What is the relationship between the size of the cube and the number of painted sides?

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The Painted Cube Problem:

A cube falls into a bucket of paint and comes out covered on all 6 faces. The cube is then cut into smaller cubes, each 1 inch on an edge.
If the original cube was 2 inches on each edge, how many pieces will there be?
How many of those pieces will have paint on 3 faces? On 2 faces? On 1 face? On zero faces?

 Step 1 Close 

Move points A or B to change the size of the square. Note the measures of square (side and its diagonal) and record them. Sort the data by the length of S, and create a table with 4 colunms.


 Step 2 Close 

Cube Size 0 Sides Paint 1 Sides Paint 2 Sides Paint 3 Sides Paint Total Cubes
2 in.
3 in.

Find the ratio of the length of the diagonal to the length of the side of each square, rounded to two digits after the decimal point.


 Observations and Remarks Close 

What is the pattern?
Can you state a rule for the pattern?