Glossary of Sylver Coinage

Canonical Representation

The representation of a Sylver Coinage position as an ascending sequence of numbers, none of which can be expressed as a sum with multiples of previous numbers in the sequence. Every position has a unique canonical representation.

Coin Problem

F. G. Frobenius's problem of finding the largest number that cannot be expressed as a sum with multiples of numbers in a given set.

Enclosure

The enclosure E(S) of a position S with g=1 is the position formed by adjoining every legal move y greater than t(S)/2 such that t(S) − y is also legal. The enclosure of a position is always an ender.

Ender

A position S in Sylver Coinage is an ender if g(S) = 1 and t(S) is eliminated by any legal move less than t(S). See quiet ender.

Hutchings's Theorem

If m and n are coprime and are not 2 and 3, then the Sylver Coinage position {m,n} is 𝓝.

Long

A position S in Sylver Coinage is long if g(S) is prime and S ÷ g(S) is not a quiet ender. See short.

𝓝

In a game in which two players move alternately, a position is 𝓝 if the player whose turn it is to move can win. See 𝓟.

𝓟

In a game in which two players move alternately, a position is 𝓟 if the player who has just moved can win. (If neither player has yet moved, this means the second player.) See 𝓝.

Pairing

In a Sylver Coinage position, two legal moves are said to be paired if each wins against the other.

Probably 𝓝

A position S is probably 𝓝 if some move in S produces a position that is probably 𝓟.

Probably 𝓟

A Sylver Coinage position S with g=2 is probably 𝓟 if it is short and has no known winning moves, and all moves in S not known to lose produce long positions. See 𝓟.

Quiet Ender

A position S in Sylver Coinage is a quiet ender if g(S) = 1 and t(S) is eliminated by any legal move u less than t(S) without using multiple instances of u. For example, let S = {4, 5, 7}; then t(S) = 6. S is an ender because 3 eliminates 6; but S is not a quiet ender because 6 cannot be expressed as a sum of numbers in {3, 4, 5, 7} without using two 3's. See ender. An ender S is quiet precisely if t(S) is odd.

Quiet End Theorem

If S is a Sylver Coinage position, and m and n are coprime, then Sn is a quiet ender if and only if (S×m)n is. Here n need not be legal in S.

Roberts's Formula

A formula discovered by J. B. Roberts for the greatest number not expressible as a sum with multiples of numbers in a given finite arithmetic progression. If the progression is a₀,…,a_s with successive difference d>0, a₀>2, and g(a₀,d)=1, then t(a₀,…,a_s) + 1 = (⌊(a₀−2)/s⌋ + 1) ⋅ a₀ + (a₀ − 1) ⋅ (d − 1). When s=1, this formula reduces to Sylvester's Formula.

Short

A position S in Sylver Coinage is short if g(S) is prime and S ÷ g(S) is a quiet ender. See long.

Single Win Theorem

For any m > 1 there is a position in Sylver Coinage in which m alone wins.

Sylver Coinage

J. H. Conway's two-player game based on the Coin Problem. Each player in turn names a number that cannot be expressed as a sum with multiples of numbers previously named. The player who names 1 loses.

Sylvester's Formula

A formula discovered by J. J. Sylvester for the greatest number not expressible as a sum with multiples of two relatively prime numbers: t(m,n) = (m − 1)(n − 1) − 1. For a proof see, for example, Greg Knese's page.