IMC 2015 Q17
17. The football shown is made by sewing together 12 black pentagonal panels and 20 white hexagonal panels. There is a join wherever two panels meet along an edge.
How many joins are there?
- A 20
- B 32
- C 60
- D 90
- E 180
Intermediate Mathematical Challenge 2015, Question 17. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D 90
The 12 pentagonal panels have a total of 12 × 5 = 60 edges. The 20 hexagonal panels have a total of 20 × 6 = 120 edges. So in total the panels have 180 edges. When the panels are sewn together, two edges form each join. So the number of joins is 180 ÷ 2 = 90.
Solution © UK Mathematics Trust, from the IMC 2015 solutions. Longer versions with investigations: Solutions and investigations (PDF).