

Magnus Eisengrim Inner circle Sulla placed heads on 1064 Posts 
This product is a 3x3 tray of cubes. In the writeup the manufacturer claims that "billions of fascinating images to be created (68,719,476,736 possible combinations)." The number they give happens to be 2^36.
This seems wrong to me. Assuming each face is distinct and has no rotational symmetry and that all 9 cubes are in play, each cube can be placed in the tray in 24 ways. (6 faces x 4 orientations). The 9 cubes can be arranged in 9! ways. If I am right, we have 24x9!=8,709,120 permutations. I find combinatorial problems difficult, so I'm sure I've made a basic error somewhere. But I have a hard time believing that 2^36 is right either. Thanks John
The blooddimmed tide is loosed, and everywhere
The ceremony of innocence is drowned; The best lack all conviction, while the worst Are full of passionate intensity.Yeats 
Magnus Eisengrim Inner circle Sulla placed heads on 1064 Posts 
On sober second thought, it seems that it should that be 24^9.
Geez.
The blooddimmed tide is loosed, and everywhere
The ceremony of innocence is drowned; The best lack all conviction, while the worst Are full of passionate intensity.Yeats 
Magnus Eisengrim Inner circle Sulla placed heads on 1064 Posts 
Sorry for thinking out loud on a public forum. My second post is just stupid. Sober second though? Ha!
The blooddimmed tide is loosed, and everywhere
The ceremony of innocence is drowned; The best lack all conviction, while the worst Are full of passionate intensity.Yeats 
S2000magician Inner circle Yorba Linda, CA 3465 Posts 
Quote:
On 20110307 15:51, Magnus Eisengrim wrote: I think it's (9!) * (24^9) ÷ 4 = 239,664,780,049,121,000. I think. (Decide which cube goes in which position: 9!. Decide the orientation of each cube: 24^9. Remove duplicates obtainable by rotating the frame: 4.) 
Magnus Eisengrim Inner circle Sulla placed heads on 1064 Posts 
Thank you. That was one of the approaches that I talked myself out of last night. I don't know why, but this kind of problem leads me all sorts of selfdeceit.
I couldn't convince myself that there were only 4 rotational duplicates with your method. John
The blooddimmed tide is loosed, and everywhere
The ceremony of innocence is drowned; The best lack all conviction, while the worst Are full of passionate intensity.Yeats 
S2000magician Inner circle Yorba Linda, CA 3465 Posts 
Quote:
On 20110308 09:06, Magnus Eisengrim wrote: If you allow that you could flip the thing over, there are eight. Imagine a simpler objet d'art: only one cube. There are 6 * 4 = 24 orientations, but the same four rotational duplicates; hence, 24 ÷ 4 = 6 orientations. That is, you pick which face is on top. 
Magnus Eisengrim Inner circle Sulla placed heads on 1064 Posts 
Thank you. Somehow the problem seem simple now
John
The blooddimmed tide is loosed, and everywhere
The ceremony of innocence is drowned; The best lack all conviction, while the worst Are full of passionate intensity.Yeats 
landmark Inner circle within a triangle 5022 Posts 
The other day I was trying to figure how many unique 4x4 sudoku answers there could be. Harder than it looks. Can't say I'm right.
Click here to get Gerald Deutsch's Perverse Magic: The First Sixteen Years
All proceeds to Open Heart Magic charity. 
landmark Inner circle within a triangle 5022 Posts 
Quote:
On 20110308 00:09, S2000magician wrote: Could be wrong, but those nine cubes look identical to me, in which case you don't need the factor of 9! Also some of the faces appear to have point symmetry (i.e. 180 degrees) and 90 degree rotational symmetry so that the orientation of some cubes may be double counted for some situations in the above. I think though that it only becomes an issue for the middle cube where all the other sides are hidden.
Click here to get Gerald Deutsch's Perverse Magic: The First Sixteen Years
All proceeds to Open Heart Magic charity. 
S2000magician Inner circle Yorba Linda, CA 3465 Posts 
Quote:
On 20110319 15:46, landmark wrote: It isn't clear to me whether the cubes are identical, so I did the calculation as if they weren't and  as you point out  there is no symmetry to each cube. If the cubes are identical or there are symmetries to consider, the number of visually distinct patterns is reduced accordingly. 
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