Calculations

Following is an explanation of all the calculations involved in the Troll Calendar.

Item	Calculation
Number of moves to move a tower one position	2⁹ - 1 = 511 moves
Number of moves to move a tower three positions (back to original position)	3 * (2⁹ - 1) = 3 * 511 = 1533 moves
Offset from one platform to the next	1533 / 7 = 219 moves
Intervals for all 7 platforms: 0, +219, +438, +657, etc. mod 511	0, 219, 438, 146, 365, 73, 292
Quint (smallest interval between consecutive tower completions from previous row)	1533 moves / (7 platforms * 3 towers/platform) = 73 moves per tower
Year (unadjusted for leap days) is 5 quints	73 * 5 = 365 moves
Year (after first adjustment where troll rests)	365 * 1534 / 1533 = 365.2381 days (365 moves)
Normal number of days between stone moves	1533 + 1 = 1534 days
Days between stone moves (after the last stone is moved off a platform)	1534 - 219 = 1315 days
Days to move all 8 stones over by one platform	7 * 1534 + 1315 = 12053 days (33 years)
Days to move all 8 stones back to original position (seven platforms)	7 * 12053 = 84371 days (231 years)
Year (after second adjustment)	365 * (7 * 1534 + (1534 - 219)) / (7 * 1533 + 1314) = 365.2424 days or simply 12053/33 = 365 8/33 days
29 position inner loop alternating with 30 position outer loop (basic lunar period)	(29 + 30) / 2 = 29.5 moves per lunation
Lunation length in days adjusted by leap year adjustments	29.5 * (12053 / 33) / 365 = 29.51959 days per lunation
Length of Troll Calendar Metonic Cycle	73 moves/quint * 19 quints = 1387 moves
Lunar moves in one Troll Calendar Metonic Cycle.	30 * 24 + 29 * 23 = 1387 moves
Days in lunar Metonic Cycle	1387 * (12053 / 33) / 365 = 1387.9212 days
Lunations in Troll Calendar Metonic Cycle	47 lunations
Days in Troll Calendar lunation before long term correction	1387.9212 / (47) = 29.530239 days
Lunar correction by 1/30 lunation once per complete circuit of stones	1/30 lunation every 231 years
Days in Troll Calendar lunation (after 231 year adjustment)	29.530239 * (84371 + 29.530239/30) / (84371) = 29.530583 days

A basic Towers of Hanoi puzzle made up of n tiles takes 2ⁿ-1 moves to move all tiles from one tower to another tower. The troll calendar uses n=9 tiles per platform, so the number of moves to move all tiles from one tower to another is 2⁹-1 = 511 moves.

There are three towers on each platform, so the total number of moves for one cycle on a platform is 511*3 = 1533 moves.

For the leap day scheme, the troll looks at the same platform once for each of 8 stones, and when it has a complete stack of 9 tiles on the closest tower, he turns around, thus spending a day moving no tiles. So 1533 moves of tiles is equated to 1534 days. That gives an average year length of 365 * 1534/1533 = ~365.238 days.

After the eighth stone is moved, the troll is now looking at a different platform. This next platform is out of phase with respect to the previous one by 219 days. So there are 7 cycles of 1533 moves in 1534 days, and one cycle of 1533-219= 1314 moves in 1534-219 = 1315 days. That's a total of 7*1533+1314=12045 moves in 7*1534+1315=12053 days, giving an average year length of 365*12053/12045 = exactly 365 8/33 days. Another way to look at it is that 12053 days are used for a complete cycle of 33 years, producing a mean year length of 12053/33 = 365 8/33 days.

According to the wikipedia article on Tropical Year, the length of the northern spring equinox year as of the year 2000 was 365.242374 days. So for the troll's calculation is in error by about one day in 20,000 years. Of course by that time, the equinox year length will have changed.

On the inside of the device are two rings of white circles. These are used to calculate the phase of the moon. The outer ring has 30 circles. The inner ring has 29 circles. They are connected at one circle (let's call it the intersection). The basic pattern is for the moon icon to alternate going around the inner and outer rings, changing rings at the intersection. This gives a basic length of 29.5 days for the lunation. This pattern of alternating 30 and 29 moves alternates for each Troll Calendar Metonic cycle.

A Troll Calendar Metonic cycle consists of 19 quints = 1387 moves = 47 lunations = 19/5 years. The end/beginning of a Metonic cycle can be recognized when the lunar marker is on the intersection when one of the towers is completed. If the marker is on this spot when a tower is completed, it always continues on the 30 move ring. This happens once in 47 lunations, so the average length of a lunation is (30 * 24 + 29 * 23) / 47 = 29.510638298 moves. The ratio of days to moves is 12053/12045, so the mean lunation length is (30 * 24 + 29 * 23) * 12053 / (47 * 12045) = 29.530238556 days.

This is still a little short, so another correction is used by the Troll Calendar. Fortuitously, there's a beautiful adjustment that corrects this, described below.

The correction for the year is performed by the movement of stones, as described above. The time between movement of stones is 1533 moves = 1534 days for 7 of the 8 stones. For the other stone, the period needs to be adjusted by the offset between platforms, which is 219 moves, or 1533 - 219 = 1314 moves = 1534 - 219 = 1315 days. So it takes 7 * 1533 + 1314 moves = 12,045 moves = 7 * 1534 + 1315 days = 12, 053 days for all stones to move from one platform to the next. This is exactly 33 years. Since there are 7 platforms, it takes 33 * 7 = 231 years for the stones to make one complete circuit around the Troll Calendar. This happens to be exactly the length of time needed for an accurate adjustment.

To facilitate this adjustment, another marker is moved around on the lunar circles. Since the adjustment being made is 1/30 lunation, it always moves along the outer ring. It is moved only when the completed tower signaling the start of a Metonic cycle is on the closest position of the north platform, and all the stones are also on that same platform. The resulting mean lunation length is L * (84371 + L/30) / 84371, where L = (30 * 24 + 29 * 23) * 12053 / (47 * 12045). Evaluating, we get 29.530583 days.

To the left is an illustration of the center of the device showing the lunar circles. North is to the right in the illustration. The tiles on the north platform are orthogonal to the platform. Additionally, the intersection of the 29 position ring and 30 position ring is on the north side of the troll.

The black circle is a reference circle. It is always moved along the 30 position ring. It is moved when all nine tiles are on the closest stack to the troll on the north platform when all 8 stones are also on that platform.

The red circle moves one position clockwise each move. When it arrives at the intersection on the north end, if it had been on the 29 position ring it switches to the 30 position ring, except at the start of a new Metonic cycle. A Metonic cycle happens when the red marker is at the intersection when one of the towers is completed.

The phase of the moon is indicated by the relative positions of the black and red markers. If they are on opposite sides of the troll, it is a full moon. If they are on the same side, it is a new moon. In this picture, the moon is waxing and nearly full.

Easter calculation

The app can be used to calculate the date of Easter. It's not always correct, but it usually is. First, a little background of the calculation of Easter is in order. Easter is the Sunday following the Paschal full moon. The Paschal full moon is intended to be the full moon on or after the spring equinox. Without going into the nuances of this definition, let's proceed.

First, we find the spring equinox. That is simply the only configuration in which all the tiles are on a single stack on one of the seven platforms during the month of March. Here it is for Gregorian year 2014.

The Paschal full moon is the full moon that is on or after this date. There is a full moon when the red dot is opposite the black dot. Swipe eastward (from left to right if north is up) until the red and black dots are on opposite sides.

Finally, we need to find the first Sunday after this date. The day of the week in the app is indicated by a red dot on the far end of each platform. Sunday occurs when the red dot is on the north platform. So we just pan eastward until the red dot is on the north platform.