Twenty-seven identical white cubes are assembled into a single cube, the outside of which is painted black. The cube is then disassembled and the smaller cubes thoroughly shuffled in a bag. A blindfolded man (who cannot feel the paint) reassembles the pieces into a cube. What is the probability that the outside of this cube is completely black?
2007-01-14 03:01:23 · 4 answers · asked by zeusdonna 1 in Science & Mathematics ➔ Mathematics
There are
27! (24^27)
ways of arranging the 27 small cubes in the 3x3x3 cube. There are
8! (3^8) 12! (2^12) 6! (4^6) 1! (24^1)
ways of arranging the painted small cubes in the 3x3x3 cube such that the outside is all black. Hence, the probabiliy is this number divided by the other.
It works out to 1 in:
546506281199945915
1238583897240371200
which is exceedingly small. But it's still larger than the IQ of Yahoo! Answers people that made it so that I can't write this number out in full in a single line without having it chopped off.
2007-01-14 03:35:46 · answer #1 · answered by Scythian1950 7 · 0⤊ 0⤋
Needless to say, it is going to be very tiny. So there are various kinds of cubes after painting them. The 8 corners have 3 sides painted, the 12 middle of the edges have 2 sides painted, the 6 center of the faces have one side painted, and the cube in the middle doesn't have any. Those cubes are not interchangeable except within their category. So the probability that they are put back in a place which is equivalent to their former place is 8! x 12 ! x 6! / 27!. Now each of them has to have its faces in the right position. For the middle one, there is no problem. For the ones in the middle of their faces, the probability that the painted faces shows is 1/6, for the ones at the middle of the edges, they have exactly one edge between two painted faces, and this edge has to coincide with the edge of the big cube. Since there are 12 possible edges, the factor will be 1/12. For the same kind of reason, but with the summits, the probability will be 1/8 for the corners. All in all we have
p={8! x 12 ! x 6! / 27!} (1/6)^6 x (1/8)^8 x (1/12)^12, which is indeed tiny.
2007-01-14 11:19:05 · answer #2 · answered by gianlino 7 · 0⤊ 0⤋
27 nPr 6
213127200 is how many different ways it could be arranged.
divide that by 27.
1.267 times 10^-9 %
2007-01-14 11:06:32 · answer #3 · answered by Ken F 3 · 0⤊ 0⤋
Probabiliy = number of events / total number of possibilities for those events
2007-01-14 11:06:54 · answer #4 · answered by Luis U 2 · 0⤊ 0⤋