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番外 · 题谱 · 1917 · P4

Amusement 22 · Bridge Walk 22

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 22 topological

A man went recently into a dairyman's shop to buy eggs. He wanted them of various qualities. The salesman had new-laid eggs at the high price of fivepence each, fresh eggs at one penny each, eggs at a halfpenny each, and eggs for electioneering purposes at a greatly reduced figure, but as there was no election on at the time the buyer had no use for the last. However, he bought some of each of the three other kinds and obtained exactly one hundred eggs for eight and fourpence. Now, as he brought away exactly the same number of eggs of two of the three qualities, it is an interesting puzzle to determine just how many he bought at each price.

一名男子最近走进一家奶制品店买鸡蛋。他想要他们具有各种品质。推销员以五便士一个的高价出售新产的鸡蛋,一便士一个的新鲜鸡蛋,一个半便士的鸡蛋,还有竞选用的鸡蛋,价格大大降低,但由于当时没有选举,买家没有用最后一个。然而,他又买了另外三种鸡蛋中的一些,并以八又四便士的价格买到了正好一百个鸡蛋。现在,由于他带走了三种品质中两种品质的鸡蛋完全相同,因此确定他在每个价格下购买了多少个鸡蛋是一个有趣的难题。

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。