- 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.)

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Col. G. L. Sicherman [ HOME | MAIL ]