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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?