August 2, 2001
Old Challenge (inspired by a walk across the campus of the University of California at Berkeley). If two houses on the side of a hill are at the same height, is there a path between them at that height, without going uphill or downhill?
Answer. Almost all respondents agree that the answer is yes, but actually the answer is no. Imagine a ring of hills surrounding a valley. If two houses are on opposite sides of a hill, at a height below the height of the ridges connected to adjacent hills, then there is no constant-height path between them. It is necessary to assume that the houses are on the same hill and in the same valley. Even then, you might have to transverse the face of a cliff with crampons and continue on a fractal path of infinite length.
New Challenge (William K. Scherer). What is the best way to cut a square pizza into three pieces, each with the same amount of pizza and with the same amount of crust?
("I came upon this whilst deciding how to divide the sole remaining Klondike bar in the freezer between myself, my wife, and my 2.5 year old son. (Un)Fortunately, my wife had already eaten the last one, so I never had to solve it! But it intrigued me just the same. I have no solution yet...")
A Mathematician at Heaven's Gate: a play by Frank Morgan, continued from last column. Our mathematician is enjoying his new life in heaven.
His house is a four-dimensional cube or tesseract, just as in Heinlein's wonderful story, "And he built a crooked house," so that all rooms are contiguous in a most surprising and convenient way. A his desk there is a huge number of surfaces, shelves, and drawers within easy reach, so that he has everything close at hand. Heaven is full of wonderful sights and sounds, but at any time he can draw the curtains and somehow block out the sounds as well as the sights. He just loves working at his desk.
Meals and rest are precisely on time. There is never any desire to eat early out of boredom, or late out of having to finish some task. Meals are modest and simple but somehow more satisfying than earthly feasts.
. . . [Our mathematician apparently found utter contentment, for we have received no further reports.]
Copyright 2001, Frank Morgan.
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.