- 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
*a*_{0},…,*a*with successive difference_{s}*d*>0,*a*_{0}>2, and*g*(*a*_{0},*d*)=1, then*t*(*a*_{0},…,*a*) + 1 = (⌊(_{s}*a*_{0}−2)/*s*⌋ + 1) ⋅*a*_{0}+ (*a*_{0}− 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.)

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