• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Investigating 'Painted cubes'.

Extracts from this document...



I am investigating about ‘Painted cubes’. I have been given the following task. ‘Imagine that there is a very large cube which measures 20 by 20 by 20 (20 x 20 x 20) small cubes. The outer surface of the cube is painted red. When it is cut up into smaller cubes there are 8000 small cubes altogether.’

The ultimate aim is to find how many small cubes have 0 faces, 1 face, 2 faces, 3 faces, 4 faces, 5 faces and 6 faces that can be seen. I have to also work out formulas for the nth term, so I can work out how many cubes have 0 faces, 1 face, 2 faces, etc for any size cube.


I counted the number of

...read more.



7 x 7 x 7






8 x 8 x 8






9 x 9 x 9






10 x 10 x 10







After filling in the table with the data I started looking for patterns so that I could work out formulas. I had to investigate and find formulas that would work out how many cubes had different amount of faces, e.g. 0 faces, 1 face, 2 faces, 3 faces.

I noticed that the columns for 3 faces had a pattern except 1 x 1 x 1. All of the cubes had 8 cubes with 3 different faces painted. All of these 8 are the vertices of the cube and so had three faces painted.

For 2 faces I noticed that each one got higher by 12, so it was + 12. This told me that somewhere in the formula that there would be +12.

For 1 face I knew that there would be a 6 in the formula, because all the numbers were multiples of 6. Whilst for 0 faces I could not see a pattern straight away, because there was a big leap for the numbers in the column for 0 faces.

The column for 4 and 5 faces was empty because on the large cube there can only be cubes with 0, 1, 2, 3 faces showing. However the cube with dimensions 1 x 1 x 1 had 6 faces showing which is an exception.  


3 faces

2 faces

1 face

0 faces







...read more.



  • I then checked to see if the formulas worked because if you add them up then it should be equal to the total number of small cubes in the large cube. Which has the formula x=n3, (x representing the total number of cubes).
  • With this it would also work out the general formula, so I added the formulas for 0 faces, 1 face, 2 faces and 3 faces.

Here are the formulas when added together:

= (n – 2)3 + 6(n – 2)2 + (12n +24) + (8)

= (n3 – 6n2 + 12n – 8) + (6n2 – 24n + 24) + (12n – 24) + (8)

= n3 – 6n2+ 12n – 8 + 6n2– 24n + 24 + 12n – 24 + 8

= n3 + 12n – 8 -24n + 24 + 12n – 24 + 8

= n3 + 12n – 8 –24n + 24 + 12n – 24 + 8

= n3 – 8 +24 –24 + 8

= n3 – 8 +24 –24 + 8

= n3 +24 –24

= n3 +24– 24

= n3


I found that the shape of the cube had a part in the formulas, like the number of cubes with painted faces was 8, because there are 8 vertices. Also on 2 faces 12 was to be multiplied by something because there are 12 edges.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Hidden Faces and Cubes essays

  1. shapes investigation coursework

    As this is quite a complex formula in comparison to previous ones I have found, I feel it would be wise to test this formula once for triangles, squares and hexagons, just to make sure. So where P=14, D=2 and the shape is T (i.e.

  2. I am doing an investigation to look at shapes made up of other shapes ...

    If you look at the ones above, the first is straightforward, the second has a 'divide by' at the beginning of the formula, and the last one has it at the end, as well as a set of brackets. I will try to get them all looking the same, with

  1. Shapes Investigation I will try to find the relationship between the perimeter (in cm), ...

    Assuming I do find a formula in terms of T=, Q= or H=, I will be able to simply rearrange them to give D= and P=. As I am trying to get a 'universal' formula that will work for all 4 shapes (rhombuses included), I will later on substitute T,

  2. gcse maths shapes investigation

    derived from is fine - I will test them just to make sure. There is no reason mathematically why the formulas should not work, unless there is no or irregular correlation between P, D and Q, but testing will iron out any mistakes I may have made.

  1. mathsI will try to find the correlations between the perimeter (in cm), dots enclosed ...

    Now I shall test it, just to make sure it works. So where P=14, D=4 and Q=10, Q=P/2+D-1 � Q=7+4-1 � Q=10 C And where P=16, D=6 and Q=13, Q=P/2+D-1 � Q=8+6-1 � Q=13 C And where P=16, D=9 and Q=16, Q=P/2+D-1 � Q=8+9-1 � Q=16 C As I expected,

  2. I am doing an investigation to look at shapes made up of other shapes.

    If you look at all these tables, you will see that where D=0, P is always 2 more than T. This can be written as P-2 +/- D=T. The reason I have written +/- D is because, as D is 0, it can be taken away or added without making any difference.

  1. Am doing an investigation to look at shapes made up of other shapes (starting ...

    However, just to make sure, I will test the new formulas once for each number of triangles. I will make sure I do not test them with a D of zero, as this would give less margin for error (I have not tested any shapes where T or P are zero)

  2. Shapes (starting with triangles, then going on squares and hexagons. I will try to ...

    C P=18 and D=0 � 18-2+0=16 C This proves that (for triangles at least) the formula P-2+2D=T works. This can be rearranged to give D=(T+2-P)/2 and T= P+2D-2. I do not need to test these two new formulas, as they have simply been rearranged from the existing one, P-2+2D=T.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work