The Big Red Painted Cube Project

By

Ruth Sutton

Contents

1. Introduction  to Part 1 The Big Painted Red Cube

2. Cube Tables

3. Explanation of why there isn’t a length of 1

4. Predictions for lengths 6

5. Explanation of the 10 x 10 x 10 cube

6. Formula explanation

7. The 10 x 10 x 10 table

8. Introduction to Part 2  The Big Painted Red Cube

9. How the cuboid works for the 5 x 4 x 3

10. Another “ examples

11. Formula Explanation

12. Explanation of how a Cuboid Formula turns into the Cube Formula

1. Introduction to Part One The Big Painted Red Cube.

A stage set for a magic act has as one prop a large cube measuring 10 feet by 10 feet by 10 feet. It is made up of a lot of little yellow cubes each 1foot in length attached together, all together there was 1000 cubes.

The big cube was dipped into a pot of red paint, and was then suspended from the ceiling on a cord, unfortunately the cord broke and it came crashing down breaking up into all the little cubes which all came apart. Matt the stage hand was given the job of collecting them all up. He noticed that some were all yellow then some had 1 red face, some had 2 red faces and so on. He wondered about how that had come about. Obviously this was because only the surface area was red.

2. Cube Tables

a. 2 x 2 x 2 Cube

                              Total                                                                     8

b.

5

6

0

0

1

8

27

64

1

0

6

24

54

96

2

0

12

24

36

48

3

8

8

8

8

8

4

0

0

0

0

0

5

0

0

0

0

0

6

0

0

0

0

0

Total

8

27

64

125

216

The formula I had at this point created was as follows.

5. Explanation of the 10 x 10 x 10 cube

This was relatively easy to work out, the total was1000 cubes, because it was 10 x 10 x 10.

The zero face was accomplished by  taking the 2 adjacent cubes away then cubed .The actual sum was 10-2 = 8 x 8 x8 = 512.

The one face was worked out by subtracting the 2 corner cubes from each length of 10 (N) then multiplying by 6.  The actual sum was 10 – 2 = 8 x 8 = 64 x 6 = 384.

The two face sum was worked out by  10 -2 =8 x 12 = 96

The 3 face corners will always be 8

6. Formula Explanation

The first formula that I worked out was the total no of cubes, this was simple as to measure the volume of a cube is to cube the length that you have. This equaled to

n x n x n = n 3.

The second formula was for 0 faces, the formula for this was (n – 2) 3. The -2 is because there are 2 cubes adjacent to every side of the internal cubes , then it would be cubed minus the outer edged  each side of the cubes, as it is also a cube albeit a smaller one.

The third formula was for one red face, the formula for this was 6[ ( n-2) 2 ].

The 3 face formula is the same as a cube as no matter what the size it will always have 8 corners no matter what size.

The total formula will be Width x Height x Length to get the volume.

12. Explanation of how a Cuboids Formula turns into the Cube Formula

The formula for o faces within a cuboid is ( L-2)(w-2)(h-2), if these were all the same length as in a cube the letters would become all the same for example N it would be (N-2)3 , this is the same formula as a cube

The formula for 1 face is  2[( w -2)(L -2)] + 2[(h –2)(L-2)] + 2[(h-2)(w-2)] ,if they were all changed to the same letter for example N it would be 2[( n -2)(n-2)] + 2[(n –2)(n-2)] + 2[(n-2)(n-2)], this would equalize to be the same as a cube ,the known formula is             6 [ (n – 2) 2].

The formula for 2 faces is 4(L-2) + 4( h-2) + 4(w-2), if these were all changed to the same letter , for example N it would look like this, 4(n-2) + 4( n-2) + 4(n-2), as in the previous formulas this also equalizes to the same as the cube formula for 2 faces which is 12( n-2).

If the formula for of total cubes is lwh within a cuboid if changed to the same letter for example n this would become n3, the same as the formula for a cube