Sim is six dots and fifteen lines. Two players colour lines, and you lose the moment three of your own lines make a triangle.
Play it: https://dev48.infy.uk/game/day77-sim-ramsey-game.html
There are no draws, and that is a theorem you can check
Every one of the 32,768 two-colourings of K₆ contains a monochromatic triangle. 0 exceptions. K₅ has exactly 12 that do not — and 12 is also the number of Hamiltonian cycles of K₅, counted a second time with no colouring anywhere in the argument.
So the game only becomes a game at exactly the Ramsey number. On five dots it is a draw; on six the second player wins; and all 15 openings lose, because K₆ is edge-transitive and the fifteen of them are one position.
The advice everyone gives decides almost nothing
There is one rule anybody ever states: never complete your own triangle. Four policies that obey it to the letter, scored on how often they keep the second player's guaranteed win:
| policy | keeps the win |
|---|---|
| best reading | 46.3993% |
| second | 22.6546% |
| third | 10.9703% |
| worst | 0.0813% |
A 570× spread between four rules that all follow the same advice — and the worst is 56× worse than a player who has never heard of it and colours at random.
Both sides playing it turns a solved second-player win into a coin flip: 50.3140% to the player who is supposed to lose every game.
You cannot attack in Sim
The structure explains it, and the structure is measured rather than argued. Over all 6,162,465 moves in the game graph:
your move raises the OPPONENT's danger by exactly 0.0000 (0 exceptions)
your move raises YOUR OWN danger by 0.9176 on average
Every mine under a player was laid by that player. The only lever is denial — and denial is worth 278×: two policies differing solely in which of your own safe lines you spend score 22.6546% and 0.0813%.
And the best cheap rule does not pull that lever
Keeping the most safe lines for yourself scores 46.3993% against denial's 22.6546% — 2.05× better. The structural argument is right about the mechanism and wrong about the move, which is the part worth carrying away.
Verifier 49,175,462 asserts, page self-check 109,039,715 assertions over 1,350,022 positions and 8,562,585 moves. 0 failures. Vanilla JavaScript, one file, no build step.
