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Each face of a soccer ball is either a pentagon or a hexagon. Each pentagonal face is adjacent to five hexagonal faces and each hexagonal face is adjacent to three pentagonal and three hexagonal faces. If the ball has 12 pentagonal faces, how many hexagonal faces are there?

A. 12 B. 20 C. 24 D. 8 E. None of the above

This is from math olympiad for middle school students. I tried to solve this but failed.

I found a solution for this problem but it is not possible to understand the solution. Could someone elaborate more about the existing solution or give an easier solution?

Each pentagon has 5 sides and each hexagon has 6 sides. Total number of sides of the pentagons = 5 × 12 = 60. Total number of hexagons= 60/3 = 20. since each hexagonal face is adjacent to 3 hexagonal faces. Answer: (B)

When it says 60 it doesn't consider double counting of sides. Also, I can't understand the last two sentences of the solution (from total number...)

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To make it a bit more explicit: Let's count all of the edges that are shared by pentagons and hexagons. We'll call such an edge a "5/6 edge". (We don't need to worry about the edges between two different hexagons for this argument.) We can count the 5/6 edges in two ways.

  • Every pentagon has five "5/6 edges", since each pentagon borders five hexagons. Since there are 12 pentagons, there are 60 "5/6 edges" on the soccer ball.

  • Every hexagon has three "5/6 edges", since each hexagon borders three pentagons. If there are $n$ hexagons, there are $3n$ "5/6 edges" on the ball.

But these two numbers must be equal, implying $3n = 60$ or $n = 20$.

Michael Seifert
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Two givens:

  1. A pentagonal face is adjacent to 5 hexagons
  2. A hexagonal face is adjacent to 3 pentagons and 3 hexagons.

Suppose there are 12 pentagonal faces in all, each having 5 sides. By 1., no two pentagonal faces share a common side. Thus, the total number of sides of the pentagons is simply 12*5=60.

Great, so there are 60 distinct pentagon sides. Now, naively using 1. in isolation, we might first guess there are 60 hexagons, one for each pentagon side. But this guess would be overcounting, because there is not a one-to-one mapping between pentagon sides and hexagons. In particular, counting three pentagon sides only corresponds with counting one hexagon (using 2.). Thus, 60 counts each hexagon thrice, so dividing our guess by 3 gives us our answer of 20 hexagons.

And in case the visual helps... enter image description here

Golden_Ratio
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