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Mathematics-Online problems:

Solution to the problem of the (previous) week


Problem:

Sepp Herberger (the famed coach of the German national soccer team in 1954) coined the sentence ``The ball is round''. Contrary to this statement, let us assume that a soccer ball is a polyhedron.

\includegraphics{ball.eps}

Determine the number $ V$ of the vertices by means of the Euler polyhedron formula.


Answer:

$ V$ =


Solution:

Let $ V$ denote the number of vertices, $ E$ the number of edges, $ A$ the number of pentagons, and $ B$ the number of hexagons of a soccer ball. These considerations lead to the following system of equations:

\begin{displaymath}
\begin{array}{rcrcrcrcr}
V & - & E & + & A & + & B & = & 2\\...
...& - & 3B& = & 0\\
1.5 V & - & E & & & & & = & 0\\
\end{array}\end{displaymath}

Solving for the number of vertices, we get $ V=60$.


[problem of the week]