# Where does 24 comes from ?



## deadalnix (Mar 14, 2009)

When we consider the numbers of emplacement for one type of piece, assuming that the same place with a different orientation is another emplacement, we find 24.

For edges, corners and centers.

Even on bigger cubes, wing, or any type of center have 24 emplacements.

24 is less probable than a multiple of 3 isn't it  ?


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## Kenneth (Mar 14, 2009)

6 sides, 4 orientations for each...


To make it clearer, Pyraminx has got 4 sides with 3 orientations so each piece can sit in 12 positions. For Megaminx it is 12*5 = 60.


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## deadalnix (Mar 14, 2009)

I didn't get something. Your proposal explain the number only for centerpieces.

Or maybe I didn't get something obvious ?


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## KubeKid73 (Mar 14, 2009)

There's 4 rotations of each 6 centers. 4x6=24.


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## qqwref (Mar 14, 2009)

The reason is simply that there are 24 ways to orient a cube. This can be expressed in many ways - 6*4, 8*3, 12*2. There are 6 centers with 4 orientations each because there are 4 different cube orientations which put each center on top. There are 8 corners with 3 orientations each because every orientation of the cube puts a different sticker on the top side of the URF corners (so 8*3 = 24 possibilities), and there are 12 edges with 2 orientations each because every orientation of the cube puts a different sticker on the top side of the UF edge (so 12*2 = 24 possibilities). There always have to be 24 possibilities because of the 24 different orientations of the cube.

There are similar numbers for other shapes of puzzle, too. You can orient a tetrahedron (Pyraminx, for example) in 12 ways, an octahedron in 24 ways, and an icosahedron or dodecahedron (Megaminx) in 60 ways. Puzzles like the Square-1 and Domino have only 8 orientations because the top side must always be on the top or bottom, but the same thing applies - the square-1 has 8 corners with 1 orientation each.


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## krazedkat (Mar 17, 2009)

Incorrect. It'd be 8*3 no matter what. There are 8 corners. Each with 3 perms...


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