# Orbo / Magic Rainbow Ball

The Magic Rainbow Ball, which is sold by Popular Playthings under the name Orbo, is a simple puzzle, both simple to solve and simple in design. It consists of a hollow sphere with 12 holes in it, arranged symmetrically as if on the vertices of an icosahedron. There are 11 balls inside it, slightly larger than the holes. These balls stick out of the holes of the sphere because at the centre there is a foam ball that pushes these balls outwards. One of the holes has no ball, and you can push any of the balls in the adjacent holes inwards until pops up at the previously empty hole. This essentially makes this a sliding piece puzzle played on the vertices of an icosahedron. The balls and the holes are coloured, so that each ball has a unique hole of the same colour where it belongs.

## The number of positions:

There are 11 distinct balls and an empty hole. This gives a total of 12! = 479,001,600 positions. Unlike the classic Fifteen sliding puzzle, all these positions can be reached.

I have calculated God's Algorithm for this puzzle. It turns out that it can be solved in at most 34 moves.

DepthPositions
01
15
220
375
4280
51,045
63,737
712,600
839,922
9119,100
10333,797
11878,761
122,164,594
134,947,085
1410,403,757
1519,943,386
1634,455,569
1752,938,896
DepthPositions
1871,008,889
1981,235,453
2077,288,040
2159,666,712
2236,624,633
2317,656,078
246,664,913
251,989,654
26488,194
27106,934
2823,566
294,910
30867
31101
3220
335
341
Total479,001,600

The single antipodal position has every ball exactly opposite to its hole. As all eleven balls are three moves away from where they belong and every move involves only one ball, it is clear that this position takes at least 33 moves. The table above shows it needs one extra move to solve.

## Solution:

This puzzle is quite simple, so there is no need for a detailed solution. First find the hole without a colour, which is where the hole eventually belongs. Then start solving the puzzle beginning with the hole on the opposite side of the puzzle. After that hole, solve the five adjacent ones. Finally work your way around the five holes next to the colourless one. You will never need to disturb any of the previously solved balls (though you may need to disturb holes that you have not explicitly solved yourself).