# Count the number of shapes in a polyhedron.

So this is a question that was asked in the International Kangaroo Math Contest 2017. The question is:

The faces of the following polyhedron are either triangles or squares. Each triangle is surrounded by $$3$$ squares and each square is surrounded by $$4$$ triangles. If there are $$6$$ square faces, how many triangular faces are there?

What I did:

Each square shares each of its four neighboring triangles with two more squares. So we can say that for 6 squares we have $$6\times4\ -\ 2 \times 6 = 12$$ triangles. However, I still know that this calculation of mine is quite wrong and based on an awkward thinking. So, what is the correct answer and how?

Thanks for the attention.

• Always hated these questions when they specifically asked for reasoning. I just used my onboard 3D engine and could 'see' 8. Why? Because look :D – Lamar Latrell Oct 27 '18 at 6:41

Each edge of the polyhedron is shared between exactly one triangle and exactly one square, as can be inferred from the question statement. Thus, given six squares, there are 24 edges ($$6×4$$), and thus eight triangles ($$24÷3$$).

The polyhedron is called a cuboctahedron.

• Thanks! I just didn't try thinking it that way....;) – Faiq Irfan Oct 26 '18 at 11:35

Each square is bordered by four triangles and $$6\times4=24$$. However every triangle is bordered by three different squares, so it was counted three times in the multiplication above. This means there are $$24/3=8$$ triangles.

This isn't rigorous, but if you don't have to write a proof, just get the right number, it's clear from the illustration. You can see "one hemisphere" of the polyhedron except for a triangle parallel with the line of sight, so the total number of sides of each type are just double what apppear in that hemisphere (i.e. what you see plus the hidden triangle).

• That would give 2*3+1=7 triangles but they are actually 8. I think you'd like to amend something? – Rad80 Oct 27 '18 at 10:39
• @Rad80: "double what you see plus the hidden triangle" should clearly be read with parentheses around all but the first word; I thought that was obvious from the text before it. – R.. GitHub STOP HELPING ICE Oct 27 '18 at 22:05
• well, at least we should be able to agree that is was not obvious what you meant. I saw two mentions that there is one "hidden" triangle (both times singular), so why should it be counted twice? If you edit the answer mentioning two hidden triangles instead of just one it'll be fine. – Rad80 Oct 29 '18 at 8:11
• @Rad80: There's only one hidden triangle in the hemisphere you're doubling. "Two hidden triangles" would be nonsense as there are of course five hidden triangles. – R.. GitHub STOP HELPING ICE Oct 29 '18 at 15:07
• @Rad80: Hopefully the change I made addresses what you found confusing about the wording. – R.. GitHub STOP HELPING ICE Oct 29 '18 at 15:08