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 such 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. 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 arithmetic progression. If the progression is a[0],...,a[s] with successive difference d>0, a[0]>2, and g(a[0],d)=1, then t(a[0],...,a[s]) + 1 = floor((a[0]-2)/d) . a[0] + (a[0] - 1) . (d - 1). If 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 Donald Hazlewood's lecture notes on linear diophantine equations. (This page contains TeX output that cannot be read with a text browser.)