard NimLeave the heaps with a nim-sum (the bitwise XOR of their sizes) of zero, with a twist in the misère endgame. | Northcott's gameTreat the gap in each row as a Nim heap and leave a nim-sum of zero, as in normal-play Nim. When the opponent moves a counter back, move your counter in that row forward by the same distance to restore the gap. |
History
Northcott's game appears in Winning Ways for your Mathematical Plays by Elwyn Berlekamp, John Conway and Richard Guy, first published in 1982, and in later references on combinatorial games such as Eric Weisstein's MathWorld. It is a favourite example of a game that looks partisan, since each player has their own counters, yet plays exactly like Nim, because a move backwards can always be cancelled by the opponent.
We have not found who Northcott was or when the game was first played. Some descriptions let the counters move only toward each other; the version here follows MathWorld and Alexander Bogomolny's Cut the Knot, which also allow moving back.