An interesting geometrical problem entails the representation of 3-dimensional solids in two dimensions, The classic 3D solids are the Platonic solids first proposed by Plato, e.g.
The five basic Platonic solids are:
1) Tetrahedron
2) Cube
3) Octahedron
4) Dodecahedron
5) Icosahedron
And each of which can be represented by a polyhedral graph which is shown in the top graphic. The problem for the industrious reader then is as follows:
Identify which of the polyhedral graphs for the solids is Hamiltonian, which are Eulerian and which are "graceful".
A graph is designated Hamiltonian if there is a closed circuit along the edges of the graph that 'hits' each node exactly once.
A graph is Eulerian if there is a closed circuit that hits each edge exactly once.
A graph is called "graceful" if:
a) Each node gets a number from 0 to n, where n is the number of edges, and no node number is repeated.
b) Each edge gets a number equal to the positive difference of the node numbers on either side of it.
c) Provided no edge number is repeated given edges are numbered 1 through n.
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