Question

6. A football (as used in soccer, not in American football) is constructed by sewing together hexagonal and pentagonal patches in a special way: it is a polyhedral graph where each face is a pentagon or a hexagon subject to two conditions: at each vertex exactly 3 polygons meet, and at each edge exactly 2 polygons meet. Write h for the number of hexagonal faces, p for the number of pentagonal faces, n for the number of vertices and m for the number of edges, and express the conditions above as two equations in these variables. Then use Euler's formula to show that the number of pentagonal faces is 12 (no matter how many hexagonal faces there are!).

Answer 1

This problem is an application of Euler's theorem from graph theory.

inoo at in a nkamdnd soccerkall at exury werten ot y edge tnnchy Enocly lgons med 2 ony maet t mumbn of henagonal facn the number o lgonol faco m+f= By Eules thuorem ehere m io Hhe. numben nurmbn for a plaon soccer ball n- oedges 2 -Cu) mumber of ventic fon hanagors = 6A begjn henagons Noco vetias for a pentgn - 56 4 edgus fon a pertgon 5. -6h From C) above wese teat we hove counted cach ventin vce vetrey 2 each edge dsice 5. nurnba of edgpan ehgis 2 numbn of enfceo 2 C howe Civ) 6ht5b 2 3 2ht1op-gh-15p tgt6 =2. 12 Naman ofpentogonal facs ut 212 n (Iv) e Fla terms wifhstil Crosres out /«. we haNe an identity does mot dspend on hSo p2 fon al any numbn of heaogonal facer pentagon ibAwrounde i adja cert to 3 Fon antandardfoth al each Comment Guo Aapung 6 munban of hengons 20 bneganal facw 7hera are 20