Columbus Egg / The X super puzzle

The Columbus Egg is an intriguing mechanical puzzle from Japan. In the side of this egg there is a little window through which you can see the edge of 5 parallel disks. If you rotate the bottom third of the egg clockwise, some of the disks rotate. There is a sliding button on the side of the egg, which influences which disks will move when you turn the bottom part of the egg. The disks each have 10 positions, and one of these positions shows red through the window, the others white. The aim is to make every disk show red, and when you do that a weight inside the egg is unlocked. This weight then allows you to stand the egg upright.

The puzzle was patented by Morichika Hatakeyama and Koichi Minami on 25 December 1984, US 4,489,944, assigned to Tomy Kogyo Co. Inc.. My egg puzzle has a logo on the back of Toybox Corp., 1983.

The number of positions:

There are 5 disks, each with 10 orientations, giving 105 = 10000 positions, and these are all reachable and solvable.

Notation:

Hold the egg with the pointy end to the left, and the window towards you.
Call the five disks from left to right A, B, C, D, and E.
The button has three states. Call them X, Y, and Z, where X means the button is the normal state, flush with the egg surface.
The move turning the bottom of the egg a single click is denoted by X1, Y1, or Z1, depending on the state of the button.
Similarly, turning more than one click is denoted by the button letter followed by the number of clicks.

Solution:

1. Set button to X, then twist until A is red.
2. Set button to Z, then twist until B is red.
3. Set button to Y, then twist until C is red.
4. If D is not red, then do X3 Z10 X6. Repeat until D is red.
5. If E is not red, then do X3 Y10 X6. Repeat until E is red.

Note: In step d or e, you can usually see whether the disk you are solving is one click past its correct position. If that is the case, then a single application of the move sequence will solve the disk. If it is not one click past its solved position, then you need to apply the sequences at least twice, and then there is a shortcut. When applying either sequence twice, you may replace each X6 by an X1 instead. So:

1. If D is one click past its correct position, then do X3 Z10 X6, otherwise do X3 Z10 X4 Z10 X1. Repeat until D is red.
2. If E is one click past its correct position, then do X3 Y10 X6, otherwise do X3 Y10 X4 Y10 X1. Repeat until E is red.

Explanation:

The disks that turn when a move is made are usually as follows:
 Button Moved Blocked X: A, D, E B, C Y: C, E A, B, D Z: B, D, E A, C

If this were the whole story, then there would only be 103=1000 positions, because the state of discs A,B,C would determine the state of D and E. If you look carefully at the disks, you will notice that disk E has a smaller radius, and that the other four disks each have two sections where they are narrower. When you make a move, all the disks move except for those that the button blocks from moving. If however a disk has one of its narrow sections at the button, then it will not be blocked. This means that if one of the disks happens to be 3 or 8 clicks past its correct orientation, then it will be moved regardless of the the state of the button.

I wrote a computer program to calculate God's Algorithm for this puzzle. The results show that it can be solved in at most 15 turns, or 52 clicks.

Turns Clicks
Moves   Positions
01
127
2413
32438
44061
56628
67384
710390
810084
913706
1013480
1115454
128768
134234
142522
15410
Moves   Positions    Moves   Positions    Moves   Positions    Moves   Positions
0 1131613263334392884
1 3141784273507402440
2 16152145283635411848
3 52162417293582421430
4 86172567303683431080
5 13918269731395044 912
6 20719283132410645 526
7 32820301233403846 314
8 47421309434396847 208
9 61922307335392048 130
10 84023321736372849 84
11109924325037339050 24
12137325327038306451 6
52 2

An interesting aspect of this puzzle is that a move applied to two different positions may both result in the same position. This could have made it possible to find a move sequence that would always solve the puzzle. Regardless of its original position, after applying such a move sequence it would be solved. Unfortunately, the mechanism cannot distinguish between one disk orientation and the disk orientation a half turn away. The best you could do would be to find a sequence that reduces the number of remaining positions from 105=10000 to 25=32.
For example, the move sequence Y1 X9 Y1 X9 Y1 X9 Y1 X1 will always solve disk A or leave it a half turn away from solved. You can follow this by similar sequences for each of the other disks.