Two players A and B are playing a game. A starts with 100 and B says a number, satisfying the equation 0 < | A - B | equal to or greater than 11. The person who reaches 0 first wins. Who can ensure that4 irrespective of his opponent's move he can win the game.
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Answers
Answer:
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.
Answer:
starts with 100 and B says a number, satisfying the equation 0 < | A - B | equal to or greater than 11. The person who reaches 0 first wins. Who can ensure that4 irrespective of his opponent's move he can win the game.