The census
Eight rule genes for a stone game on a hexagon: board size, a winning line, a losing line, custodial or surround capture, a stone cap, adjacency and stones per turn. That is 6,750 rule sets. An engine played every one of them against itself and ran each through six gates: does the game end, is the first player fair, does a strong player beat a weak one, do games have a shape, does anything ever happen, does it stay legible. 2,357 passed all six. Every tile below is a game you can play, gate failures included. The map is the piece; the best tile is the game I played by hand.
The map
Every tile is one rule set. A row is a board size with a winning line length and a losing line length; along a row the other five genes vary: what connects, what captures, how many captures win, where you may place, and how many stones you place per turn. Dark tiles died at one of the six gates; the amber ones survived, brighter for a higher score. Tap the map to open a row, then tap a tile.
Play
The opponent is the same search that scored the census, compiled to WebAssembly and running in your browser. Any rule set can also be opened directly: . Format: s board side, w line length that wins (0 off), l line length that loses (0 off), c connection (0 off, 1 opposite sides, 2 any three sides), k capture (0 off, 1 single custodial, 2 custodial line), K captures needed to win (0 off), a placement (0 anywhere, 1 touching any stone, 2 touching your own), p stones per turn.
What the census says
6,750 rule sets went in. 2,357 came out of the six gates. The gates, in the order they were applied:
| gate | meaning | removed |
|---|---|---|
| 1. Random play holds its own | the engine with 1,000 playouts per move fails to beat a random mover 90% of the time: the rules leave nothing to think about | 145 |
| 2. Thinking harder doesn't help | ten times the search budget wins fewer than 70% of games: the outcome is mostly luck | 906 |
| 3. Too many draws | more than one game in five ends without a winner | 290 |
| 4. Lopsided | one colour wins more than three games in four | 1,644 |
| 5. Over too fast, or never over | average game fills less than a quarter of the board, or more than three quarters | 1,331 |
| 6. Foregone conclusions | in more than a third of games the winner is already at 90% before the game is 60% done | 77 |
Survival by gene
| gene | value | rule sets | survived | rate | best score | best rule set |
|---|---|---|---|---|---|---|
| board side | 4 (37 cells) | 2,250 | 828 | 37% | 0.946 | s4w0l3c1k1K0a0p2 |
| 5 (61) | 2,250 | 927 | 41% | 0.959 | s5w5l3c0k0K0a1p2 | |
| 6 (91) | 2,250 | 602 | 27% | 0.979 | s6w6l3c0k0K0a1p2 | |
| line that wins | off | 1,890 | 784 | 41% | 0.962 | s6w0l0c1k1K0a0p2 |
| 3 | 486 | 10 | 2% | 0.639 | s4w3l0c0k2K0a1p1 | |
| 4 | 972 | 155 | 16% | 0.954 | s5w4l3c0k0K0a1p1 | |
| 5 | 1,458 | 562 | 39% | 0.960 | s6w5l3c0k0K0a1p1 | |
| 6 | 1,944 | 846 | 44% | 0.979 | s6w6l3c0k0K0a1p2 | |
| line that loses | off | 2,376 | 586 | 25% | 0.964 | s6w6l0c0k2K0a0p1 |
| exactly 3 | 1,944 | 771 | 40% | 0.979 | s6w6l3c0k0K0a1p2 | |
| exactly 4 | 1,458 | 600 | 41% | 0.972 | s6w6l4c0k2K0a0p1 | |
| exactly 5 | 972 | 400 | 41% | 0.936 | s6w6l5c1k0K0a0p1 | |
| connection | off | 2,214 | 668 | 30% | 0.979 | s6w6l3c0k0K0a1p2 |
| opposite sides | 2,268 | 954 | 42% | 0.962 | s6w0l0c1k1K0a0p2 | |
| three sides | 2,268 | 735 | 32% | 0.943 | s6w0l3c2k0K0a1p2 | |
| capture | off | 738 | 307 | 42% | 0.979 | s6w6l3c0k0K0a1p2 |
| single custodial | 3,006 | 1,103 | 37% | 0.962 | s6w0l0c1k1K0a0p2 | |
| custodial line | 3,006 | 947 | 32% | 0.972 | s6w6l4c0k2K0a0p1 | |
| captures win | never | 2,214 | 770 | 35% | 0.979 | s6w6l3c0k0K0a1p2 |
| 3 | 1,512 | 431 | 29% | 0.917 | s6w0l0c0k1K3a0p2 | |
| 5 | 1,512 | 528 | 35% | 0.908 | s6w0l0c2k1K5a0p2 | |
| 8 | 1,512 | 628 | 42% | 0.949 | s6w0l0c0k2K8a0p2 | |
| placement | anywhere | 2,250 | 1,378 | 61% | 0.972 | s6w6l4c0k2K0a0p1 |
| touching any stone | 2,250 | 631 | 28% | 0.979 | s6w6l3c0k0K0a1p2 | |
| touching your own | 2,250 | 348 | 15% | 0.883 | s6w0l3c0k1K3a2p2 | |
| stones per turn | one | 3,375 | 1,089 | 32% | 0.972 | s6w6l4c0k2K0a0p1 |
| two | 3,375 | 1,268 | 38% | 0.979 | s6w6l3c0k0K0a1p2 |
Published games as controls
| rule set | what it is | fate | light : dark : draws | strong beats weak |
|---|---|---|---|---|
| s5w4l3c0k0K0a0p1 | Yavalath (Browne, 2007) | died at gate 5 (over too fast, or never over) | 12 : 12 : 0 | 100% |
| s5w5l0c0k0K0a0p1 | five in a row on the Yavalath board | rank 734 of 2357, score 0.820 | 10 : 12 : 2 | 100% |
| s6w5l0c0k0K0a0p1 | five in a row, side 6 | died at gate 5 (over too fast, or never over) | 16 : 8 : 0 | 100% |
| s5w0l0c1k0K0a0p1 | connect opposite sides, side 5 | rank 892 of 2357, score 0.804 | 10 : 14 : 0 | 100% |
| s6w4l3c0k0K0a0p1 | Yavalath's rules on the side-6 board | died at gate 5 (over too fast, or never over) | 13 : 11 : 0 | 100% |
| s6w0l3c0k0K0a2p1 | my first pick this morning, before the reframe | died at gate 5 (over too fast, or never over) | 12 : 12 : 0 | 100% |
The gates kill Yavalath, and that is a finding about the gates. The engine's 24 games of it averaged 12.2 stones on 61 cells, under the quarter-board floor of gate 5, because at 1,000 playouts a move somebody blunders a three early. People play it longer. The same gate counts stones, not turns, so a two-stone game fills the board twice as fast and survives more easily: that is part of why two stones a turn does so well in the table above. I left the gate as it was rather than move it after seeing which control it killed.
Best of each board
| board | rule set | score | in words |
|---|---|---|---|
| side 4 | s4w0l3c1k1K0a0p2 | 0.946 | side 4, exactly 3 loses, connect opposite sides, custodial capture (single), two stones per turn (first turn one) |
| s4w5l3c0k2K0a1p2 | 0.924 | side 4, 5-in-a-row wins, exactly 3 loses, custodial capture (line), must touch a stone, two stones per turn (first turn one) | |
| s4w0l0c2k0K0a0p2 | 0.914 | side 4, connect three sides, two stones per turn (first turn one) | |
| side 5 | s5w5l3c0k0K0a1p2 | 0.959 | side 5, 5-in-a-row wins, exactly 3 loses, must touch a stone, two stones per turn (first turn one) |
| s5w4l3c0k0K0a1p1 | 0.954 | side 5, 4-in-a-row wins, exactly 3 loses, must touch a stone | |
| s5w0l3c1k0K0a0p2 | 0.946 | side 5, exactly 3 loses, connect opposite sides, two stones per turn (first turn one) | |
| side 6 | s6w6l3c0k0K0a1p2 | 0.979 | side 6, 6-in-a-row wins, exactly 3 loses, must touch a stone, two stones per turn (first turn one) |
| s6w6l4c0k2K0a0p1 | 0.972 | side 6, 6-in-a-row wins, exactly 4 loses, custodial capture (line) | |
| s6w6l3c0k0K0a0p1 | 0.966 | side 6, 6-in-a-row wins, exactly 3 loses |
The film
watch on YouTube · the Short · Both Hands on itch.io
The film walks the gates, the finalists and one full engine game of Both Hands, with the split-four trick that decides it.