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### The 'First to 100' Game

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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?

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

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