A limit that changes every turn
After a player removes x counters, the next player may take from one through 2x counters, capped by the remaining pile. A small take restricts the reply; a large take opens more options.
Fibonacci Nim
Take the final counter while controlling the changing upper limit on each move.
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How to play
Take the final counter while controlling the changing upper limit on each move.
- On the first turn, take fewer than all counters.
- Afterward, take at most twice the number removed on the previous turn.
- Continue until one player takes the final counter and wins.
Strategy tips
- A small move can sharply restrict the computer's reply.
- Use Fibonacci decompositions to identify useful opening takes.
- Before taking a large group, verify that the expanded reply limit is safe.
Complete guide
Rules, scoring, and strategy
Fibonacci Nim begins with one pile. The first player may not take the whole pile; afterward, each take may be no larger than twice the previous take. The final counter wins.
The special opening
The first move must leave at least one counter. This prevents an immediate trivial win and creates the first response limit from the number you choose to take.
Fibonacci decomposition
Every positive integer can be represented as a sum of nonconsecutive Fibonacci numbers. The smallest term in that representation helps identify useful moves and explains the game's name.
Control before greed
Removing many counters is not always progress because it expands the computer's legal range. Compare the pile remaining with twice your proposed take and avoid granting an immediate finish.
Questions players ask
Why can I not take the whole pile first?
The opening rule requires a proper take, so at least one counter must remain after the first move.
What limits later moves?
A player may take at most twice the number removed on the immediately preceding turn.
Who wins on the last counter?
The player who legally removes the final counter wins.