Help TOm to find the number of black, white and green pieces?

Question

TOm has a spongy boom ball that is made of 32 pieces of polygon figures: 12 black pentagons and 20 white hexagons. Each pentagon adjoins 5 hexagons and each hexagon adjoins 3 pentagons and 3 hexagons. Boogy drew a polygon (i.e. a closed line without intersections) along the edges of the pieces. The polygon divided the ball into two parts and Boogy painted one of them green. He is curious if given a description of the polygon you are able to compute the number of black, white and green pieces?

Answer 1

If the two parts are equivalent then presumably

• half the original black pentagons are still black $(\frac{12}{2}=6)$;
• half the original white hexagons are still white $(\frac{20}{2}=10)$; and
• half the original polygons are now green $(\frac{32}{2}=16)$.

Otherwise you would have to give a description of the dividing polygon.

Answer 2

Clearly not, because if the polygon cuts the ball into two pieces of different sizes, the answer depends on which piece was painted green - the polygon is the same.