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

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# Painted Cube Problem

This 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.