Math Game
Math Game
How does this math game work?
In this game we have to (2 players) pick a # from 1-10. once you pick that number, you add it up to the total (0 at the start) and try to reach 100. the first one to ge to 100 wins.
Thank you.
2 players. Pick a # 1-10 and try to get to a hundred first, by adding the # to 0 when they start, and then to the total every turn. How are you guranteed to win?
The winning strategy is to choose your number such that the total becomes 1+11k for some integer k. If you go first (or your opponent deviates from this strategy at any time), you are guaranteed to win using this strategy. Here’s why:
You know that if, at the beginning of your turn, the total is any of 90-99, you are guaranteed to win (since you can make the total 100 on that very turn). If, at the start of your turn, your total is 89, then your opponent wins, because no matter what move you make, it will leave him with a total between 90 and 99. Thus, 89 is a losing number — if you start the turn with this number and your opponent knows the strategy you are guaranteed to lose. Now suppose that the current total is among the 10 preceding any losing number – then you can force your opponent to lose by choosing your number so as to make the total the losing number. Now, if the total is exactly 11 less than a losing number, no matter what move you make, he will be able to choose a number that leaves you with the losing number, guaranteeing his victory. Thus, 11 less than any losing number is a losing number. So every number less than 100 and congruent to 89 mod 11 is a losing number, and 89≡1 mod 11. Thus, by choosing your number so as to leave your opponent with a number congruent to 1 mod 11, you can always ensure your opponent loses.
It follows that, if you move first, your first move should always be 1.
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Tags: education, game, games, interactive, math, math games, math games for 6th graders, math games for kids, math games for kindergarten, math games for third grade, math games online, maths, themathgames.com
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