Topology: Polygonal Meshes, Euler's Formula Firstly, I would like to tell this is my first question on stackexchange and I am a senior school student. Please bear with me, if I am not able to express myself clearly.

I am trying to solve a question based on euler's formula: V - E + F = 2 (1 - G)

I have been given a closed mesh of a sphere. Number of Edges and Faces and number of Vertices with valence 4 are given.


Now, I have to compute V, x_1, x_2 and x_3. The vertices for valence 4 are some how not printed in my book and I assume them to be 137 (274/2) relating to table on the right. I know F = 2 V, but 275 = 2V gives a decimal point number which is not possible !

Pardon for my brevity and correct me if I am not clear :)
Let g be the number of vertices of valence 4. A closed sphere has G=0 and we have the equation V-E+F=2. Now V=x1+g+x2 and F=275. 

As a first step let us count the edges. If we multiply each face by its valence and sum those up, we get twice the number of edges (as each edge is shared by two faces), so:

(1)   2*E = (274*4 + x3*1)


Immediately we can see that 2 divides x3.

Similarly if we multiply each vertex with its valence we get again twice the number of edges, as each edge is shared by two vertices, so:

(2)   2*E = 3*x1 + 4*g + 5*x2


And again we see that x1 and x2 must have the same parity.

Plugging the first of this equation (with the very first ones) into V-E+F=2 we get

(3)   2 = V-E+F = x1 + g + x2 - (1096+x3)/2 + 275 = x1 + x2 - x3/2 + g + 548


Equating the two equations we derived above we get 

(4)   1096 + x3 = 3*x1 + 4*g + 5*x2


Now the system of equations (3)&amp;(4) has all the information encoded (excluding the parity information), so it appears you don't already have everything you need to know to determine these four unknowns.

I made a small Haskell program that does some bruteforcing with these conditions: Try it online!

The first solution it does get is (0,732,2564,0) = (x1,x2,x3,g)