Math Investigation of Painted Cubes
Introduction
============
I was given a brief to investigate the number of faces on a cube,
which measured 20 small cubes by 20 small cubes by 20 small cubes (20
x 20 x 20)
To do this, I had to imagine that there was a very large cube, which
had had its outer surface painted red. When it was dry, the large cube
was cut up into the smaller cubes, all 8000 of them. From there, I had
to answer the question, 'How many of the small cubes will have no red
faces, one red face, two red faces, and three faces?'
From this, I hope to find a formula to work out the number of
different faces on a cube sized 'n x n x n'.
Solving the Problem
-------------------
To solve this problem, I built different sized cubes (2 x 2 x 2, 3 x 3
x 3, 4 x 4 x 4, 5 x 5 x 5, 6 x 6 x 6, 7 x 7 x 7, 8 x 8 x 8, 9 x 9 x 9)
using multi-links.
I started by building a cube sized '2 x 2 x 2'. As I looked at the
cube, I noticed that all of them had three faces. I then went onto a
'3 x 3 x 3' cube. As I observed the cube, I saw that the corners all
had three faces, the edges had two, and the faces had one. I looked
into this matter to see if this was true…
As I went further into the investigation, I found this was true. This
made it much easier for me to count the cubes, and be more systematic.
Now I could carry on building the cubes, and be more confident about
not missing any out.
Whilst building the cubes, I also drew them and decided to colour code
the different faces (Red = Three faces, Green = Two faces, Blue = One
face). As I built and drew more and more cubes, it became much more
6x6x6 cube and see if I can find a pattern. When I have found a
this solution we had to weigh out 5g of KHP, which is the amount needed to
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