June 22, 2000<\/p>\n
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What do you think is the best voting system among three or more candidates? Rank the following from 1 (best) to 4 (worst) and return to\u00a0mackenzi@cruzio.com<\/a>\u00a0by June 27.<\/p>\n _____Plurality (whoever gets the most votes wins)<\/p>\n _____Runoff (between two highest vote getters)<\/p>\n _____Approval voting (voter votes for all acceptable candidates and the candidate so approved by the most voters wins)<\/p>\n _____Ranking (voter ranks the candidates from 1 to n and the candidate with the smallest sum of rankings wins)<\/p>\n All four methods have paradoxical consequences, in which a majority of the voters would prefer another candidate over the winner. (See\u00a0“Ask the Experts”<\/a>\u00a0and\u00a0Math Chat<\/a>\u00a0of November 8, 1996.) Results of this poll will appear in the next Math Chat July 6 and in the November issue of\u00a0Discover<\/i>\u00a0magazine.<\/p>\n New Challenge.<\/b>\u00a0Design your own voting system and submit it to MathChat <Frank.Morgan@williams.edu<\/a>>. The winning answer will appear in the next Math Chat July 6.<\/p>\n Old Challenge.<\/b>\u00a0(Salvador Segura Gomis). What is the shortest line segment fencing off prescribed area 0<\/p>\n Answer<\/b>\u00a0(Joseph DeVincentis). We may as well assume that A is at most 1\/2, since a larger area on one side corresponds to a smaller area on the other side. For A up to 1\/4, fence off a corner diagonally. For A from 1\/4 to 1\/2, just use a horizontal fence.<\/p>\n Similarly in the unit cube, for small volume slice off a corner; for intermediate volume slice off an edge; for larger volume use a horizontal plane. The transitional values turn out to be 28<\/sup>\/37<\/sup>\u00a0and 1\/4.<\/p>\n For a 4D cube, for small volume slice off a corner; for somewhat larger volume slice off an edge; for somewhat larger volume slice off a 2D face; for larger volume use a horizontal hyperplane. The transitional values are 317<\/sup>\/231<\/sup>, 28<\/sup>\/37<\/sup>, and 1\/4.<\/p>\n Joe Shipman reports that the transitional volumes for general dimensions are (1-1\/n)1.5n(n-1)<\/sup>\u00a0nn<\/sup>\/n! Plugging in n = 2, 3, 4 yields 1\/4, 28<\/sup>\/37<\/sup>, and 317<\/sup>\/231<\/sup><\/p>\n <\/p>\n Send answers, comments, and new questions by email to\u00a0<\/a>Frank.Morgan@williams.edu,<\/a>\u00a0to be eligible for\u00a0Flatland\u00a0<\/i>and 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 THE MATH CHAT BOOK,<\/a>\u00a0including a $1000 Math Chat Book\u00a0QUEST,\u00a0<\/a>questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).<\/p>\n Copyright 2000, Frank Morgan.<\/p>\n","protected":false},"excerpt":{"rendered":" June 22, 2000 What do you think is the best voting system among three or more candidates? Rank the following from 1 (best) to 4 (worst) and return to\u00a0mackenzi@cruzio.com\u00a0by June 27. _____Plurality (whoever gets the most votes wins) _____Runoff (between two highest vote getters) _____Approval voting (voter votes for all acceptable candidates and the […]<\/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-3682","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3682","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=3682"}],"version-history":[{"count":3,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3682\/revisions"}],"predecessor-version":[{"id":3776,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3682\/revisions\/3776"}],"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=3682"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
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