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番外 · 题谱 · 1917 · P4

Amusement 10 · Boundary Path 10

拓扑趣题 · P1/P4 · 起手题

题面摘自 Henry E. Dudeney 公版文本;英文为古腾堡原文整理,中文为本站自译,提示、解答骨架和闲谈保留本站原创结构。

Amusements in Mathematics 1917 10 topological

"Whom do you think I met in town last week, Brother William?" said Uncle Benjamin. "That old skinflint Jorkins. His family had been taking him around buying Christmas presents. He said to me, 'Why cannot the government abolish Christmas, and make the giving of presents punishable by law? I came out this morning with a certain amount of money in my pocket, and I find I have spent just half of it. In fact, if you will believe me, I take home just as many shillings as I had pounds, and half as many pounds as I had shillings. It is monstrous!'" Can you say exactly how much money Jorkins had spent on those presents?

“威廉弟兄,你认为我上周在城里遇见了谁?”本杰明叔叔说。 “那个老顽固的乔金斯。他的家人一直带他到处买圣诞礼物。他对我说,‘为什么政府不能废除圣诞节,让送礼物受到法律惩罚呢?今天早上我口袋里揣着一些钱出来,我发现我只花了一半。事实上,如果你相信我,我带回家的先令和我有的英镑一样多,也有我有先令的一半。太可怕了!’”你能确切地说出乔金斯在这些礼物上花了多少钱吗?

提示 1

先只保留连接关系,把多余长度和角度擦掉。

提示 2

数端点、奇偶穿越和连通块,看看有没有不变量。

提示 3

最后再把抽象关系放回原图,确认没有偷换路径。

完整解答

一笔画回到起点需要所有路口度数为偶数;若有四个奇度路口,就不可能每条路恰好走一次并回到原点。至少要改变两条端点关系,或增加一条连接两个奇度点的路,把奇度点数降到 0。

Dudeney 的趣题常把难点藏在“看起来可以试”的地方。别急着猜答案;先把图、表或状态画出来,再问哪些限制一直没有变。这也是它和 Carroll 逻辑题互补的地方:一个拆句子,一个拆结构。