GeekDad Puzzle of the Week Solution – Ladybug Box Paths

Reading Time: 3 minutes

This past week’s puzzle as previously posited:

One of our neighbors moved last year, but still inadvertently selects their local (to me) address for the occasional Amazon purchase. This means that every so often we package up a few boxes for them and forward them on.

ladybug_story
NOTE: Drawing definitely NOT to scale.

During a recent repackaging, one of the larger boxes (32″ x 20″ x 10″) fell onto the floor, on it’s largest face so that it was only 10″ tall. It happened to fall directly in the path of a ladybug, landing such that she was at one of the corner edges of the box at floor-level.

This week’s GeekDad Puzzle of the Week is straightforward: If the ladybug wants to get to the opposite corner of the box (the corner at floor-level), what is her best path? Is there an “overbox” path that is shorter than a simple end-run? How long is this shortest path?

The simplest route to calculate is having our dear ladybug stay on our floor, and traverse the floor-level edges of the box. By either route, they walk 32″ + 20″ = 52″, a great contender for the shortest route.

simple_endrun

Walking up the side to any point that hits the front edge of the box (and doesn’t go up top) is sub-optimal; any “height” gained during this trip makes this path longer than floor-level trip.

But what happens when we flatten the box in our minds, and look at those paths? As both the starting point and the ending point are at the intersection of two edges, there are four different possibilities:

The first option is the hypotenuse of a (10″ + 32″) by (20″ + 10″) or 42″ by 30″ right triangle. Per Pythagoras, this path is 51.614″ in length, and is shorter than the end run route.422 + 302 = 1764 + 900 = 2664, sqrt(2664) = 51.614path01
path02The second option is the hypotenuse of a (10″ + 10″ + 32″) by (20″) or 52″ by 20″ right triangle, resulting in a 55.714″ path.522 + 202 = 2704 + 400 = 3104, sqrt(3104) = 55.714
The third option is the hypotenuse of a (32″) by (10″ + 10″ + 20″) or 32″ by 40″ right triangle. This path comes out to be 51.225″ in length, our shortest so far.322 + 402 = 1024 + 1600 = 2624, sqrt(2624) = 51.225path03
path04The fourth option is the hypotenuse of a (32″ + 10″) by (10″ + 20″) or 42″ by 30″ right triangle, and like the first option comes out to be 51.614″, tying the first option, but not beating the third option.422 + 302 = 1764 + 900 = 2664, sqrt(2664) = 51.614

Congratulations to Steven Hecht who correctly found the shortest path of 51.225″. While this is just under only 1.5% shorter than the simple “stay on the floor” or end run path of 52″, such small increments in the life of a ladybug are indeed meaningful. This week’s prize of a $50 Gift Certificate from ThinkGeek will be on its way shortly.

Thanks to everyone that submitted an answer this week, and special thanks once again to ThinkGeek for providing our fabulous prizes.

Happy puzzling!

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