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Problem of the Week 847

Cover the Triangle

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 not be 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.

The Math Forum

2 October 1998