Interactive research game

Zero Forcing Game

Exact optimal play•Current paper rules

Spend fewer seeds. Force the rest.

Learn the rule, play the game, then watch optimal strategy unfold.

Alice and Bob build competing forcing chains. A seed costs one. A legal force is free. The player who spends fewer seeds wins.

Lesson 1 of 4Alice: follow the highlighted move
1

Start a forcing chain

Alice has no coloured vertices, so her first move must be a seed. Select v0.A gold ring marks a seed. It adds one to Alice's score.
AliceBobSeedOptimal

Alice has no coloured vertices, so her first move must be a seed. Select v0.

The rule that makes the game

Forces are protected, but your own force always comes first.

1

Seed

Colour any permitted white vertex with your colour. This costs one seed.

2

Force

If one of your vertices has exactly one white neighbour, colour it for free.

3

Preserve

Do not seed an opponent's force target while a non-blocking move exists.

4

Compete

Your own legal force is always allowed, even when it removes an opponent's force.