The Problem

A solid block of wood in the shape of a right-rectangular prism measures a inches by b
inches by c inches. The block is dipped into red paint and is then cut into one-inch cubes.
How many of the one-inch cubes will have no paint on them? Give your answer in terms
of a, b, and c.

(Source: Adapted from Mathematics Teaching in the Middle School, Jan 1998)

The Solution

First, we need to consider what the total volume of the rectangular prism will be, then we can determine how many cubes will not have paint on them. Since we are given the dimensions of the cube to be a inches by b inches by c inches, we know that the volume, or total number of 1 inch cubes contained in the prism is equal to abc.

Next, we need to consider what each face of the prism will look like. The prism will have 6 faces. The faces have the following dimensions:

a x b

a x c

b x c

There are two faces for each of the above listed dimensions, so the surface area of the prism is equal to the following:

2ab + 2ac + 2bc

One might be tempted to think that this would be the number of cubes that will be painted; however, this is an incorrect assumption. Since all of the faces share edges, some of the cubes will be painted on more than one side. The eight cubes located at the verticies will be pained on three sides, and each cube along an edge will be painted twice, once for each face it is a part of.

Thus, we must now determine how many cubes will really be pained. First we can consider the top and bottom of the prism. We can use the two faces that measure axb for the top and bottom. For the top we get the following number of cubes:

2a + 2(b - 2)

The 2a comes from the length of the prism. This figure will account for all of the cubes along the length of the top, all four of the top verticies, and the a part of the side faces. We need to subtract 2 from each width of b to account for the cubes located at the vertices that we accounted for by counting a. Finally, we can multiply this figure by 2 to give us the number of cubes along the edge for the top and the bottom:

4a +4(b-2)

Next, we can determine the number of cubes that will be pained along the sides. Keep in mind that we already counted all of the cubes that lie along the length of a, so we only need to account for the height of the prism, but we must remember not to double count the vertices. So, we know that we will have 4 edges of height c to count, less the two points on each end of side c which lie in the vertices that have been counted, giving the following:


We can combine these two figures to determine the number of painted cubes along the edges of the prism:

4a +4b +4c -16

Next, we must determine how many cubes that are not edges have a face on the surface of the prism. We already stated that the sides of the prism are ab, ac, and bc. But we took off one cube on each side to account for the edges. This gives us new dimensions for each side:




As before, each of these rectangles appears twice on the prism, so we have the following number of cubes that are painted only on one face:




By comining the above number of cubes we get:


Next we want to add this number to the number of cubes we determined were located along the edges of the prism:

(2ab+2ac+2bc-4a-4b-4c+24) +(4a+4b+4c-16) = 2ab+2ac+2bc+8

Now we know how many cubes were indeed painted, so we must subtract amount from the total number of cubes contained in the rectangular prism: