Contents:
Repeating impartial clobber patterns are "easy", in the sense that the number of possible/relevant subgames is limited. After most moves, the board divides into two subgames of "known" nim values. "Known" means they are already stored in the transposition table, from solving boards with smaller n). For each new n, only a few new larger subgames need to be computed and added to the table. In contrast, random starting positions allow less re-use.
History:
(xo)n, n=1..42, except n= 17, 21, 39, 40, by (Dai and Chen 2022).
(xo)n, n=1..2000 computed Nov 20, 2025. All values from (Dai and Chen 2022) confirmed.
(xxo)n, n=1..1000 computed Nov 20, 2025.
(xxxo)n, n=1..1000 computed Nov 20, 2025.
We are not aware of any previous published results on this game. In the table, symmetric boards with rows > columns are not shown. 1×n values listed separately below.
| Rows\Columns | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|
| 2 | *1 | *1 | *0 | *0 | *0 | *0 | *1 | *1 |
| 3 | *1 | *0 | *0 | *1 | *0 | |||
| 4 | *0 | *0 | *0 |
History:
All boards computed Nov 22, 2025 with Lemoine - Viennot algorithm.
| Board Size | NoGo Position | Nim Value | Notes |
|---|---|---|---|
| 1 | . | *0 | |
| 2 | .. | *1 | |
| 3 | ... | *0 | |
| 4 | .... | *1 | |
| 5 | ..... | *2 | |
| 6 | ...... | *0 | |
| 7 | ....... | *1 | |
| 8 | ........ | *0 | |
| 9 | ......... | *1 | |
| 10 | .......... | *2 | |
| 11 | ........... | *3 | |
| 12 | ............ | *1 | |
| 13 | ............. | *0 | |
| 14 | .............. | *3 | |
| 15 | ............... | *1 | |
| 16 | ................ | *0 | |
| 17 | ................. | *2 | |
| 18 | .................. | *2 | |
| 19 | ................... | *3 | |
| 20 | .................... | *1 | |
| 21 | ..................... | *3 | |
| 22 | ...................... | *3 | |
| 23 | ....................... | *1 | |
| 24 | ........................ | *0 | |
| 25 | ......................... | ? | Timed out (1hr limit) on Nov 21, 2025 |
History:
Boards 1-19 computed May 1, 2025 with MEX algorithm.
Boards 20-24 computed Nov 21, 2025 with Lemoine - Viennot algorithm.
It is well-known that all even × even boards are *0, second player wins, by following a mirroring strategy. All even × odd boards are first player wins, by playing in the middle and then mirroring.
1×n Cram is the same as Dawson’s Kayles, the take-and-break octal game .07 . It has period 34 with only few exceptions near the start (https://oeis.org/A002187).
MCGS has verified the values up to n=127, using the 2-d grid implementation of impartial domineering. Grid has a 7 bit limit on dimensions (width and height).
2×n boards are *0 for all even n, and *1 for all odd n (Winning Ways, Uiterwijk 2020). MCGS has solved all boards up to 2×26 on 2025-11-23.
3×n, n=3..20, from (Lemoine and Viennot 2013). 3×n is a win if n is even. 3×21 is a first player win (Uiterwijk 2020) Some boards also verified with CGSuite. MCGS has solved all boards up to 3×13 on 2025-11-23.
| Columns | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| *0 | *1 | *1 | *4 | *1 | *3 | *1 | *2 | *0 | *1 | *2 | *3 | *1 | *4 | *0 | *1 | *0 | *2 | *≥1 |
| Rows\Columns | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 4 | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 1 | 0 | 3 | 0 | 4 |
| 5 | - | 0 | 2 | 1 | 1 | 1 | 2 | 0 | 3 | |||
| 6 | - | - | 0 | 5 | 0 | 1 | 0 | 0 | 0 | |||
| 7 | - | - | - | 1 | 3 | 1 |
Sources: From (Wikipedia).
4×5, 4×7, 4×9, from (Lemoine and Viennot 2013).
4×11, 4×13, 4×15 from (Beling).
5×n, n=5..9, from (Lemoine and Viennot 2013), n=10-12 (Beling).
4×n = *0 if n is even.
4×n is a win if n is odd
5×n is a win if n is even
MCGS results so far (no database, no move ordering): MCGS has confirmed 4×5 and 4×7 on 2025-11-23. It also computed even boards up to 4×8. It confirmed 5×5 and 5×6 on 2025-11-23.