## 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 b c.

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

Hence, d1 d2 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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