Advanced book on Mathematics Olympiad

740 Combinatorics and Probability

F= 2 +E−V= 5.

We have reached a contradiction, which shows that the answer to the problem is negative.

850.With the standard notation, we are given thatF≥5 andE=^32 V. We will show that
not all faces of the polyhedron are triangles. Otherwise,E=^32 Fand Euler’s formula
yieldsF−^3 F 2 +F=2, that is,F=4, contradicting the hypothesis.
We will indicate now the game strategy for the two players. The first player writes
his/her name on a face that is not a triangle; call this faceA 1 A 2 ...An,n≥4. The
second player, in an attempt to obstruct the first, will sign a face that has as many common
vertices with the face signed by the first as possible, thus claiming a face that shares an
edge with the one chosen by the first player. Assume that the second player signed a
face containing the edgeA 1 A 2. The first player will now sign a face containing the edge
A 3 A 4. Regardless of the play of the second player, the first can sign a face containing
eitherA 3 orA 4 , and wins!
(64th W.L. Putnam Mathematical Competition, 2003, proposed by T. Andreescu)

851.Start with Euler’s relationV−E+F=2, and multiply it by 2πto obtain 2πV−
2 πE+ 2 πF= 4 π.Ifnk,k≥3, denotes the number of faces that arek-gons, then
F =n 3 +n 4 +n 5 + ···.Also, counting edges by the faces, and using the fact that
each edge belongs to two faces, we have 2E= 3 n 3 + 4 n 4 + 5 n 5 +···.Euler’s relation
becomes

2 πV−π(n 3 + 2 n 4 + 3 n 5 +···)= 4 π.

Because the sum of the angles of ak-gon is(k− 2 )π, the sum in the above relation is
equal to. Hence the conclusion.

Remark.In general, if a polyhedronPresembles a sphere withghandles, then 2πV−
= 2 π( 2 − 2 g). As mentioned before, the number 2− 2 g, denoted byχ(P), is called
the Euler characteristic of the polyhedron. The difference between 2πand the sum of
the angles around a vertex is the curvatureKvat that vertex. Our formula then reads
∑

v

Kv= 2 πχ(P).

This is the piecewise linear version of the Gauss–Bonnet theorem.
In the differential setting, the Gauss–Bonnet theorem is expressed as
∫

S

KdA= 2 πχ(S),

or in words, the integral of the Gaussian curvature over a closed surfaceSis equal to the
Euler characteristic of the surface multiplied by 2π. This means that no matter how we
deform a surface, although locally its Gaussian curvature will change, the total curvature
remains unchanged.