How to solve cube problems, like a big cube is divided into 64 smaller cubes. Some sides are painted in blue, red or green. Can you plz. guide me how to approach these types of problems?

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Sooraj Prabhakaran

  • Jan 1st, 2007

The given question is not fully correct .

I will give u another eg ...

If a cube is painted red and  cut into 64 pieces .

Approach this question as ,,

64 pieces = 4 *4* 4   so take  n=4

just use this formula

1. no sided painted  =(n-2)^3

2. 1  sided painted =6 * (n-2)^2

3. 2  sided painted = 12 *(n-2 )

4.  3 sided painted = always 8 (a cube has 8 corners)

It is very easy to solve cube problems. there is a simple solution to solve cube problems if we use formulae.

Before solving the color cube problem first we have to solve the no of 1 side,2 side, 3 side and central cubes in a huge cube.

If we cut the cube into x number of halves, the total number of small cubes is (no. of halves) power 3.

the number of central cubes is -----------------" (number of Middle columns) power 3"
the number of 1 sides is -------------------------" Number of faces * central cubes"
the number of 2 sides is -------------------------"Number of two sides in one face* 3"
the number of 3 sides is -------------------------"Number of Corners" i.e.. 8 .

For eg
One cube is divided into 5 halves then
the number of central cubes == ( number of Middle columns) power 3== 9 power 3 ==27
the 1 side cubes == Number of faces * central cubes == 6 * 9 ==54
the 2 side cubes == Number of two sides in one face* 3 == 3 * 12 ==36
the 3 side cubes == Number of Corners == 8
Total number of cubes ==125

After calculating this based on the color information we have to decide the answer for the questions.

for eg

If a cube is colores red on one face, green on the opposite face,yellow on another face and blue on a face adjacent to the yellow face. the other two faces left uncolored. it is cut into 125 smaller cubes of equal size.

Q: How many cubes are uncolored on all faces?
A: Here two faces are uncolored
So the total number of 1 side cubes on two sides is == Central cubes+ 1 side for two faces+ uncolored common cubes for those two faces.
i.e.. 27+(2*9)+3 === 48

vikrant dwivedi

  • Aug 9th, 2011

There is no particular formula used to solve this type of question but better understanding of the question helps to to respond quickly

ANSWER:suppose the initially the length of the side of the cube is x.After cutting the cubes into 64 smaller sub cubes the sides of each cubes will be x/4(since 64=4*4*4. in case of 125 smaller sub cubes length of each side will be x/5.......)
now you can solve the question easily according to the question that which face is painted in which color

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  • Oct 17th, 2011

In above example the value of n should be 3 but not 4. Because when you give 4 cuts in same direction the cube will be divided into 5 parts. So n(the number of cuts) will be always one less than the total number of parts on each side.

4x4x4=64 so 4 parts on each side.
3 cuts will make 4 parts.
3x3x3 so 3 cuts on each side of cube.

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  • Jan 7th, 2012

Instead of analyzing the properties of cube its very easy 2 remember the formulas...
we can get answer very fast too.......

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  • Mar 29th, 2012

write 1 to 5 numbers on a paper(1,2,3,4,5). Now strike the number 1 and 5. then count the numbers u r having at d middle. there are 3 numbers. In the same way take a cube and eliminate left side corner boxes and right side corner boxes. Then count any 1 line boxes horizontally.

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  • Jun 11th, 2012

I solved the cube in another manner and was unable to get ur answer? How is it so?


  • Sep 4th, 2012

Mr Sooraj... Second formula should be 6*(n-2)^2

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  • Dec 11th, 2012

The mean of middle column is the column which are not in boundary. thats why we here use (n-2)

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  • Apr 24th, 2014

This single image can answer your quarries:


  • Apr 24th, 2014

Verbal Reasoning: Cube & Dice Quantitative Aptitude Test

Geometry of Cube
A cube is a three-dimensional solid object bounded by six sides, with three meeting at each vertex. It features all right angles and a height, width and depth that are all equal ( length = width = height). It has two types: 1. Standard Cube; and 2. Non-Standard Cube

Important Facts:

1. A cube has 6 square faces. (Ref. Image 1)
2. A cube has 8 points (vertices). (Ref. Image 1)
3. A cube has 12 edges. (Ref. Image 1)
4. Only 3 sides are visible at a time (called "Joint Sides"). (Ref. Image 1)
5. Joint sides can never be opposite to each other.
6. Things that are shaped like a cube are often referred to as ‘cubic’.
7. Most dice are cube shaped, featuring the numbers 1 to 6 on the different faces.
8. Addition of number of dots (pips) from opposite sides of a standard cube or dice is always 7.
9. Total of two adjacent faces of cube can never be a 7.
10. 11 different ‘nets’ can be made by folding out the 6 square faces of a cube. (Ref. Image 2)

(Image 1)

(Image 2)

:: Problem Solving ::

Things to remember before stepping ahead:

Image 3: Painted Sides of a Cube
* We can categorise a cube (or a colour cube) after cutting it, in these four categories: (See Image 3)

a.) Central cube (Yellow): In middle of a faces & has only one coloured side.

We can find out the total number of cubes with singe colour on any side with this formula:


b.) Middle Cube (Green): In middle of edges and have two coloured sides.

We can find out the total number of cubes with singe colour on any side with this formula:


c.) Corner cube (Blue): Cubes on corners and have three coloured sides.

A cube can have only 8 cut-corner cubes with colours on three sides. Hence answer will be always the same - 8.

d.) Inner Cube (No Colour): Cubes inside faces & has no coloured side.

We can find out the total number of cubes without any colour on any side (inner cube) with this formula: (X-2)^3

***Note: To find out total number of cubes we use this formula- (X)^3


guru desai

  • Nov 17th, 2014

I think ans should be 45....27 + ( 2*9 )

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Alan Alex

  • Apr 12th, 2017

Actually sir I have a doubt to clarify: A cube is coloured red on to opposite faces, blue on two adjacent faces and yellow on the two remaining faces. It is then cut into 2 halves along the plane parallel to red faces. One piece is then cut into 4 equal cubes and the other one into 32 equal cubes. How many cubes do not have any coloured face?

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  • Aug 18th, 2017

Cube means all sides with equal size
As pr the data, you got 64 small cubes(assume all are with equal size)
here all sides are painted
1. cube have 8 corners (all 3 faces are present)
2. 12 edges (2 faces meet)
3. 6 faces (only one face is there).
The below formula applicable only the cube is (6 faces are painted) fully painted we have 8 smaller cubes with 3 face with color.
smaller cubes with 2 painted faces = 12*(x-2)
here x is the cubes in the edge of the big cube
small cubes with one face paint = 6*(x-2)^2
cubes without color = (x-2)^3
Answer for ur question is:
3 painted cubes =8
2 painted cubes = 12*2 = 24
1 painted cubes = 6*(4-2)^2 = 6*4 = 24
cubes without paint = (4-2)^3 = 2^3 = 8

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