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# painted sides of a cube

Extracts from this document...

Introduction

Painted Sides of a Cube

By Connor McInnes

Here is a 3 x 3 x 3 cube:

These 3 cubes all represent 3 x 3 x 3 cubes, the first one has the blocks shaded (pink) that will be painted 3 sides, the second cube has the blocks shaded (blue) that will have 2 sides painted, and the third cube has blocks shaded (green)

Middle

There is a definite pattern for the cube and the sides painted.
After looking at the first 4 cubes, the sides painted look as such:

In a 2 x 2 x 2 cube there are:
0 blocks with 0 sides painted.
0 blocks with 1 side painted.
0 blocks with 2 sides painted.
8 blocks with 3 sides painted.

In a 3 x 3 x 3 cube there are:
1 blocks with 0 sides painted.
6 blocks with 1 side painted.
12 blocks with 2 sides painted.
8 blocks with 3 sides painted.

In a 4 x 4 x 4 cube there are:
8 blocks with 0 sides painted.

Conclusion

For the cubes with 3 sides painted, it will always be 8. The eight painted cubes are the 8 corners of any and all cubes.

The spreadsheet below takes you through the first nine cubes with n x n x n sides.

Now, we have found these four values for our cubes with n sides:

(n-2)3, 6(n-2)2, 12(n-2), 8

Looking at the four values, one notices that the values are the products of a binomial expansion.

n3= ((n-2) +2)3

The expansion of the binomial looks as such:

n3= (n-2)3 + 6(n-2)2 + 12(n-2) + 8

This student written piece of work is one of many that can be found in our GCSE Hidden Faces and Cubes section.

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