Easy, Fun Games With Numbers
by Malcolm Blake
Games with numbers can be great fun and excercise the mind! Below are a few number games with directions on how to play:
Who Can Reach 100 First? This game is easy to understand, and many people can tQ taken in by it. The game may be played several times without the opponent guessing the trick of winning. Two players are needed. The winner is the one who reaches 100 first.
The person who starts - let's call him George - chooses a number. His partner - say, Ken - adds any number between i and 10. Now it's George's turn. The two players take turns in adding a number between 1 and 10 to the previous total. Who will win? Who will get to 100 first?
The winner is the one who can make his opponent reach 99. If Ken reaches 89, then he has already won, because the largest number George can reach is 99. In order that Ken can reach 89 first, he must also reach 78 first. If we continue along these lines, Ken must also reach 67, 56, 45, 34, 23, and 12 first. If George does not know the game and he starts with any number other than 1, then Ken replies with a number which gives a total of 12. Already the winner has been decided, because George can increase this total by 10 at the most, giving a total of 22. Ken makes it 23, and so on, until the total of 100 is reached.
Different rules can also be used. It is possible to fix a target figure different from 100; the upper and lower limits for the numbers to be added can also be changed. We can win all the games if the sum of the largest and smallest number is subtracted from the target figure. In this way we get a series of numbers that ensures victory. If, for instance, the target is 80 and the numbers added must lie between 2 and 7, then the winning series will be 82-9 = 71, 71-9 = 62, then 53, 44, 35, 26, 17, and 8.
The same game has an even more exciting variation. Count out 40 matches. The two players take turns, removing at least 2 and not more than 5 at a time. The player removing the last match is the winner.
This problem is easy to solve: the winner is the one who leaves 7 on the table. Since his opponent must remove at least 2 and not more than 5, in the first case 5 and in the second 2 would be left. So that 7 matches will be left, the winner must also leave 14 previously. Similarly, this applies when the number left on the table is 21, 28 and 35. These are multiples of 5 + 2 = 7 - therefore, if the person who starts knows the game, he simply removes 5 matches and the game is as good as won.
If, however, the person who knows the game does not start, he may not be able to win. Suppose that Ken knows the game, but George insists on starting. George takes 4 matches, leaving 36. In this case Ken cannot reach 35. It is possible that George will play into Ken's hands on the next move, but if George thinks about it logically, then whatever Ken's second move is, George can prevent Ken reaching 28. Therefore, Ken has lost the game.
If the rules are changed so that the loser removes the last two matches, the one who starts must try to leave two matches after his last move. This he can achieve if the number left after his next-to-last move is 2 + 2 + 5 = 9. We get this number by adding, to the last number to be left, the smallest and the largest number that can be taken away. The number 9 can be reached if the previous total is 9 + 7 = 16. Then, the previous totals are 23, 30, and 37. The person who starts must, therefore, remove 3 matches. Naturally, only the one who starts can be certain of winning.
Think of a Number. This is a well-known game with many variations. Let's have a closer look at some of the more interesting ones. Katie says to Valerie, "Take a piece of paper and a pencil, think of a number, and write it down. Multiply it by 10 and take away the number you first thought of. Add 36 and cross out one of the figures in the final number (except the last one, if it is o). Tell me the figures that make up the final number, in any order, and I will tell you the figure you crossed out". Valerie writes down 312. She multiplies it by 10: 3,120. 3,120 - 312 = 2,808. 2,808 + 36 = 2,844. She crosses out the 8 and rearranges the remaining numbers: 4, 2, 4.
Katie adds these together, 4 + 2 + 4 = 10. She subtracts the sum from the nearest number larger than 10 that is divisible by 9, that is, 1 8. The remainder is the number Valerie crossed out.
Why does this work? Valerie, when she had finished the sum set, arrived at a number divisible by 9. If any number is multiplied by 10 and the original is subtracted, then the remainder is 9 times the original number. If we add to this 36, that is, 4X9, then the sum will also be divisible by 9. We know that if the figures of a number divisible by 9 are added together, the sum will also be divisible by 9. So the missing figure, with the sum of the other figures, gives a number divisible by 9.
Valerie did not want to be outdone and asked Katie to think of a number. Then she said, "Double it, add 4, divide by 2, add 7, multiply by 8, subtract 12, divide the remainder by 4, subtract n, and tell me the result. I'll tell you the number you thought of".
Katie thought of n. n X 2 = 22, 22 + 4 = 26, half of 26 = 13. 13 + 7 = 20, 20 x 8 = 160, 160 - 12 = 148, 148 /4 = 37. 37 - 11 = 26. Katie tells Valerie the result: 26.
Valerie subtracts 4 (22), halves that, and says, "The number you thought of was n". Naturally, Katie is very curious. Valerie explains, "You take four from the final result, halve that, and you have the original number". They try it several times, with different numbers, and the answer is always right.
Malcolm Blake has written articles on all manner of games, including popular flash PSP games.
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