Unit Cubes 1 – Counting Faces

Probability Level 2

Consider the outside of cubes made out of $1 \times 1 \times 1$ unit blocks, with side lengths $n \times n \times n$ , for $n \geq 2$ , which we have painted red on the outside.

The following table shows the number of unit cubes which are colored on a given number of faces.

Cube Size	3 faces red	2 faces red	1 face red
$2 \times 2 \times 2$	8	0	0
$3 \times 3 \times 3$	8	12	6
$4 \times 4 \times 4$	?	?	?

What is the sum of the numbers missing from the row for the $4 \times 4 \times 4$ cube?

60 52 56 64

2 solutions

Scott Immel
May 24, 2014

4x4x4 cube has 64 cubes total to build it, with a 2x2x2 cube of 8 small cubes on the inside. Since we aren't asked the individual pieces, but the sum, the question really is how many are painted with at least one side.

Thus, 64-8 = 56 "painted" cubes.

Indeed sir. This is the shortest way to solve it.

V A - 6 years, 2 months ago

Victor Paes Plinio
May 23, 2014

3 painted faces ---> only $8$ from the corners.

2 painted faces ---> is the edges cubes. Have 2 edges cubes with 2 faces painted in each edge. Any cube have 12 edges. $2 \times 12 = 24$

1 painted faces ---> is the middle pieces. We can get the number of middle pieces in a cube with a side measuring $x$ cubes, using $6 \times {(x-2)}^{2}$ . We have $6 \times {(4-2)}^{2}$ ===> $6 \times 4 = 24$

The Answer is $8+24+24=56$

Mine answer was right but i accidently click the other one....

rishabh singhal - 6 years, 12 months ago

Unit Cubes 1 – Counting Faces

2 solutions

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