Question 1193210: Imagine that you have used centimeter cubes to build a larger cube with side length 4 centimeters. You then paint your large cube blue on every side. If you take your cube apart, how many of the small cubes will have
3 blue faces?
2 blue faces?
1 blue face?
No paint at all?
Repeat question 1 for the cubes with side lengths of 1, 2, 3, and 5 centimeters. Be sure to organize your work, perhaps by using a table.
What do you notice in your table? Use any patterns you notice to predict the
outcome for n = 6. What about n = 100?
Generalize your results by finding algebraic expressions to represent the nth case.
Answer by ikleyn(52786)   (Show Source):
You can put this solution on YOUR website! .
For n x n x n cube
- the number of cubes that have 3 blue faces is 8 // the corner cubes;
- the number of cubes having 2 blue faces is 12*(n-2) // the cubes along 12 edges, without counting the corner cubes;
- the number of cubes having 1 blue face is n^3 - (n-2)^3 - 12*(n-2) - 8.
In the last formula, n^3 - (n-2)^2 is the number of all small cubes minus the number of interior cubes,
that comprise the smaller (n-2)*(n-2)*n-2) cube;
12*(n-2) is the number of cubes along 12 edges that have two faces painted blue;
and 8 is the number of corner cubes, having 3 faces painted blue.
Solved.
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