Monday, October 28, 2024

Mensa Polyhedral Graphs Brain Buster Problem

 




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.

Platonic solid - Wikipedia

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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