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The 'First to 100' Game


Date: 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/   
    
Associated Topics:
High School Number Theory
High School Puzzles
Middle School Puzzles

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