 # Regular Icosahedron ACT Math Problem from 2019

Our Video explanation of how to solve the problem.

## Difficult Math Questions Can Be Made Easy

The April ACT from 2019 had several questions that were unusually difficult, and we can see based on the curve for this test that students across the country struggled with this Math section more than usual. For reference, missing 10 out of 60 Math questions on a 2015 ACT would earn you a section score of 28. Missing 10 out of 60 on a 2018 ACT would earn you a score of 30. On the April 2019 test, missing 10 out of 60 earned students a score of 32!

#### The Question: A regular icosahedron is a solid that has 20 congruent faces, each of which is an equilateral triangle. Each vertex is shared by 5 faces, and each edge is shared by 2 faces.

How many edges does the specialty die have?

A. 10

B. 18

C. 20

D. 30

E. 36

### How to Approach The Problem

First, Vocabulary is very important. It helps you rule out bad choices.

The best way to solve this problem is to visualize the parts of the icosahedron that you can’t see in the picture. This will help you figure out how many edges are not depicted. We know that there are only 10 faces shown, and since the icosahedron is regular (all angles and side lengths are the same) we can infer that the ‘invisible’ part of the icosahedron has the same structure as the part we can see. In other words, there must be edges on the back of the icosahedron that mirror the edges on the front.

#### Our Notes:

• We know it’s a 3-D figure, so we have to figure out how many edges we can’t see and then add those to the edges that we can see.
• There are 20 faces and we can see 10, plus each vertex must connect to 5 faces, so we can make some deductions about the parts of the icosahedron that we can’t see.
• We can see that the top vertex is connected to 3 faces, meaning there must be 2 more faces that we can’t see “behind” the figure.
• To draw those in, we can notice that the bottom of the icosahedron has 2 faces and copy that structure.
• The bottom vertex, as we just noticed, is connected to 2 faces, so we need to draw in the 3 “missing” faces that are “behind” the figure.
• To draw those in, we can copy the structure at the top of the icosahedron, where we saw that the vertex connects to 3 faces.
• Now we must connect the two parts we’ve drawn. To do so, we can copy the structure in the middle of the icosahedron, but we must flip it upside down. This is because the front of the icosahedron has 3 faces at the top and 2 at the bottom, but the “back” of the figure has 2 faces at the top and 3 at the bottom.
• Once we’ve drawn this middle section, we have accounted for every face of the icosahedron that was not initially visible.
• All that remains is to count the number of edges we’ve drawn and add that number to the number of edges we could see. This is 12 “invisible” edges + 18 visible edges, totaling 30 edges. If you can visualize the parts of the icosahedron represented by the dotted lines in the image above, you are well on your way to solving the problem!

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