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番外 · 题谱 · 1907 · P3

Canterbury Puzzle 9 · Square Table 9

几何趣题 · P3/P6 · 压轴题

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 9 geometric

The Carpenter produced the carved wooden pillar that he is seen holding in the illustration, wherein the knight is propounding his knotty problem to the goodly company (No. 4), and spoke as follows: "There dwelleth in the city of London a certain scholar that is learned in astrology and other strange arts. Some few days gone he did bring unto me a piece of wood that had three feet in length, one foot in breadth and one foot in depth, and did desire that it be carved and made into the pillar that you do now behold. Also did he promise certain payment for every cubic inch of wood cut away by the carving thereof. "Now I did at first weigh the block, and found it truly to contain thirty pounds, whereas the pillar doth now weigh but twenty pounds. Of a truth I have therefore cut away one cubic foot (which is to say one-third) of the three cubic feet of the block; but this scholar withal doth hold that payment may not thus be fairly made by weight, since the heart of the block may be heavier, or perchance may be more light, than the outside. How then may I with ease satisfy the scholar as to the quantity of wood that hath been cut away?" This at first sight looks a difficult question, but it is so absurdly simple that the method employed by the carpenter should be known to everybody to-day, for it is a very useful little "wrinkle."

木匠拿出了插图中他拿着的雕刻木柱,其中骑士正在向好朋友提出他的棘手问题(第4号),并说道:“伦敦城里住着一位学者,精通占星术和其他奇怪的艺术。几天前,他确实给我带来了一块长三英尺,宽一英尺,深一英尺的木头,并希望将其雕刻并制成你们现在做的柱子。看哪,他还承诺为雕刻时砍下的每立方英寸木材支付一定的费用。事实上,我因此切掉了三立方英尺的块中的一立方英尺(也就是说三分之一);但这位学者确实认为,按重量支付可能不公平,因为块的中心可能比外部更重,或者可能更轻。那么,我怎样才能轻松地让学者满意所砍伐的木材数量呢?”乍一看,这是一个困难的问题,但它是如此简单,木匠所采用的方法今天应该为每个人都知道,因为这是一个非常有用的小“皱纹”。

提示 1

先把图画成可量的对象,标出边、格点或切割线。

提示 2

找面积、长度、对称轴或不变量,不急着剪。

提示 3

若是重排题,检查每一块在移动前后是否保持形状和面积。

完整解答

把题目先放到方格纸上。若四块面积相等,每块面积是总面积的四分之一;“轮廓相同”要求两道切割线在旋转或翻折后能互相对应。解题主线是先画出中心对称或四分旋转的候选,再用方格计数核对每块面积。

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。