Name: Katie

Who is asking: Student
Level: Secondary

Question:
This problem is about finding the number of cubes visible and hidden in a cube.

In a cube that is 3x3, 19 cubes can be seen. 8 are hidden.
In a cube that is 4x4, 37 cubes can be seen. 27 are hidden.
In a cube that is 5x5, 61 cubes can be seen. 64 are hidden.
In a cube that is 6x6, 91 cubes can be seen. 125 are hidden.

The question is:
Explain how you could find the number of small cubes that are visible and hidden in a cube of any size.

I have no clue how to do this. Please help me figure it out. Thank You.

Katie

A cube is three-dimensional so the cubes you mention are 3x3x3, 4x4x4, 55x5 and 6x6x6. Now I can see where the numbers come from.

A 3x3x3 cube is made from 27 cubes and a 2x2x2 cube is made from 8 cubes. 27 - 8 = 19 and hence the numbers in the first row.

A 4x4x4 cube is made from 64 cubes and a 3x3x3 cube is made from 27 cubes. 64 - 27 = 37 and hence the numbers in the second row.

In each row the ARITHMETIC pattern is

If you have 27 identical blocks and you start to construct a 3x3x3 cube you immediately see that there is 1 block that is hidden.

If you have n³ identical blocks and you construct an nxnxn cube there are (n-2)³ block that is hidden.