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Spider Fly Distance

Problem

A spider, S, is in one corner of a cuboid room, with dimensions a by b by c, and a fly, F, is in the opposite corner.

Find the shortest distance from S to F.


Solution

There are three straight line routes from S to F.

Let the distances from S to F1, F2, and F3, be d1, d2, and d3 respectively.

Using the Pythagorean Theorem we get:

d12 = (a+b)2 + c2 = a2 + b2 + c2 + 2ab
d22 = (a+c)2 + b2 = a2 + b2 + c2 + 2ac
d32 = (b+c)2 + a2 = a2 + b2 + c2 + 2bc

Without loss of generality, let us assume that a greater than or equal b greater than or equal c.

As b greater than or equal c, ab greater than or equal ac, and it follows that d1 greater than or equal d2.
Similarly, as a greater than or equal c, ab greater than or equal bc, and d1 greater than or equal d3.
And finally, as a greater than or equal b, ac greater than or equal bc, giving d2 greater than or equal d3.

Hence, d1 greater than or equal d2 greater than or equal d3 and, of the three routes, the shortest distance would be from S to F3; that is, the journey from S to the longest edge.

What is the smallest cuboid for which the shortest route is integer?
What about the smallest cuboid for which all three routes are integer?

Problem ID: 201 (10 Jan 2005)     Difficulty: 3 Star

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