Problem number 14. Each of the shapes shown is made by gluing together four cubes of the same size. The shape will be painted. Which shape has the smallest area to be painted? Each shape is made up of the same number of cubes, and each cube has six faces. But some of these faces are covered up by being glued to other faces. One way to solve this problem is to look at each individual cube, find the number of faces that are not covered up, and add those up to figure out how many of the square faces need to be painted on each figure. The smallest number will be the smallest area. In the first shape, the first cube has just one face covered up right here where it joins the next one, so there are five faces that are not covered up. In the next one, one face is covered up on this end and one on this end, so there are four that are not covered up. The next cube is exactly the same, four, and the last one has one face covered up, which means there are five that are not covered up. 5 plus 5 is 10. 4 plus 4 is 8, which gives us a total of 18. In the next shape, each cube is joined to two other cubes. The one in the corner here is joined here and here, so it has only four faces that are not covered up. This one, again, will have four faces that are not covered up. So will this one on the bottom and the last bottom one. 4 plus 4 is 8. 4 plus 4 is 8. That gives us 16. In this C, the leftmost cube is only joined to one other cube, so it has five faces that are not covered up. The next one is joined to two cubes, so there are four faces that are not covered. On the next one, there also are four faces that are not covered because it's joined to two more cubes. And the last one is only joined to one cube, which means that there are five faces that are not covered. Again, 5 plus 5 is 10. 4 plus 4 is 8, which gives us 18. In answer D, the cube on the left is joined to one other cube, so it has five faces that are not covered. The one next to it is joined to two cubes, so there are four faces not covered. Then again, this next one is joined to two cubes, giving four uncovered faces. And the last one has five faces that are not covered. And that again gives us 18. In E, the top cube is joined to one other cube, so it has five faces that are not covered. The cube on the far left is joined to one other cube, so it has five faces that are not covered. Then the cube in the middle is joined to three other cubes, so there are only three faces here that are not covered. And the last cube on the bottom right has five faces that are not covered because it only touches one other cube. 5 plus 5 is 10, plus 5 more is 15. 15 plus 3 is 18. Now looking at the numbers, we see that 16 is the smallest number, so this answer, which is answer B, has the smallest area to be painted.

smallest paintable surface

four connected cubes

uncovered faces

shape B

surface area calculation