GeekDad Puzzle of the Week Solution: Planetary Paths

This past week's puzzle was an interesting study in math and human nature. While we did not have nearly as many solutions offered as we do most weeks,* all* of the solutions offered were correct!

Here is the puzzle as presented:

Just past UDFj-39546284, one of the most distant galaxies discovered to date by the Hubble telescope, is a small, flat/disc universe with six planets that ring an asteroid field. Over time, robot traders mining this section of space have hewn each of the fifteen direct trading routes from each planet directly to every other through the field, creating a total of thirteen intersections throughout the field.

The law in this part of space is not only strict, but also strictly geared to robot pilots.

• A ship can only travel along a route that is getting it closer to its final destination.
  • A ship can only revisit a complete route from one planet to another after exhausting every other possible complete route between those two planets.

For example, there are exactly five (5) different paths from any planet to one of the planets immediately adjacent to it. (Go ahead and check, I'll wait.) The set of robot pilots that fly between those two planets only visit each complete route every five trips. In this part of space, each complete path takes the same amount of time to traverse – one Earth day. It happens that these "adjacent planet" pilots complete an integer number of sets of trips in their Local year — that is, the Local year is a multiple of five earth days.

If all sets of robot pilots in this part of space complete integer numbers of sets of trips during their Local year, how long is their Local year in Earth days?

Perhaps a handy dandy diagram would have solicited more solutions. If we look at the six planets the vertices of a regular hexagon, and draw each of the paths from every planet to every planet, it would look something like this:

If you were to measure the distance from each vertex to the target planet, using either geometric formulas or a ruler, you could easily see which points were or were not getting a robot pilot closer to their final destination.

Pilots traveling from any given planet to an adjacent neighbor have 5 different routes to take. Pilots traveling from any given planet to a planet "two doors down" have 41 different routes to take. Pilots traveling from any given planet to the planet directly across from them have a whopping 121 different routes to take.

As all pilots happen to complete any given path in an Earth day, and as all pilots complete an integer number of sets of trips within the Local year, the Local year is a common multiple of 5, 41, and 121. The lowest common multiple of these numbers is also their product, 5x41x121 = 24,805 Earth days (or just shy of 68 Earth years!)

Congratulations to readingipod for submitting one of the correct solutions, and for winning the $50 ThinkGeek Gift Certificate. For the rest of us, we can use GEEKDAD72JL to get $10 off a ThinkGeekorder of $50 or more.