Cube A 3 × 3 × 3 cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure. (a) Determine how many unit cubes of the 3 × 3 × 3 cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces. (b) Repeat part (a) for a 4 × 4 × 4 cube, a 5 × 5 × 5 cube, and a 6 × 6 × 6 cube. (c) Write formulas you could use to repeat part (a) for an n × n × n cube.
Cube A 3 × 3 × 3 cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure. (a) Determine how many unit cubes of the 3 × 3 × 3 cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces. (b) Repeat part (a) for a 4 × 4 × 4 cube, a 5 × 5 × 5 cube, and a 6 × 6 × 6 cube. (c) Write formulas you could use to repeat part (a) for an n × n × n cube.
Solution Summary: The author explains how the number of unit cubes of a 3times, 3, 1 and 2 blue faces is determined.
Cube A
3
×
3
×
3
cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure.
(a) Determine how many unit cubes of the
3
×
3
×
3
cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces.
(b) Repeat part (a) for a
4
×
4
×
4
cube, a
5
×
5
×
5
cube, and a
6
×
6
×
6
cube.
(c) Write formulas you could use to repeat part (a) for an
n
×
n
×
n
cube.
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