Jaap's Puzzle Page

Marusenko Sphere

The Marusenko Sphere is a ball-shaped puzzle which is an extension of the 2×2×2 puzzle balls. The ball is divided into eight octants by three perpendicular cuts and moves just like a 2×2×2 Rubik's Cube. Unlike such puzzle balls, this puzzle also has 24 pieces arranged in 6 circular faces. Each octant has three such pieces, in three different faces. This puzzle was invented by Alexandr Marusenko, and has the patent WO2007/028837, published on 15 March 2007. The pictures above show a prototype version, which has painted colours, and a two-colour production version.

The number of positions: The 8 octant pieces have 3 orientations each, giving a maximum of 8!·38 positions. There are also 24 face segments which can be arranged in 24! ways, resulting in a total of 24!·8!·38. Just as with a 2·2·2 cube, this limit is not reached because: The total twist of the octant pieces is fixed (3)

the orientation of the puzzle does not matter (24) The 8 octant pieces have 3 orientations each, giving a maximum of 8!·3positions. There are also 24 face segments which can be arranged in 24! ways, resulting in a total of 24!·8!·3. Just as with a 2·2·2 cube, this limit is not reached because: This leaves 24!·8!·37/24 = 2,279,626,699,712,199,018,518,937,600,000 or approximately 2.2·1030 positions.

That number is based on the assumption that all pieces are distinct and that orientations of the octant pieces are distinguishable. This is not the case on most colour schemes. There are 5 types of colour scheme available, labelled with difficulty levels 1 to 5. The lower levels have more identical pieces, and so fewer possible positions. To get the exact number of distinct positions you then have to take into account that some positions have symmetry by applying Burnside's Lemma. Level Picture Name Octants Segments Approximate Exact 1 Black and White 4+4 12+12 8!·24! / (24·4!2·12!2) 7,889,304 2 Circular 8 4+4+4+4+4+4 8!·24! / (24·8!·4!6) 135,277,941,853,080 3 Flag 4+4 4+8+8+4 8!·24! / (24·4!4·8!2) 1,932,542,454,240 4 Classic 2+2+2+2 6+6+6+6 8!·24! / (24·2!4·6!4) 242,418,070,100,640 5 Triangular 1+1+1+1+1+1+1+1 3+3+3+3+3+3+3+3 8!·24! / (24·3!8) 620,590,250,933,452,800,000