Bachet的问题：尽可能少的重量来衡量它们
Bachet's Problem: as few weights to weigh them all
摘要：
一个经久不衰的问题是：“如果砝码可以放在任何一个秤盘上，那么在一个秤盘上可以用来称量从1到40（包括1和40）的任何整数磅的砝码的最少数量是多少？”“W. W. Rouse Ball将这个问题的第一次记录归因于17世纪初的Bachet，称之为“Bachet的重量问题”。然而，Bachet的问题一直延伸到1202年的Fibonacci，使其成为第一个整数分割问题的可行候选人。值得注意的是，考虑到Bachet问题的年龄，一个优雅而简洁的解决方案，当我们用任何整数代替40时，这个问题只是在过去的15年左右才出现。我们希望在这里阐述这一普遍性，只是用我们敏锐的智慧和归纳的意愿来武装。在这样做的过程中，我们将发现整数划分和枚举组合学的一些乐趣。这篇临时文章，而感兴趣的研究人员在组合，整数分区和数学史，是写一个易受影响的本科观众铭记。
英文摘要：
A problem that enjoys an enduring popularity asks: "what is the least number of pound weights that can be used on a scale pan to weigh any integral number of pounds from 1 to 40 inclusive, if the weights can be placed in either of the scale pans ?" W.W. Rouse Ball attributes the first recording of this problem to Bachet in the early 17th century, calling it "Bachet's Weights Problem". However, Bachet's problem stretches all the way back to Fibonacci in 1202, making it a viable candidate for the first problem of integer partitions. Remarkably, given the age of Bachet's problem, an elegant and succinct solution to this problem when we replace 40 with any integer has only come to light in the last 15 or so years. We hope to expound on this generalization here armed only with our sharp wits and a willingness to induct. In doing so we will discover some of the joys of partitions of integers and enumerative combinatorics. This expository article, while of interest to researchers in combinatorics, integer partitions and the history of mathematics, is written with an impressionable undergraduate audience in mind.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
