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番外 · 题谱 · 1907 · P4

Canterbury Puzzle 4 · Knight’s Passage 4

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。

The Canterbury Puzzles 1907 4 topological

This worthy man was, as Chaucer tells us, "a very perfect, gentle knight," and "In many a noble army had he been: At mortal battles had he been fifteen." His shield, as he is seen showing it to the company at the "Tabard" in the illustration, was, in the peculiar language of the heralds, "argent, semée of roses, gules," which means that on a white ground red roses were scattered or strewn, as seed is sown by the hand. When this knight was called on to propound a puzzle, he said to the company, "This riddle a wight did ask of me when that I fought with the lord of Palatine against the heathen in Turkey. In thy hand take a piece of chalk and learn how many perfect squares thou canst make with one of the eighty-seven roses at each corner thereof." The reader may find it an interesting problem to count the number of squares that may be formed on the shield by uniting four roses.

正如乔叟告诉我们的那样,这个值得尊敬的人是“一位非常完美、温柔的骑士”,“他在许多高贵的军队中:在殊死的战斗中,他十五岁。”正如插图中他在“战袍”中向连队展示的那样,他的盾牌用先驱者的特殊语言来说是“银色,玫瑰色,红色”,这意味着在白色的地面上散布着红玫瑰,就像用手播种的种子一样。当这位骑士被要求提出一个谜题时,他对同伴们说:“当我与帕拉丁领主一起在土耳其对抗异教徒时,一个异鬼问过我这个谜语。你们手里拿着一支粉笔,看看你能用粉笔的每个角上的八十七朵玫瑰中的一朵画出多少个完美的正方形。”读者可能会发现计算四朵玫瑰组合在一起可以在盾牌上形成的方块数量是一个有趣的问题。

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。