Multiples, not arbitrary amounts
If the values are a and b with a no larger than b, a legal move replaces b with b − ka for an integer k of at least one. The chosen multiple may make the new value zero, which wins immediately.
Euclid's Game
Make one of the two values zero by choosing legal multiples on alternating turns.
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How to play
Make one of the two values zero by choosing legal multiples on alternating turns.
- Inspect the smaller and larger positive values.
- Choose a multiple of the smaller value that does not exceed the larger value.
- Subtract it from the larger value; making zero wins immediately.
Strategy tips
- A move may subtract the smaller value more than once.
- Consider the quotient before choosing the maximum legal multiple.
- Positions near the golden ratio often determine whether a winning reply exists.
Complete guide
Rules, scoring, and strategy
Euclid's Game turns the subtraction step of the Euclidean algorithm into a two-player contest. On each move, subtract a positive multiple of the smaller value from the larger without making it negative.
The quotient defines your options
The integer quotient of the larger value divided by the smaller is the largest legal multiple. A large quotient offers several choices; a quotient of one forces a single subtraction.
Greedy subtraction can fail
Always choosing the largest multiple reaches zero when it is available, but otherwise may hand the opponent a favorable ratio. Compare the resulting pair after each candidate multiple instead of treating the game as automatic arithmetic.
Ratios and forced play
Positions near the golden ratio often force a chain of quotient-one moves. When the quotient is at least two, the current player frequently has more control over which side of that boundary the next pair occupies.
Questions players ask
May I subtract any number?
No. The amount must be a positive whole-number multiple of the smaller current value.
What happens when a value reaches zero?
The player who made that move wins immediately.
Are the values reordered after a move?
Yes. The interface presents the resulting pair from smaller to larger so the next legal multiples are clear.