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Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 ofWinning Ways for Your Mathematical Plays. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers. The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3.

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  • Acuñación de Sylver (es)
  • Sylver coinage (en)
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  • La acuñación de Sylver es un juego matemático para dos jugadores, inventado por John H. Conway. Se trata en el capítulo 18 de Winning Ways for your Mathematical Plays.​ Los dos jugadores se turnan para nombrar números enteros positivos mayores que 1 que no son la suma de múltiplos no negativos de números enteros previamente nombrados. El jugador que no pueda nombrar tal número pierde. Por ejemplo, si el jugador A abre con 2, B puede ganar nombrando 3. (es)
  • Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 ofWinning Ways for Your Mathematical Plays. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers. The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3. (en)
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  • La acuñación de Sylver es un juego matemático para dos jugadores, inventado por John H. Conway. Se trata en el capítulo 18 de Winning Ways for your Mathematical Plays.​ Los dos jugadores se turnan para nombrar números enteros positivos mayores que 1 que no son la suma de múltiplos no negativos de números enteros previamente nombrados. El jugador que no pueda nombrar tal número pierde. Por ejemplo, si el jugador A abre con 2, B puede ganar nombrando 3. La acuñación de Sylver lleva el nombre de James Joseph Sylvester, quien demostró que si a y b son números coprimos enteros positivos, entonces (a -1) (b - 1) - 1 es el número más grande que no es una suma de múltiplos no negativos de a y b. Por lo tanto, si a y b son los dos primeros movimientos en un juego de Sylver, esta fórmula da el número más grande que todavía se pueden reproducir. De manera más general, si el máximo común divisor de las jugadas jugadas hasta ahora es g, entonces solo un número finito de múltiplos de g puede quedar por jugar, y después de que se hayan jugado todos, g debe disminuir en el siguiente movimiento. Por lo tanto, todo juego de acuñación de sylver debe terminar eventualmente. Cuando un juego de monedas de Sylver tiene solo un número finito de movimientos restantes, el número más grande que todavía se puede jugar se llama número de Frobenius, y encontrar este número se llama el problema de la moneda.​ (es)
  • Sylver coinage is a mathematical game for two players, invented by John H. Conway. It is discussed in chapter 18 ofWinning Ways for Your Mathematical Plays. This article summarizes that chapter. The two players take turns naming positive integers greater than 1 that are not the sum of nonnegative multiples of previously named integers. The player who cannot name such a number loses. For instance, if player A opens with 2, B can win by naming 3. Sylver coinage is named afterJames Joseph Sylvester, who proved that if a and bare 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. 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, ifthe greatest common divisor 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. (en)
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