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The Spider and the Fly

A hungry spider is sitting on the floor in the corner of a rectangular room. She sees a tasty fly on the ceiling in the far corner. What is the shortest route from the spider to the fly?

Suppose the room is a metres long, b metres wide and c metres high. To achieve the shortest route, the spider must keep to a straight line on the floor, ceiling or walls, changing direction only when moving across a floor-wall, wall-ceiling or wall-wall boundary. The spider has several different options: across the floor then up the right wall, up the front wall then up the right wall, etc.

To find the shortest route, think of the room as a box unfolded in different ways. The spider's route must then be a straight line, whose length is given by the Theorem of Pythagoras.

For example, if the spider chooses the route across the floor and up the right wall, the length of the route will be

sqrt((a+b)^2 + c^2)

What are the lengths of the other routes? Which is the shortest? This problem could form the basis of an Expo project, combining elementary but interesting mathematics with the opportunity of illustrating the theory with three-dimensional models, for example, elastic bands stretched around blocks of wood. The problem can be generalised to considering other initial positions for the spider and the fly.

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