February 4, 1999<\/p>\n
OLD CHALLENGE.<\/b>\u00a0You have two one-hour fuses: lighting one end of a fuse will cause it to burn down to the other end in exactly one hour’s time. You know nothing else about the fuses; in particular you don’t know how long any segment of a fuse will burn, only that an entire fuse takes one hour. How can you tell when exactly 45 minutes have passed?<\/p>\n
ANSWER.\u00a0<\/b>This is possible, but it takes two good ideas. The first good idea is to light both ends of a fuse at once. It will burn in a half hour. The second good idea is to light one end of the other fuse at the same time, and its other end after a half hour (as measured by the first fuse). It will finish in 45 minutes: a half hour at normal speed and 15 minutes from both ends. The two fuses need not be identical.<\/p>\n
The best, winning entries also consider other time periods. Joe Shipman points out that with a single fuse you can time any fraction 1\/n of an hour, by snipping it into pieces and always keeping n ends burning. Similarly with two fuses, any fraction 1\/n + 1\/m. John Robertson notes that even without snipping, with two fuses you can time any fraction of the form n\/4 [except perhaps 1\/4?], and with three fuses any fraction of the form n\/8, except perhaps 5\/8? Elliot Kearsley warns against trying these experiments in your living room. Len VanWyk suggests that with enough fuses, you could trade them in for a watch.<\/p>\n
Todd Feitelson points out that two basic theorems of calculus, the Intermediate Value Theorem and the Mean Value Theorem, have applications to fuses: “Each fuse burns continuously from time T=0 to T=60, which means there is a point on each fuse where T=30. This is a ‘Time Center’ rather than a ‘Geometric Center,’ but the Intermediate Value Theorem guarantees that it exists. . . . These poorly made fuses which burn irregularly take exactly one hour to burn. The Mean Value Theorem guarantees that there is a point on each fuse where the rate of burning is exactly 1\/60 of the total length of the fuse per minute.”<\/p>\n
JOKE CHALLENGE.<\/b>\u00a0Readers are invited to submit mathematical jokes. Winning jokes will appear in future Math Chats.<\/p>\n
NEW CHALLENGE<\/b>\u00a0(Joe Shipman). Select the best occurrence in the world of each number from 1 to 10. For example, 12 is the number of eggs in a dozen or the number of months in a year.<\/p>\n
Send answers, comments, and new questions by email to:<\/p>\n
Frank.Morgan@williams.edu<\/a>, to be eligible for\u00a0Flatland<\/i>\u00a0and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan’s homepage is at\u00a0www.williams.edu\/Mathematics\/fmorgan\/<\/a>.<\/p>\n Copyright 1999, Frank Morgan.<\/p>\n","protected":false},"excerpt":{"rendered":" February 4, 1999 OLD CHALLENGE.\u00a0You have two one-hour fuses: lighting one end of a fuse will cause it to burn down to the other end in exactly one hour’s time. You know nothing else about the fuses; in particular you don’t know how long any segment of a fuse will burn, only that an entire […]<\/p>\n","protected":false},"author":2965,"featured_media":0,"parent":3459,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-3592","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3592","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/2965"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=3592"}],"version-history":[{"count":1,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3592\/revisions"}],"predecessor-version":[{"id":3593,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3592\/revisions\/3593"}],"up":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3459"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=3592"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}