All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
A flat ball? (Posted on 2004-04-08) Difficulty: 2 of 5
Soccer balls are usually covered with a design based on regular pentagons and hexagons.

How many pentagons/hexagons MUST there be, and why?

  Submitted by Federico Kereki    
Rating: 3.7500 (4 votes)
Solution: (Hide)
There must be 12 pentagons, but the number of hexagons may vary, and even be zero. (A quick way of solving this: assuming there IS one answer, if we use no hexagons, we would have a regular polyhedron built out of pentagons: a dodecahedron, with 12 pentagons.)

A proof: according to Euler's theorem, E+2=F+V (E=edges, F=faces, V=vertices). In our case, if there are H hexagons and P pentagons, F=H+P, E=(6H+5P)/2 (since every edge belongs to two faces) and V=(6H+5P)/3 (since vertices appear at the junction of three faces). Substituting and equating, you get P=12.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Puzzle AnswerK Sengupta2022-09-03 22:13:53
Some ThoughtsSoccer ballsbroll2016-04-16 02:38:46
An interesting link (non-regular)Jer2004-05-03 09:43:05
re(2): Johnson solidsJer2004-04-27 14:24:22
re: Johnson solidsThalamus2004-04-15 12:32:08
Johnson solidsJer2004-04-15 12:03:24
re(3): The 720 degree deficitTristan2004-04-13 16:36:27
re: A related question...Brian Smith2004-04-13 14:26:51
Hints/TipsA related question...SilverKnight2004-04-13 12:20:41
Questionre(3): The 720 degree deficit - Euler's formulaSilverKnight2004-04-12 12:48:21
Solutionre(2): The 720 degree deficitTristan2004-04-12 10:32:31
re(2): The 720 degree deficit - Euler's formulaBrian Smith2004-04-12 10:26:10
re: The 720 degree deficitJer2004-04-12 09:47:25
Pentagons and HexagonsBrian Smith2004-04-12 09:15:46
The 720 degree deficitLarry2004-04-09 19:34:23
The 720 degree deficitLarry2004-04-09 19:34:19
Some ThoughtsSo far, no cigar...Federico Kereki2004-04-09 18:51:00
re(2): Hint in the title?Jer2004-04-09 14:10:40
re: Hint in the title?Brian Wainscott2004-04-09 13:06:20
Hint in the title?Jer2004-04-09 09:37:29
SolutionBend it like BeckhamPenny2004-04-08 16:13:08
SolutionanswerBilly Bob2004-04-08 15:37:57
Solutionre: My thoughtsThalamus2004-04-08 15:30:20
Some ThoughtsMy thoughtsBryan2004-04-08 15:28:25
Some ThoughtsMy thoughtsBryan2004-04-08 15:26:09
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (12)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information