Difference between revisions of "Fifteen"

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Two friends, Sam and Lloyd, play the following game: each boy, on their turn, removes a number from 1,...,9 without replacement. The winner of the game is the one who first obtains 3 numbers that sum to 15. Does Sam have a winning strategy assuming he goes first?
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==Puzzle==
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Two friends, Sam and Lloyd, play the following game with nine ping-pong balls, numbered 1 to 9.  Each boy, on their turn, takes a ball (without replacement). The winner of the game is the first one to obtain 3 balls that sum to 15. Does either player have a winning strategy in this game?
  
 
==Help==
 
==Help==
  
 
{{Hint| Try to reinterpret the game as taking place on a 3x3 grid.}}
 
{{Hint| Try to reinterpret the game as taking place on a 3x3 grid.}}
{{Answer| No. At most he can guarantee a draw.}}
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{{Answer| No. At best Sam (going first) can guarantee a draw.}}
 
{{Solution| This game is actually "equivalent" to Tic Tac Toe on a 3x3 magic square (i.e. each row, column and diagonal sums to 15). Since there is no winning strategy for Tic Tac Toe neither is there for this game.}}
 
{{Solution| This game is actually "equivalent" to Tic Tac Toe on a 3x3 magic square (i.e. each row, column and diagonal sums to 15). Since there is no winning strategy for Tic Tac Toe neither is there for this game.}}
  
 
[[Category:Game theory]]
 
[[Category:Game theory]]

Current revision as of 13:10, 18 November 2010

Puzzle

Two friends, Sam and Lloyd, play the following game with nine ping-pong balls, numbered 1 to 9. Each boy, on their turn, takes a ball (without replacement). The winner of the game is the first one to obtain 3 balls that sum to 15. Does either player have a winning strategy in this game?

Help

Hint
Try to reinterpret the game as taking place on a 3x3 grid.
Answer
No. At best Sam (going first) can guarantee a draw.
Solution
This game is actually "equivalent" to Tic Tac Toe on a 3x3 magic square (i.e. each row, column and diagonal sums to 15). Since there is no winning strategy for Tic Tac Toe neither is there for this game.