Test Your Math Skills

How To Win at Ant Racing

Joe and Rita are ant friends. They love to race each other but always tie, since they crawl at the exact same speed. One day, they designed a race that one of them could actually win.

For this race, they both start at the bottom corner of a cuboid, then crawl as fast as they can to reach a crumb at the opposite corner. The measurements of their cuboids are as pictured:


If they both take the shortest possible route to reach their crumb, who will reach their crumb first? (Don’t forget they’re ants, so of course they can climb anywhere on the edges or surface of the cuboid.) Think outside the box!

Solution

The key to solving this problem is to come up with the length of each of their shortest routes. The simplest way to find this shortest route is to flatten the box. Once the box is flattened, it’s very easy to find the shortest route between the ant and their crumb. The shortest route between two points is a straight line!
In Joe’s case, flattening the box is straightforward. Since it’s a cube, it doesn’t matter which way you flatten it. If you flatten the top front fold, you’ll see the following rectangle:

Clearly, Joe’s fastest route will be a straight line to the crumb. Using the pythagorean theorem (or graph paper and a good ruler), you can determine that his shortest path is √45 inches, or 6.71 inches.

Rita’s is slightly trickier to figure out, as there are three possible ways that you could “unfold” her cuboid:

Again, the pythagorean theorem—or a very good ruler—will tell us the diagonal of the second rectangle is the shortest at √41 inches, or 6.40 inches. So Rita will achieve this route if she crawls on the left most (unseen surface) and then onto the top: