Maths Investigation -Painted Cubes

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Maths Investigation – 

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 apparent, which reinforced my previous hypothesis.

Results Table

Patterns and Formulas

From my results, I tried to find patterns, which would enable me to find a formula.  I looked closely for patterns throughout the numbers.  I found that to find the number of cubes with three faces painted, it was always a constant number (8).

I then looked for a pattern in my results to find how many cubes had two faces painted.  I wrote the numbers down, to see if they had a pattern between them.

0       12       24       36       48

 

                                 12      12       12      12

As shown above, I did find a pattern in the numbers. I found the numbers increased by 12.  I assumed the formula was ‘12n’.  I tried out the formula by using ‘n = 2’ and multiplied it by 12, totalling up to 24.  This was incorrect as I needed the answer 0.  This made me realise I would have to minus a number in order to get my formula right.

I tried subtracting 2 from ‘n’ to find it was the correct method.  I tested it on 2 x 2 x 2.

n = 2                2 – 2 = 0  

                12 x 0 = 0

This was correct.  I now went on to see how many cubes had one face painted.  Again, I looked for patterns between the numbers.

    0        6       24       54       96

        6        18        30         42

   12            12             12

This time I found that there was a second difference, which was constant.  This told me that my formula contained ‘n²’.

Using my previous knowledge of patterns, I knew I had to half the 12 to make it work with the n².  Therefore this became ‘6n²’.

Taking 2 as my ‘n’, I tested this formula to see if it was correct.

n = 2

6 x (2²) = 24

The answer was supposed to be 0, which meant my formula was yet again wrong.  As this was the same problem I had before, (the final answer was too big), I knew I had to subtract a number.  I looked at the formula I was previously working on, and noticed I used (n – 2).  I decided to use it in my present formula to see if it would help or not.

Join now!

6(2 – 2)²        

6(4²) = 24

Now my formula was complete and correct.  Finally I moved onto cubes which had no faces painted.

 0         1          8          27          64

1        7        19        37

6        12        18

6        6

This time I found that the third difference was constant.  This meant the formula contained ‘n³’.  I noticed the original numbers were all cube numbers, so I tried the formula n³ by using ‘n = 2’.

n = 2

2 x 2 x 2 = 8

This was wrong.  The answer I was looking for was ...

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