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Cover an equilateral triangle of side-length 1 with four disks, so that the total area of the disks as small as possible. The disks need

notbe congruent.To save you trying to improve on the proved optimum, I

[Stan]will be happy to reveal it on request. "Triangle" includes interior.Of course, you are welcome to try this for 2 disks, 3 disks, 5 disks, etc. This problem is due to Hans Melissen (Eindhoven, The Netherlands; full reference next week) and is a variation on a trick that was popular at English fairs around the turn of the century: the tourist was asked to cover a large disk with 5 copies of a smaller disk, but he could not move the disks once placed. Hmmm . . . I saw this very game at an amusement park in Montreal a couple of years ago.

[I played (and lost!) this game at a county fair in Tennessee when I was a teenager. -Jeff]

Source:Hans Melissen, Reference: Mathematics Magazine 70 (April 1997) 118-124.

PS: Here is one that is perhaps too hard for a PoW but seems quite nice:Find a triangle of perimeter 2 that cannot be covered by an equilateral triangle of side 1.This result, due to John Wetzel (ibid, p. 125), goes against the belief expressed in print (1922) that every perimeter-2 triangle can be covered by a unit equilateral.

Misc News:My book with Victor Klee, Unsolved Problems in Plane Geometry and Number Theory, has just appeared in German. Published by Birkhauser. ISBN 3-7643-5308-2.I am just back from the Mathematica Developers Conference in Champaign. We saw highlights of version 3.1 which is due out in a couple months. Much faster arithmetic, both at machine precision and high precision.

© Copyright 1997 Stan Wagon. Reproduced with permission.

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2 October 1998