Published on December 14th, 2010 In category Education | Maths

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Painted Cube Problem

Painted Cube Cutting ProblemThis is classic problem and how to solve the question of how many cubes are colored on one, two or three sides when the cube of size x  is cut eaqually in 1 unit size.

For example lets take this question:  A cube of 9 cm size is given, all the face of this cude is first painted and then cube is cut into 1 cm size small cubes. How many cubes will have their one side painted. It may also be asked how many cubes will have 2 sides painted or how many cubes are painted on all sides or three sides etc. The trick to solve such problems is quite easy :

The total numbers of smaller cubes that will formed will always be – N ^3 – in the above example it would 9^ 3 = 9 x 9 x 9 = 729 cubes

The cubes that are in the corners will have 3 sides painted, cubes that are on the edges will have 2 sides painted and cubes that are on the face will have a side painted.

To Calculate :How many cubes obtained from cutting large cube are painted on single face

In one face of a large cube there are 9×9 = 81 squares , ( N-2) x(N-2), If we leave cubes in the edges we get (N-2) in each row in our case (9-2), are on NOT on the edge hence in our case (9-2) x (9-2) = 49 have one side painted,

Now since there are 6 faces of a cube we have 49 x 6 = 294 cubes painted on one side

To Calculate :How many cubes obtained from cutting large cube are painted on 2 or More sides or face

The cube with 2 or more side painted = Total no. of cubes by cutting large cubes – Cubes with one side painted . In our case it is 729 – 294 =   435

Mind you only there would be only 8 cubes that will have 3 sides painted and those will be the corner ones. So getting the numbers of cubes with only 2 sides painted will not be tough.

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  1. anuja says:

    @VAIBHAV no sides painted is simply the cube of total no of cubes on 1 side -2..i.e just taking out the outer cover of cube…

  2. vaibhav says:

    thank u . but u have left the cubes with no sides painted, thats where most students face problem.

  3. Vicky says:

    Valuable info. Lucky me I found your site by accident, I bookmarked it.

  4. Marc says:

    Thank you for idea and good work.