Previous Puzzles of the Month + Solutions

Since 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!

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

 

The Winners of the Puzzle of the Month are:
Mauro Zoffoli, Italy
Krisje Storm, the Netherlands

Congratulations!

© 2006 G. Sarcone, www.archimedes-lab.org
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More Math Facts behind the puzzle

Leonhard 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.