Bachet's game

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A strategy game for two players, described as early as 1612 by Claude Gaspard Bachet de Méziriac , is known as a Bachet game (also: Ziel 100 ) , which is a special case of the Nim game . The game of bachet is representative of all so-called one-pile Nim games .

regulate

  • You start with a random number less than 30.
  • The players take turns adding a self-chosen whole number between 1 and 10 to this number.
  • The first player to reach 100 wins.

A quasi game

Actually, the game of bachet and its variations are not "real" games with an open result, since usually in the first (or second) move it is decided who wins.

analysis

The object of the game is to be the first to reach 100. This means that the last move can only be reached if the number is between 90 and 99. If you give your opponent 89 as a number, he can't win the game. The 89 is a key number. The other key numbers are separated by a difference of 11: 78, 67, 56, 45, 34, 23 and 12 (and 1).

example

  • You start with a 21.
  • Player A adds a 2 to it. The number is 23 and player A can no longer lose with skillful play.
  • Player B adds a 5. The number is 28.
  • Player A adds a 6. The number is 34 .
  • Player B adds an 8. The number is 42.
  • Player A adds a 3. The number is 45 .
  • Player B adds a 7. The number is 52.
  • Player A adds a 4. The number is 56 .
  • Player B adds a 5. The number is 61.
  • Player A adds a 6. The number is 67 .
  • Player B adds a 3. The number is 70
  • Player A adds an 8. The number is 78 .
  • Player B adds a 9. The number is 87
  • Player A adds a 2. The number is 89 .
  • Player B adds a 1. The number is 90
  • Player A adds a 10. The number is 100 and Player A is expected to win.

observation

As you can see, after player A has reached the key number 23, player B always adds an 11 to his move.

Unfavorable starting situation

It becomes unfavorable for the first player if the randomly chosen starting number is a key number.

generalization

The principle can be applied to modified rules. With a target z and a train width of 1 to n, the key numbers are z- (n + 1), z-2 (n + 1), z-3 (n + 1), ...

literature

  • Claude Gaspard Bachet de Meziriac: Problemes plaisans et delectables, qui se font par les nombres. Paris 1612, 2nd ed. 1624, 3rd and 4th ed. von Labosne 1874 and 1879, chap. 1, p. 115, problem XXII.
  • Wilhelm Ahrens : Mathematical conversations and games. Teubner, 1901, p. 72.
  • Maurice Kraitchik: Mathematical Recreations. 2nd Edition. Dover, New York 1953, pp. 83-86 (The Battle of Numbers).

Individual evidence

  1. ^ Wilhelm Ahrens : Mathematical conversations and games . Teubner, 1901, p. 72