• 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. Maths Investigation -Painted Cubes

    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.

  2. 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.

  1. "With reference to theories of visual object recognition outline the ways in which faces ...

    Expression analysis takes a person's facial features and uses them to extract their emotional temperament informing the perceiver of the mood of the person. Facial speech analysis is where the observer looks at the persons lip movements in order to identify speech sounds and patterns.

  2. An experiment to find out if seeing the eyes of a well known persons ...

    I only marked the answers correct if the participants wrote the full name (or in some cases a nickname or well-known character name) of the celebrity. Procedure: This experiment took place on 21st January 2005 in lab 5 at Barton court grammar school in Canterbury.

  1. The Painted Cube - Maths Investigations

    If I take a length of small cubes, as long as they are not all made up from the outer layer they will look like this. Because there are two small cubes on the ends of the strand it will make the formulae n-2 and because the 0 sided faces

  2. Cubes and Cuboids Investigation.

    Exceptions 'font-size:14.0pt; '>There are though times when my formulae will not work. These can be divided into two categories, cuboids with one dimension of one and the other is for cuboids with two dimensions of one. Again as a cube is a special cuboid you can also add a third category of a cube with dimensions of 1.

  1. Hidden Faces Investigation

    I will prove that it is the same as h=6xyz-2xy-2yz+xz and is therefore correct. First I'll take an example arrangement, where x and z equal 2, and y equals 1. In this cuboid there are 4 hidden faces with dimensions xy, 4 with dimensions yz, and 4 with dimensions xz, as shown.

  2. Shapes made up of other shapes

    D=3� D=(15+2-11)/2� D=3 T=11+6-2 � T=15 And where T=16, P=10 and D=4� D=(16+2-10)/2� D=4 T=10+8-2 � T=16 And there T=20, P=12 and D=5� D=(20+2-12)/2� D=5 T=12+10-2 � T=20 Without going any further, I would say this is sufficient evidence to prove that my three formulas work for triangles.

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