How many cubes touch the face?
I was recently asked this question during an interview, and found it quite fun to solve. I will give you a hint that I was not given: There are two ways of visualizing the solution, and one may seem less obvious than the other.
Suppose you have a cube which is composed of n x n x n smaller cubes. How many cubes are exposed on the outside of the structure?
Seth Morabito
Wednesday, April 14, 2004
First visual:
Number of cubes exposed on the outside of the structure
= total number of cubes  number of cubes not exposed
= total number of cubes  number of cubes on the inside
= n^3  (n2)^3 , n >= 2
Second visual:
Volume of big cube is n^3, which implies
Length of one side is n.
Surface area of big cube is 6n^2
Blocks on the 12 edges are counted twice, subtract the duplicate count from Surface area
6n^2  12*(n2)
Blocks on the 8 corners are counted three times, subtract the 2 duplicate counts from previous total
6n^2  12*(n2)  2*8
which simplifies to 6n^2  12n + 8
which equals n^3  (n2)^3 as the reader can verify.
JHY
Wednesday, April 14, 2004
Exactly right.
I solved it the second way, and then the interviewer said "There's another way too," and gave me the first answer.
I think it's an interesting look into how people tackle problems, and I must say I'm a little embarassed I immediately saw the second, more complex way, but not the first, easier way. To most people, I think the first way is more "obvious". Apparently I like taking things apart!
Seth Morabito
Wednesday, April 14, 2004
I remember doing something like this as a GCSE maths investigation as a 15 year old or whatever.
All you need to do is think about the binomial expansion of
[(n2) + 2]^3
the terms of that give you the number of 'corner' pieces, 'edge' pieces, 'face' pieces, and inside pieces. You then go on to generalise it to ndimensional hypercubes and come up with a pretty formula :)
Matt
Sunday, April 18, 2004
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