Stripping A Cube

Probability Level 2

During arts and crafts, you made a cube of side length 10, by using 10 x 10 x 10 = 1000 10 x 10 x 10=1000 smaller cubes. You enjoyed your handiwork and panted the exterior surface of your large cube red. The class bully came along and decided that he wanted all those painted cubes, so he removed (only) the outer layer of cubes.

How many cubes will remain?

Image credit: Flickr A. Drauglis


The answer is 512.

Yashasvini Sharma by Yashasvini Sharma , 28, India

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5 solutions

Bhagirath Mehta
Apr 20, 2014

One removes one block from the top and bottom of each row and column. This means that instead of 10 blocks in a given row or column, we subtract 2 to receive 8 blocks in a row/column of the new cube. This leaves us with a cube with a side length of 8.

8 * 8 * 8 unit blocks= 512 blocks

13 Helpful 0 Interesting 0 Brilliant 0 Confused

where did you get "2" from?

Sarfraz Ahmed Razi - 7 years, 1 month ago

yes i also want to know this

Manav Goyal - 7 years, 1 month ago

@Sarfraz Ahmed Razi and @Manav Goyal : We remove one block each from the top and bottom of each row/column. 1 block from the top + 1 block from the bottom = 2.

Bhagirath Mehta - 7 years, 1 month ago

its imagination only.. just remove the cubes from exterior 6 faces and get 8 8 8=512 ans.

Magan Singh - 7 years, 1 month ago

1000-(200+160+128)=512

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Jayver de Torres
Apr 22, 2014

Imagine a rubik's cube. Obviously,if the exterior part of 1by 1 by 1 and 2 by 2 by 2 cubes are painted, all cubes are painted. And if we do this process in 3 by 3 by 3 cube, there is 1 cube which is not painted and it is the innermost cube. Again, doing this in 4 by 4 by 4 cube, there are 8 cubes which are not painted. There is a pattern. In 3 by 3 by 3, there is 1 which is the cube of 1. In 4 by 4 by 4, there are 8 cubes which is the cube of 2. And by this, there are 512 cubes which is the cube 8.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Ryan Redz
May 30, 2014

remove one layer of cubes from each face to get the remaining dimension 8x8x8=512

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Saurabh Mallik
Apr 26, 2014

To find the number of unpainted cubes we can use the formula:

Unpainted cubes = ( n − 2 ) 3 , n = = (n-2)^{3}, n = Edge of cube

= ( 10 − 2 ) 3 = 8 3 = 512 = (10-2)^{3} = 8^{3} = 512 small unpainted cubes

So, the answer is: 512 \boxed{512} small cubes

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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