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Well, I appreciate your help so far, but my grounding in mathematics is from an engineering perspective rather than a pure maths background. I *didn't* know that all numbers could be expressed as a product of primes, but I can see that now.
I have tried (with 12, 16, 10, 7, 9, 5, 49, & 20 for 'n') deriving the number of prime factors, but they don't add up to the number of cuboid arrangements that I have arrived at empirically.
12 [2, 2, & 3] - 3 prime factors - but 4 cuboid shapes (3x2x2, 1x1x12, 1x2x6, 1x3x4)
16 [2, 2, 2,& 2] - 4 prime factors - and 4 cuboid shapes (4x4x1, 1x1x16, 1x2x8, 2x2x4)
10 [2,5] - 2 prime factors - and 2 cuboid shapes (1x1x10, 1x2x5)
etc etc, but it doesn't work for 12 or 64 for example.
I must be missing something (other than a good mathematics background!)
J
wow! That was quick... many thanks
OK, so I am close with the prime factor thing....I'm not sure why this is the case, but I was thinking about factors in general, hit the web and saw the 'reverse division' method for determining prime factors and liked the numbers it was coming up with, so tried it. It worked for most cases, except the one with 6 cubes. I don't have a lot of hair on my head in the first place, so tearing more out in the pursuit of this was not doing my image any good - hence my question here!!
To address your combinatorics point, I am not too concerned about the complete set of possible arrangements, just shape types, so 1 X 2 X 3 is the same as 3 X 2 X 1 as far as I am concerned...
Is there a logical reason why prime factors are the key here, or is it just one of those things?
J
My daughter has been set a challenge in 3D!!
Given a number of cubes, how many different sized cuboid shapes can be made. How is this related to the number of cubes used?
OK, so for example given 2 cubes, there is only one arrangement: 1 X 2
with 6 cubes, there can be: 1 X 1 X 6, 2 X 3 X 1
with 49 cubes, there can be: 7 X 7 X 1, 1 X 1 X 49
with 7 cubes, there is only: 1 X 1 X 7
but with 20 cubes, there can be: 1 X 1 X 20, 1 X 2 X 10, 1 X 4 X 5, 2 X 2 X 5
I am really struggling with a relationship between the number of cubes and the number of cuboid arrangements - can anyone help?
It looks to be something to do with factors, or even prime factors, but it doesn't work for all cases.
Thanks in advance
John
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