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推理智力题 Logic Puzzles

1. 五顶帽子 Five Hats

从三顶黑帽子和两顶白帽子中任意选出三顶帽子戴在三个非常非常聰明的人士头上。三个人都看不到自己头上帽子的颜色 --其实，他们中有一个压根看不到颜色的 --因为他是盲人。

每个人都被问同一个问题：

“你的帽子是什么颜色的？”

第一个人仔细地看了看其他两个人的帽子然后说到：“我不知道。”当然，他非常聰明，他的回答也是非常正确的；他无法确定他的帽子的颜色。

第二个人也回答说：“我不知道。“他也非常聰明，他的回答也没错

第三个人--即那个盲人--听到问题后很快说出了他的帽子的颜色，而且他的回答很正确。他是怎么知道他的帽子的颜色的？

    From a group of three black hats and two white hats, three hats are chosen at random and placed on the heads of three very, very wise men. The men cannot see the color of their own hats -- in fact, one of the men cannot see at all --he is blind.
    Each man is asked, in turn.
    "What is the color of your hat?"
    The first man looks carefully at the hats of the other two men and says, "I don't know." He is, of course, very wise , and he is quite right; he can't determine the color of his hat.
    The second man also answers the question "I can't know." He, also a very wise man, is also right.
    The third man -- the blind man -- is quick to answer when he is asked. And he is correct. How does he know?

                                        Answer to the Puzzle
    The first wise man looked at the other two wise men's hats. If he had seen two white hats, he would have known that his own hat was black, for there are only two white hats. The first man did not see two white hats, so he said he didn't know the color of his own hat.
    The second wise man realized that the first had not seen two white hats. He looked at the third man's hat. If the third man had had a white hat, the second man would immediately have known that his own hat could not be white, too. But he said that he didn't know the color of his own hat, so he must have seen a black hat on the head of the third man.
    The third wise man -- the blind one --then knew that his own hat was black, and said so.

2. 九个铁球

现有9个同样大小铁球，其中8个铁球的重量相同，只有一个重量或轻或重，不得而知。现仅有一个天平，请问怎样使用天平找出这个重量不同的铁球，而且只能使用 3次天平。

Now there exsist 9 rion balls of the same size, 8 of which are of the same weight, except one which is heavier or lighter than the other eight. Now please find the ball different from others in weight with only one (weighing) balance (scale) and weigh only 3 times.

3. 计算物品

现在有一些物品，我们不知道有多少。如果我们每三个为一单位来数则剩余两个，如果每五个为一单位来数则剩余三个，而如果每七个为一单位来数的话则剩余两个。请问总共有多少物品？

We have things of which we do not know the number. If we count them by threes, they leave a remainder of two. If we count them by fives, they leave a remainder of two. If we count them by sevens, they leave a remainder of two. How many things are there?

（文言文原文：“今有一物，不知其数，三三数之余二；五五数之余三；七七数之余二。问物几何？”

Original in Classic Chinese: "We have a number of things which, counted by threes, leave two; by fives, leave three; and by sevens, leave two. Find the number."）

    答案：设该数为X 则： (Answer: Let the number be X, thus)
             (X-2)/3=0
             (X-3)/5=0
             (X-2)/7=0
            所以先找3、7的最小公倍数：21 (So first find the minimum common multiple, namely 21, which plus 2 is just the answer 23.)
            21+2=23
            23除以5余3 (23/5=4...3)
            故：问物几何？二十有三也。 (So: How many things? Twenty plus three.)