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Problem of the Week 806

The Invisible Loop

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Three red and three blue disjoint loops are drawn in the plane. A square piece of paper covers one of the loops entirely, but leaves parts of the other five loops visible. All we can see are two semicircles protruding from each of the top, bottom, and two sides of the paper.

These semicircles are all oriented so that their two open ends go straight into the piece of paper.

In each case, one of the semicircles is red, the other blue. Moreover, in each of the four cases, one of the semicircles is contained in the other.

On the bottom, left, and right, the exterior semicircle is red (and interior blue). On the top, the exterior is blue and the interior is red.

What is the color of the invisible loop?

- from V. Proizvolov, Quantum, May/June 1995.

For a nice figure, run the Mathematica code:

Needs["Graphics`Colors`"]
Show[Graphics[{AbsoluteThickness[4],
               {Blue, Circle[{0,2}, {1,1.4}, {0, Pi}]},
               {Red, Circle[{0,2}, {.5,.7}, {0, Pi}]},
               {Blue, Circle[{2,0}, {.7,.5}, {-Pi/2, Pi/2}]},
               {Red, Circle[{2,0}, {1.4,1}, {-Pi/2, Pi/2}]},
               {Red, Circle[{0,-2}, {1,1.4}, {Pi, 2 Pi}]},
               {Blue, Circle[{0,-2}, {.5,.7}, {Pi, 2 Pi}]},
               {Blue, Circle[{-2,0}, {.7,.5}, {Pi/2, 3 Pi/2}]},
               {Red, Circle[{-2,0}, {1.4,1}, {Pi/2, 3 Pi/2}]},
               {GrayLevel[0.9], 
                Polygon[{{-2,-2}, {2,-2}, {2,2}, {-2,2}}]},
               {AbsoluteThickness[1],
                Line[{{-2,-2}, {2,-2}, {2,2}, {-2,2}, {-2,-2}}]}}],
     AspectRatio->Automatic, 
     DisplayFunction -> $DisplayFunction,
     PlotRange->All];

© Copyright 1996 Stan Wagon, except for the picture, which is in the public domain. Reproduced with permission.

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The Math Forum

2 October 1998