July 21, 2012

Today I added some stones to my troll calendar yard art. I collected the stones from Shoal Creek. They're actually pieces of pyrite. Without the stones, the yearly cycle is 365 days. The stones are used to implement a leap day scheme. Here is how it works.

Each of the seven platforms has a Towers of Hanoi puzzle, which the troll solves optimally by moving one tile each night. After one tile has been moved on each of the seven platforms, the troll surveys his handiwork to see if there are any towers of 9 tiles.

Each platform has three towers which contain anywhere from 0 to 9 tiles. If the closest tower contains a stack of all nine tiles after the troll makes his move, then he does his leap day calculation:

First he checks if the platform with a stack of nine tiles has stones on it. If it doesn't have stones, the calculation is complete and the troll goes back to sleep.

Next he checks if the platform to the left of that platform has stones on it. If it does, then the calculation is complete and the troll goes back to sleep.

At this point, if the troll is still awake, then the platform in front of him has at least one stone. The troll takes one stone from the platform and places it onto the platform to the right. He then turns around to face away from the platform with the completed tower and goes to sleep.

The troll sleeps until midnight of the following night. If, when he wakes up, he can't see any platforms with stones, he turns around to face a platform with stones and then goes back to sleep. No tiles are manipulated. This simple procedure introduces just the right amount of leap days to cause the device to track the spring equinox.