Glossary of Sylver Coinage

2000-12-16.
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. 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,…,as with successive difference d>0, a0>2, and g(a0,d)=1, then t(a0,…,as) + 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 Donald Hazlewood's lecture notes on linear diophantine equations. (This page contains TeX output that cannot be read with a text browser.)

Back to Sylver Coinage
Col. G. L. Sicherman [ HOME | MAIL ]