mathschallenge.net

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 $d$₁, $d$₂, and $d$₃ respectively.

Using the Pythagorean Theorem we get:

$d$₁² = ($a$+$b$)² + $c$² = $a$² + $b$² + $c$² + 2$ab$
$d$₂² = ($a$+$c$)² + $b$² = $a$² + $b$² + $c$² + 2$ac$
$d$₃² = ($b$+$c$)² + $a$² = $a$² + $b$² + $c$² + 2$bc$

Without loss of generality, let us assume that $a$ $b$ $c$.

As $b$ $c$, $ab$ $ac$, and it follows that $d$₁ $d$₂.
Similarly, as $a$ $c$, $ab$ $bc$, and $d$₁ $d$₃.
And finally, as $a$ $b$, $ac$ $bc$, giving $d$₂ $d$₃.

Hence, $d$₁ $d$₂ $d$₃ 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?