Problem number 29. The second figure below shows the net of an octahedron. Each face of the octahedron is divided into three parts. The octahedron is colored with three colors, black, dark gray, and light gray, in such a way that the parts that come out of the same vertex or out of an opposite vertex are the same color. Which color should the part marked with the dot be colored? So to solve this problem, we have to figure out how the solid octahedron unfolds into the net of the octahedron shown in the second figure. One way to do this is to start to label the octahedron. For example, let's label it like shown. In the first picture, you can see that we've labeled the octahedron with ABC, and we've also marked A', B', and C'. The small apostrophe is pronounced prime. If we've done this on the original octahedron, let's look at the net now. We've marked the original triangle as ABC. Using just this triangle, we can actually figure out the labels of all the other vertices. In the original octahedron, notice that every triangle has three letters that are different, ignoring primes. For example, we might have A', B', C', or we might have A, B, C'. All of the triangles have three different letters. Using this, we can actually solve the entire net, as in we can figure out the labels of the vertices at the net. For example, the original triangle has ABC, and using this, we can figure out the other triangle that shares edge AC must be a B' C, because we already know two of the vertices are A and C, and the third one has to be different, and it cannot be B, since B and B' are opposite. Similarly, we can figure out A', B', C', A, B, C, and A' as we go along the whole net, and using this, we can fill out the whole net as shown. Now, remember the problem statement. It says that the parts that come out of the same vertex or the parts that come out of the opposite vertex are the same color. If we look at the way we've labeled the octahedron, opposite vertices are labeled with the letter and the same letter prime. For example, A and A' are opposites, B and B' are opposites, C and C' are opposites. Now, let's look at the net. The part next to the dot is vertex A', and we already know that vertex A has a part that's colored light gray. In particular, since we know that all the parts that come out of the same vertex or the opposite vertex are the same color, that means that A and A' must have all parts next to them the same color. We already know that A is next to a light gray part, so that means that A' must also be next to light gray parts, and that must be the part that the dot is in. That means that the answer is C, only light gray.

octahedron

color distribution

vertices

net

light gray