# Problem of the Week Problem E and Solution Bug on the Outside

## Problem

A ladybug walks on the surface of the $$2$$ by $$3$$ by $$12$$ rectangular prism shown. The ladybug wishes to travel from $$P$$ to $$Q$$.

What is the length of the shortest path from $$P$$ to $$Q$$ that the ladybug could take?

## Solution

We fold out the sides of the prism so that they are laying on the same plane as the top of the prism. The diagram below shows the two-dimensional shape that results. As a result of folding out the sides, vertex P of the prism is a vertex of two different faces in the diagram. We call the second instance $$P^{\prime}$$. We let $$X$$ be the vertex adjacent to $$P$$ along the side of length $$3$$, and we let $$Y$$ be the vertex adjacent to $$P^{\prime}$$ along the side of length $$12$$.

The shortest distance for the ladybug to travel is in a straight line from $$P$$ to $$Q$$ or from $$P^{\prime}$$ to $$Q$$.

$$PQ$$ is the hypotenuse of right-angled triangle $$PXQ$$. Using the Pythagorean Theorem, $PQ^2=PX^2+XQ^2=3^2+14^2=205$ Thus, $$PQ=\sqrt{205}\approx 14.3$$, since $$PQ>0$$.

$$P^{\prime}Q$$ is the hypotenuse of right-angled triangle $$PYQ$$. Using the Pythagorean Theorem, $(P^{\prime}Q)^2=(P^{\prime}Y)^2+YQ^2=12^2+5^2=169$ Thus, $$P^{\prime}Q=13$$, since $$P^{\prime}Q>0$$.

Since $$P^{\prime}Q<PQ$$, the shortest distance for the ladybug to travel is $$13$$ units on the surface of the block in a straight line from $$P^{\prime}$$ to $$Q$$.