20 regular hexagons and x numbers of regular pentagons are stitched together to make a soccer ball. Find x
I tried making a soccer ball pattern but quickly realized that, due to the average mathematician's poor art skills, there has to be a way around from drawing the pattern. I'd go ahead and try to do 20*6 to get the number of vertices but I know it isn't that simple because those vertices will be shared amongst other shapes, so I'm unsure as to how to proceed with this problem. Also, am I right to assume that this has something to do with V+F-E=2?
You should recognize that no more than three faces can fit at each vertex, because the angles in the polygons are too large to fit four faces together.
Then we have the following:
Faces =$x+20$
Edges = $(5x+120)/2$. Each pentagon has five edges and each hexagon has six; adding up all the polygons counts every edge twice because EA h edge is shared by two faces
Vertices = $(5x+120)/3$. Each Pentagon has give vertices, each hexagon has six; adding up all polygons counts each vertex three times because only three faces meet at each vertex (see above).
Now try $V+F-E=2$ with the above renderings for each term and solve for $x$.
You might say that with this method you can get answers to this kind of problem by the dozen.