Well, you're in a real pickle. You see, a few years ago, the king decided your life would be forfeit unless you tripled the gold coins in his treasury. Fortunately for you, a strange little man appeared and magically performed the feat. He placed handfuls of coins in and out of a magical bag, and sang a strange rhyme:“The more gold goes in and more comes out, as sure as I am me. And in again, and out again, and now it’s it times three!”
好吧，你真的遇到麻烦了。你看，几年前，国王决定你的生命将被没收，除非你将他国库中的金币增加三倍。对你来说幸运的是，一个奇怪的小男人出现了，他神奇地完成了这一壮举。他把一把硬币从一个神奇的袋子里取出来，并唱着一首奇怪的韵律：“金子进得越多，出得越多，我就是我。又进又出，现在是三倍了！”
Incredibly, that tripled the coins and saved your life. Were you grateful? Yes. Were you desperate? Yes. Did you promise him your first-born child in exchange for his help? Yes.Fast forward to today. No sooner have you given birth to a beautiful baby boy than the little man shows up to claim his prize. You cry and beg him not to take the baby. Softening, he begins, “If you can guess my name—” “Banach-Tarski?” you say. “It’s on the front of your shirt.”
令人难以置信的是，这使硬币增加了三倍并挽救了您的生命。你感激吗？是的。你绝望了吗？是的。你答应他你的第一个孩子以换取他的帮助吗？是的。快进到今天。你刚生下一个漂亮的男婴，小个子就来领奖了。你哭着求他不要带孩子。软化，他开始说，“如果你能猜出我的名字——” “Banach-Tarski？”你说。 “它在你衬衫的前面。”
“What! That won’t do. Aha. My bag,” he explains, “increases the number of gold coins placed inside it in a very special way. If I take any number of coins and place them in, more will come out. And if I place those in the bag again, the total that comes out will be three times whatever I began with.”He takes 13 coins and places them in the bag, then removes the contents. “I’ve used the magic once, not twice,” he says. “Tell me how many coins are in my hand and I’ll have mercy.”How many coins is he holding?
“什么！那不行。啊哈。我的包，”他解释道，“以一种非常特殊的方式增加了里面放置的金币数量。如果我拿出任意数量的硬币并将它们放入，就会有更多硬币出来。如果我再把它们放进袋子里，出来的总数将是我开始时的三倍。”他拿出 13 枚硬币放在袋子里，然后取出里面的东西。 “我用过一次魔法，不是两次，”他说。 “告诉我我手上有多少硬币，我手下留情。”他手上有多少硬币？
The bag’s magic works just like what in mathematics is called a “function,” and it’s convenient in both cases to use an arrow to denote the transformation. We can write what we know like this.We want to know what goes in this particular blank.Maybe the bag just multiplies the number of coins by some number. In that case, multiplying by that number twice would be the same as multiplying by 3, which means the multiplier would be the square root of 3. That’s not a whole number, though. And we don’t have bits of gold coins coming out of the bag. Something else is going on.
这个袋子的魔法就像数学中所谓的“函数”一样，在这两种情况下都可以方便地使用箭头来表示转换。我们可以像这样写下我们所知道的。我们想知道这个特定的空白中有什么。也许袋子只是将硬币的数量乘以某个数字。在那种情况下，乘以该数字两次与乘以 3 相同，这意味着乘数将是 3 的平方根。不过，这不是整数。而且我们没有从袋子里出来的金币。其他事情正在发生。
Well, if filling in the blank between 13 and 39 is too hard, maybe we can start with something easier. Can we figure out what’ll happen to 1 coin?If you use the bag on a single coin twice, you end up with triple; that’s three gold pieces.Because the bag always increases the number of gold coins, the blank must be between 1 and 3, so 2. It’s a start. What’s next?Let’s think about a few other possible starting places.
好吧，如果填写 13 到 39 之间的空白太难了，也许我们可以从更简单的开始。我们能算出 1 个硬币会发生什么吗？如果你在一个硬币上使用袋子两次，你最终会得到三倍；那就是三个金币。因为袋子总是增加金币的数量，所以空白必须在1到3之间，所以2。这是一个开始。下一步是什么？让我们考虑一些其他可能的起点。
We already know 2 becomes 3 and that lets us fill in the next blank as well. Now we’re getting somewhere!We just need to extend this out to 13. Remember the other rule, though: when you put more coins in, you get more coins out. That means the numbers in every column must go in increasing order as well. In other words, because 6 coins become 9, it’s not possible for 4 coins to become 10. Nor could 4 become 5, since 3 becomes 6.So 7 and 8 fill those blanks on the right of 4 and 5, which in turn gives the answer for two more blanks.
我们已经知道 2 变成 3，这也让我们可以填补下一个空白。现在我们有所进展！我们只需要将其扩展到 13。不过请记住另一条规则：当你投入更多的硬币时，你会得到更多的硬币。这意味着每一列中的数字也必须按递增顺序排列。换句话说，因为 6 个硬币变成 9，4 个硬币不可能变成 10。4 也不可能变成 5，因为 3 变成 6。所以 7 和 8 填补了 4 和 5 右边的空白，这反过来给出两个空白的答案。
Knowing that the numbers go in increasing order in every column, the only choices for the remaining blanks are 19, 20, 22, and 23.And look! We have our answer! There must be 22 gold coins in his hand.“I’ll give you three guesses,” the little man begins to say.“22 coins,” you respond.“What?! How did you know?”“I enjoy a good riddle,” you say. “Also, it’s on the back of your shirt.”
知道每一列中的数字都是按递增顺序排列的，剩余空白的唯一选择是 19、20、22 和 23。看！我们有答案！他手里一定有 22 个金币。“我猜你猜 3 次，”小个子开始说。“22 个金币，”你回答。“什么？！你怎么知道的？” “我喜欢猜谜语，”你说。 “还有，它在你衬衫的背面。”
