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Problem of the Week 1183

A Long Walk on a Box

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Consider a box that spans the region from (0,0,0) to (a,b,c). Thus the x-axis edge goes from (0,0,0) to (a,0,0). Suppose an ant starts at the origin and walks on the exterior of the box so that whenever it lands on an edge it continues on the new face so as to make a 45-degree angle with that edge. To start, the ant stays horizontal, heading to (a,a,0) or (b,b,0), depending on which of a, b is larger. There are two ways to follow the 45-degree rule, and the ant chooses the direction that corresponds to following a straight line if the two sides were folded into one plane; in short, the path is (locally) a geodesic on the surface of the box: the shortest path between two points.

This figure shows the case of a (1,2,3) box.

These paths always will terminate when they hit a corner of the box. The "length" of the path is the number of linear segments. The path for (1,2,3) has 5 segments.

Find a box (a,b,c) for which the length of the path is 1183.

Note: It is not hard to prove that any integer box leads to a finite path. An amusing warmup problem is to examine the path for the box (sqrt(2), pi, e).

Note: I feel that just about anything is fair in these problems. I am now retired from Macalester College (living in Colorado) and so am not posting these for our students. So the problems can vary widely in difficulty, sophistication of techniques used, and so on. Of course, one of the joys of this project is that some of you often find simpler approaches than I used.

Source: Inspired by the recent paper: "The mystery of the sealed box" by Jim and Fred Henle, The Mathematical Intelligencer 36:2 (2014) 18-26. Details of problem 1183 were worked out by me, in collaboration with Michael Elgersma and Jim Henle.

© Copyright 2014 Stan Wagon. Reproduced with permission.

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11 June 2014