The 'First to 100' GameDate: 03/12/2001 at 04:35:29 From: Shanthi Subject: Getting a strategy for this question The "First To 100" Game ----------------------- This is a game for two players. Players take turns choosing any whole number from 1 to 10. They keep a running sum of all the chosen numbers. The first player to make this total reach exactly 100 wins. Sample game: Player 1's choice Player 2's choice Running Total 10 10 5 15 8 23 8 31 2 33 9 42 9 51 9 60 8 68 9 77 9 86 10 96 4 100 Player 1 wins. Play the game a few times with your neighbor. Can you find a winning strategy? This is the question given to our group. Will you please help us? From, Shanthi and group Date: 03/12/2001 at 10:19:49 From: Doctor Rob Subject: Re: Getting a strategy for this question Thanks for writing to Ask Dr. Math, Shanthi. If the total T left by player A is between 90 and 99 inclusive, player B can take the rest (100 - T) and win. It should be the goal of each player not to let that happen. That means that a target number for each player is 89. If the running sum is 89, for whatever number N player A chooses, player B chooses 11 - N. Then the total will be 100, so B wins. Notice that: 1 <= N <= 10 implies that 1 <= 11 - N <= 10 so B's choice is allowed by the rules. Now if the total T left by player A is between 79 and 88, B can choose a number 89 - T, which is between 1 and 10, and make the total 89. Then B can win. That means that another target number for each player is 78. Similarly, other target numbers are 67, 56, 45, 34, 23, 12, and 1. (See a pattern here?) Either player who makes any of these totals can win. Since 1 is a target number, and the first player can choose 1, he should do so. Then he can win by this strategy: when the second player chooses N, he responds by choosing 11 - N. 11 appears here because the sum of the smallest and largest numbers each player can choose is 1 + 10 = 11. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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