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番外 · 题谱 · 1917 · P2

Amusement 8 · Innkeeper’s Bill 8

算术 · P2/P5 · 中段题

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 8 arithmetic

A charitable gentleman, on his way home one night, was appealed to by three needy persons in succession for assistance. To the first person he gave one penny more than half the money he had in his pocket; to the second person he gave twopence more than half the money he then had in his pocket; and to the third person he handed over threepence more than half of what he had left. On entering his house he had only one penny in his pocket. Now, can you say exactly how much money that gentleman had on him when he started for home?

一位慈善家的先生一天晚上回家路上,连续接到三个有需要的人的求助。他给第一个人一分钱,超过了他口袋里一半的钱;第二个人给了他一分钱。他给了第二个人两便士,比他当时口袋里的钱多了一半。他把剩下的一半以上的三便士交给了第三个人。进家门时,他口袋里只有一分钱。现在,你能准确地说出这位先生出发回家时身上带了多少钱吗?

提示 1

先列小例,不要凭直觉猜最大或最小。

提示 2

看奇偶、整除、余数和总量是否已经锁住选择。

提示 3

最后用上下界排除所有漏网情形。

完整解答

设第二人得 x 枚,则第一人得 2x - 8,第三人得 x + 11。三人总和给出 4x + 11 - 8 = 55,所以 x = 13。三份分别是 18、13、24 枚。

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。