Three kinds of legal move
A turn may reduce only the left heap, only the right heap, or both heaps equally. A diagonal move cannot remove different amounts, and no heap may go below zero.
Wythoff Nim
Take the final counter by moving the two heaps away from the opponent's cold positions.
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How to play
Take the final counter by moving the two heaps away from the opponent's cold positions.
- Choose the left heap, right heap, or both heaps.
- Remove any positive amount; a move on both must remove the same amount from each.
- The player who empties both heaps wins.
Strategy tips
- Compare the smaller heap with the difference between the two heaps.
- Diagonal moves preserve the heap difference; single-heap moves change it.
- Look for positions related to consecutive multiples of the golden ratio.
Complete guide
Rules, scoring, and strategy
Wythoff Nim uses two heaps and adds a diagonal move: remove counters from one heap, or remove the same positive amount from both. The final counter wins under normal play.
Cold pairs
The losing positions form pairs related to the golden ratio. If the smaller heap is a certain multiple of the difference between heaps, every legal move leaves that family and a correct reply can return to it.
Use the heap difference
Removing equally from both heaps preserves their difference. A single-heap move changes it. Comparing the smaller heap and the difference often reveals whether to move diagonally or alter only one side.
Finish exactly
The winner empties both heaps, either with one diagonal take or by clearing the final surviving heap. Before a large move, check whether it gives the computer an immediate diagonal finish.
Questions players ask
Must I remove from both heaps?
No. You may remove from either one heap or from both heaps equally.
Can the two heaps start with different sizes?
Yes. Unequal heaps are the normal case and make the choice between straight and diagonal moves meaningful.
What does a cold position mean?
It is a position where the player to move loses against correct play because every legal move gives the opponent a winning reply.