Sylver coinage
Encyclopedia
Sylver Coinage is a mathematical game
for two players, invented by John H. Conway. It is discussed in chapter 18 of
Winning Ways for Your Mathematical Plays
. This article summarizes that chapter.
The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers.
After 1 is named, all positive integers can be expressed in this way:
1 = 1, 2 = 1 + 1, 3 = 1 + 1 + 1, etc., ending the game. The player who named 1 loses.
A sample game between A and B:
Each of A's moves was to a winning position.
Sylver Coinage is named after
James Joseph Sylvester
, who proved that if a and b
are relatively prime positive integers, then (a − 1)(b − 1) − 1 is the largest number that is not a sum of nonnegative multiples of a and b. This is a special case of the Coin Problem
.
Unlike many similar mathematical games, Sylver Coinage has not been completely solved, mainly because many positions have infinitely many possible moves. Furthermore, the main theorem that identifies a class of winning positions, due to R. L. Hutchings, is nonconstructive: it guarantees that such a position has a winning strategy but does not identify it. Hutchings's Theorem states that any of the prime number
s 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves. Complete winning strategies are known for answering the losing openings 1, 2, 3, 4, 6, 8, 9, and 12.
Mathematical game
A mathematical game is a multiplayer game whose rules, strategies, and outcomes can be studied and explained by mathematics. Examples of such games are Tic-tac-toe and Dots and Boxes, to name a couple. On the surface, a game need not seem mathematical or complicated to still be a mathematical game...
for two players, invented by John H. Conway. It is discussed in chapter 18 of
Winning Ways for Your Mathematical Plays
Winning Ways for your Mathematical Plays
Winning Ways for your Mathematical Plays by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy is a compendium of information on mathematical games...
. This article summarizes that chapter.
The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers.
After 1 is named, all positive integers can be expressed in this way:
1 = 1, 2 = 1 + 1, 3 = 1 + 1 + 1, etc., ending the game. The player who named 1 loses.
A sample game between A and B:
- A opens with 5. Now neither player can name 5, 10, 15, ....
- B names 4. Now neither player can name 4, 5, 8, 9, 10, or any number greater than 11.
- A names 11. Now the only remaining numbers are 1, 2, 3, 6, and 7.
- B names 6. Now the only remaining numbers are 1, 2, 3, and 7.
- A names 7. Now the only remaining numbers are 1, 2, and 3.
- B names 2. Now the only remaining numbers are 1 and 3.
- A names 3, leaving only 1.
- B is forced to name 1 and loses.
Each of A's moves was to a winning position.
Sylver Coinage is named after
James Joseph Sylvester
James Joseph Sylvester
James Joseph Sylvester was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory and combinatorics...
, who proved that if a and b
are relatively prime positive integers, then (a − 1)(b − 1) − 1 is the largest number that is not a sum of nonnegative multiples of a and b. This is a special case of the Coin Problem
Coin problem
The coin problem is a mathematical problem that asks what is the largest monetary amount that cannot be obtained using only coins of specified denominations. For example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units...
.
Unlike many similar mathematical games, Sylver Coinage has not been completely solved, mainly because many positions have infinitely many possible moves. Furthermore, the main theorem that identifies a class of winning positions, due to R. L. Hutchings, is nonconstructive: it guarantees that such a position has a winning strategy but does not identify it. Hutchings's Theorem states that any of the prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves. Complete winning strategies are known for answering the losing openings 1, 2, 3, 4, 6, 8, 9, and 12.
External links
- Some recent findings about the game appear at http://www.monmouth.com/~colonel/sylver/.