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 mackenzi@cruzio.com by 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 candidate so approved by the most voters wins)

_____Ranking (voter ranks the candidates from 1 to n and the candidate with the smallest sum of rankings wins)

All four methods have paradoxical consequences, in which a majority of the voters would prefer another candidate over the winner. (See "Ask the Experts" and Math Chat of November 8, 1996.) Results of this poll will appear in the next Math Chat July 6 and in the November issue of *Discover* magazine.

**New Challenge.** Design your own voting system and submit it to MathChat Frank.Morgan@williams.edu>. The winning answer will appear in the next Math Chat July 6.

**Old Challenge.** (Salvador Segura Gomis). What is the shortest line segment fencing off prescribed area 0 **Answer** (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.

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 2^{8}/3^{7} and 1/4.

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 3^{17}/2^{31}, 2^{8}/3^{7}, and 1/4.

Joe Shipman reports that the transitional volumes for general dimensions are (1-1/n)^{1.5n(n-1)} n^{n}/n! Plugging in n = 2, 3, 4 yields 1/4, 2^{8}/3^{7}, and 3^{17}/2^{31}.

Send answers, comments, and new questions by email to Frank.Morgan@williams.edu, to be eligible for* Flatland *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 www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2000, Frank Morgan.