general math knowledge.
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.
question is: how many holes have only five neighbors?
Give a geometrical proof and explain steps and reasoning
animali acquatici unicellulari, hanno uno scheletro/guscio
poliedrico. Ne abbiamo trovato uno che possiede
un guscio perfettamente sferico con 384 buchi circolari
disposti sulla superficie in modo triangolare.
Ognuno dei buchi è circondato da 6 altri,
tranne alcuni che sono circondati da soli 5 altri
La domanda è: quanti sono i buchi circolari
sulla sfera circondati da soli 5 altri buchi?
organismes aquatiques unicellulaires, possèdent
des squelettes / capsules en forme de polyèdre.
Nous en avons trouvé un avec un squelette
parfaitement sphérique percé de 384
trous circulaires, disposés en surface de
façon triangulaire. Chaque trou est entouré de
6 autres trous, sauf quelques-uns qui sont entourés
de 5 trous seulement.
Question: combien y a-t-il de trous circulaires
sur la sphère à être entourés
de 5 trous seulement?
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).
the sphere is tiled with a combination of both
polygons, we obtain the following equality: a)E = (6H +
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]
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 +
(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!
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
60 + 32 - 90 = 2
Winners of the Puzzle of the Month are: Mauro Zoffoli, Italy
Krisje Storm, the Netherlands
encouraged to expand and/or improve this article. Send
your comments, feedback or suggestions to Gianni
A. Sarcone. Thanks!
Math Facts behind the puzzle
Euler (1707 - 1783), icosahedron and polyhedron
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
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.