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Previous Puzzles of the Month + Solutions

July-September 2009, Puzzle nr 122
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Puzzle # 122 Difficulty level: bulbbulbbulb, general math knowledge.


Radiolarian's shell
A radiolarian is a single-celled protozoa living in all the world’s ocean. Most of them have a spherical shell that survives as a fossil. We discovered a radiolarian with a perfect spherical shell having 384 circular holes arranged in a triangular pattern. Most of the holes are surrounded by six other holes, but some are surrounded by ONLY five.
  The question is: how many holes have only five neighbors? Give a geometrical proof and explain steps and reasoning

Keywords: polyhedron, sphere, Euler's formula.

Related puzzles:
- Euler's Graeco-Latin squares.

Source of the puzzle:
BrainTrainer, issue #28. © G. Sarcone.
You cannot reproduce any part of this page without prior written permission.


solution fig. 1Since each hole is surrounded by either 5 or 6 neighbors, we can consider the puzzle 122 as a tiling problem of a sphere: in fact, we can imagine that every polygonal face tiling the sphere contains a hole in its center, as well as one facing each of its sides (fig. 1).
If the sphere was tiled only with hexagons, the numbers of edges (E) would be (6 x H)/2, in total (since each hexagons’ side/edge shares 2 faces).
If the sphere was tiled only with pentagons, the numbers of edges (E) would be (5 x P)/2, in total (since each pentagons’ side/edge shares 2 faces).

As the sphere is tiled with a combination of both polygons, we obtain the following equality:
a) E = (6H + 5P)/2

And since each vertex (V) shares 3 edges, we obtain also this second equality:
b) V = (6H + 5P)/3; that is [(6 edges x number of Hexagons) + (5 edges x number of Pentagons) divided by 3]

The Euler’s polyhedron formula states for any convex polyhedron, the number of vertices (V) and faces (F) together is exactly 2 more than the number of edges (E): V + F = 2 + E
Substituting the variables of the Euler's formula with a) and b) above, and taking into account that F = H + P, we get:
c) (6H + 5P)/3 + (H + P) = 2 + (6H + 5P)/2
After reorganizing:
(6H + 5P)/3 + (H + P) - (6H + 5P)/2 = 2
and simplifying, we finally obtain:
P = 12
We must then have 12 pentagons; so of the 384 holes on the radiolarian, at least 12 have only 5 neighbors. Surprisingly, the result is INDIPENDENT of the number of holes!

truncated icosahedron
A convex polyhedron very similar to the one previously discussed, made of 12 pentagons and 20 hexagons only, is called truncated icosahedron and is well known by soccers, see fig. opposite. For the soccer ball the Euler's formulla gives:
60 + 32 - 90 = 2



cup winnerThe Winners of the Puzzle of the Month are:
Mauro Zoffoli, Italy italian flag
Krisje Storm
, the Netherlands netherlands flag


© 2006 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!

More Math Facts behind the puzzle

euler stampLeonhard Euler (1707 - 1783), icosahedron and polyhedron formula
The stamp issued in 1983 by the German Democratic Republic honoured Euler on the occasion of the 200th anniversary of his death and shows an icosahedron, one of the five Platonic solids, along with the Euler's polyhedron formula.
In 1988, mathematicians worldwide were asked to vote for their favourite theorems. Euler's polyhedron formula finished second, Euclid's theorem "There are exactly five Platonic solids" finished 4th.

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