Sylver coinage is a

mathematical game A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematics, mathematical parameters. Often, such games have simple rules and match procedures, such as tic-tac-toe and dots and boxes. Generally, mathemati ...

for two players, invented by John H. Conway. The two players take turns naming positive

integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

that are not the sum of nonnegative multiples of previously named integers. The player who names 1 loses. For instance, if player A opens with 2, B can win by naming 3 as A is forced to name 1. Sylver coinage is an example of a game using misère play because the player who is last able to move loses. Sylver coinage is named after

James Joseph Sylvester James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ...

, who proved that if ''a'' and ''b'' are

relatively prime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

positive integers, then (''a'' − 1)(''b'' − 1) − 1 is the largest number that is not a sum of nonnegative multiples of ''a'' and ''b''. Thus, if ''a'' and ''b'' are the first two moves in a game of sylver coinage, this formula gives the largest number that can still be played. More generally, if the

greatest common divisor In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers , , the greatest co ...

of the moves played so far is ''g'', then only finitely many multiples of ''g'' can remain to be played, and after they are all played then ''g'' must decrease on the next move. Therefore, every game of sylver coinage must eventually end. When a sylver coinage game has only a finite number of remaining moves, the largest number that can still be played is called the Frobenius number, and finding this number is called the coin problem.

Example

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. (Example: 27 = 4·3 + 5·3) * A names 11. Now the only remaining numbers are 2, 3, 6, and 7. * B names 6. Now the only remaining numbers are 2, 3, and 7. * A names 7. Now the only remaining numbers are 2, and 3. * B names 2. Now the only remaining number is 3. * A names 3, leaving nothing for B, and wins. Each of A's moves was to a winning position.

Analysis

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, guarantees that such a position has a winning strategy but does not identify the strategy. Hutchings's Theorem states that any of the

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

s 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves: these are the only winning openings known. When the

of the moves that have been made so far is 1, the remaining set of numbers that can be played will be a

finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...

, and can be described mathematically as the set of gaps of a

numerical semigroup In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number of integers and the binary operation is the operation of addition of integers. Also, the integ ...

. Some of these finite positions, including all of the positions after the second player has responded to one of Hutchings' winning moves, allow a special move that Sicherman calls an "ender". An ender is a number that may only be played immediately: playing any other number would rule it out. If an ender exists, it is always the largest number that can still be played. For instance, after the moves (4,5), the largest number that can still be played is 11. Playing 11 cannot rule out any smaller numbers, but playing any of the smaller available numbers (1, 2, 3, 6, or 7) would rule out playing 11, so 11 is an ender. When an ender exists, the next player can win by following a strategy-stealing argument. If one of the non-ender moves can win, the next player takes that winning move. And if none of the non-ender moves wins, then the next player can win by playing the ender and forcing the other player to make one of the other non-winning moves. However, although this argument proves that the next player can win, it does not identify a winning strategy for the player. After playing a prime number that is 5 or larger as a first move, the first player in a game of sylver coinage can always win by following this (non-constructive) ender strategy on their next turn. If there are any other winning openings, they must be 3-

smooth number In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number in which every prime factor is at most 7. Therefore, 49 = 72 and 15750 = 2 ...

s (numbers of the form for non-negative integers and ). For, if any number that is not of this form and is not prime is played, then the second player can win by choosing a large prime factor of . The first few 3-smooth numbers, 1, 2, 3, 4, 6, 8, 9, and 12, are all losing openings, for which complete strategies are known by which the second player can win. By

Dickson's lemma In mathematics, Dickson's lemma states that every set of n-tuples of natural numbers has finitely many minimal elements. This simple fact from combinatorics has become attributed to the American algebraist L. E. Dickson, who used it to prove a resu ...

(applied to the pairs of exponents of these numbers), only finitely many 3-smooth numbers can be winning openings, but it is not known whether any of them are. In 2017, offered a $1000 prize for determining who wins in the first unsolved case, the opening move 16, as part of a set of prize problems also including Conway's 99-graph problem, the minimum spacing of Danzer sets, and the thrackle conjecture.

References

External links

The Sylver Coinage Page
Mathematical games Combinatorial game theory

Example

Analysis

References

Further reading

External links