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## Problem of the Week

Spring 96 Archive || MacPOW Home || Math Forum POWs || Search MacPOW

Next year, my seventh at Macalester, I am on sabbatical and the problems will be handled by my colleague Tom Halverson (halverson@macalester.edu).

We are in the process of changing our node name from macalstr.edu to macalester.edu. But it should not be necessary for you to resubscribe (unless you yourself are from Macalester!) Our old address will work for at least one year in case problems arise with new address.

As usual, both Tom and I like to hear about interesting, elementary mathematical problems for possible use.

I occasionally run into problems that are either too difficult or require a little too much background for use as a Problem of the Week, so I will probably never use them. Here are a few to ease your transition to a problem-free summer. These two both appear in our forthcoming book (available in August) that consists of 190 PoW-type problems (and solutions, variations, references, etc), almost all of which have actually been used at Macalester between 1968 and 1995. Title is: Which Way Did the Bicycle Go? by Joe Konhauser ((deceased), Dan Velleman (Amherst C.), and Stan Wagon, Dolciani Series, Math. Association of America, 800-331-1622).

Here is a favorite: Is there a solution that does not use calculus, and perhaps does not yield the areas exactly, but with airtight logic as to which is larger? I do not know of a calculus-free solution.

Alice and Bob own roughly rectangular pieces of land on the planet Earth, which is assumed to be a perfect sphere of radius 3950 miles. Alice's land is bounded by four fences, two of which run in an exact north- south direction and two of which run in an exact east- west direction. Her north-south fences are exactly 10 miles long; her east-west fences are exactly 20 miles long. Bob's land is similarly bounded by four fences, but his north-south fences are 20 miles long and his east-west fences are 10 miles long. Whose plot of land has the greater area?
---L. R. Ford
A figure eight is a curve in the plane obtained from the basic "8" shape by any combination of translation, rotation, expansion, or shrinking; the lines forming the 8s are assumed to have no thickness. Is it possible to fit uncountably many disjoint figure 8s into the plane?
---Folklore
Show that, excluding 1s, any two entries in the same row of Pascal's triangle have a common factor.
---P. Erdös & G. Szekeres