Problem number 25 states, the solid shown in the diagram has 12 regular pentagonal faces, the other faces being either equilateral triangles or squares. Each pentagonal face is surrounded by 5 square faces and each triangular face is surrounded by 3 square faces. John writes 1 on each triangular face, 5 on each pentagonal face, and minus 1 on each square. What is the total of the numbers written on the solid? So here we have our shape, and the 3D solid that we have here is called a rhombococcydodecahedron, which is a cool fact, but it really won't help us solve the problem necessarily, unless if we have memorized how many of each face a rhombococcydodecahedron has. So we're told that this shape consists of 12 pentagons, and we know that a pentagon has five sides each. So in total, in this shape, there are 60 pentagonal sides. If we look at a square face on the solid, we can notice that each square touches two pentagons. This means that we have half as many square faces as pentagons, so 30. Now let's look at the pentagonal vertices, where the triangles are located. Each pentagon has five vertices each, which means in total there are 60 pentagonal vertices, and we can see that each triangle touches three of these vertices, so every three vertices there is one triangle. So 60 divided by 3 means that there are 20 triangles in this solid. So now let's count up all the points. Every square is worth minus 1, every pentagon is plus 5, and every triangle is plus 1. So adding them all up, we get minus 30, 60, and 20. Summed up, that's 50, which is our answer. So the question asked us, what is the total of the numbers written on the solid? The answer is 50. Letter B.

rhombococcydodecahedron

geometric solid

face scoring

pentagon triangle square

total score calculation