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What is the probability that a block of cheese...?
So, this question was given to me years ago when I was still at school. I am still not sure how to solve it so here goes...
You have a block of cheese in a tight fitting perspex container.You paint the outside of the cheese black and then divide it into 27 equal cubes (3x3x3) then randomize the cubes in a bag. You drop them back into the container directly from the bag. What is the probability that they will land so that all 6 faces of the block of cheese display entirely black faces?
The little cubes lands in the container assembled in the form of the larger 3x3x3 cube. So we are looking for the number of ways that all the pieces land so that all visible faces are black. This number divided by *all* the possible ways that the little cubes can assemble to form the 3x3x3 cube should give the desired probability. We need to include positions of cubes in the container *and* each little cube's orientation in this. Each little cube may occupy 27 different positions and have 24 (4x6) different orientations once in position.
Meant to say "The little cubes land" not "lands". Anyway Chris, your answer seems pretty good to me. That number (27!)(24)^27 is what *you* get for your denominator if you do a little jiggery pokery with your numbers in your final probability. i.e. if you write your correct orientation probabilities (1/8)^8 = (3/24)^8, (1/12)^12 = (2/24)^12, (1/6)^6 = (4/24)^24 and 1 = (24/24)^1. I'll leave the question open a bit longer just in case anyone else is interested.
2 Answers
- Anonymous9 years ago
Well - - interesting - I presume you have a cube of cheese, you paint it black on the outside, then divide it up 3x3x3 into smaller cubes (27). The 8 corner cubes will have 3 black faces and 3 white faces, the 12 centre-edge cubes will have 2 black faces (4 white), the 6 centre-face cubes will have one black face (5 white), and one internal central cube will be unpainted (no black, 6 white).
Now - what do you mean by "land" - do the small cubes just land randomly on a flat surface (in which case is it the bottom face next to the perspex that is displayed?) or are they reassembled into a large cube in the perspex box. The probabilities can probably be worked out depending on this - but it looks like the question is inadequately defined. If it is just small cubes face down on a flat surface then it can never be achieved because one small (inner) cube has no black paint (that's maybe the answer they wanted).
- Anonymous9 years ago
Well when cut up each chunk has 6 faces looked at in layers the faces that are black are 21, 12, 21 respectively from a total of 27 cubes with 6 faces each meaning the painted faces are 54/162