Spider Fly Distance
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.
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?