A Year in Numbers by Kyle D. Evans

A Year in Numbers by Kyle D. Evans

Author:Kyle D. Evans [Evans, Kyle D.]
Language: eng
Format: epub
Published: 0101-01-01T00:00:00+00:00


Most people can visualize a solution to this one quite quickly: start from any node, go around the outside of the pentagon and then around the five points of the internal star.

So if a graph has all even nodes you can start anywhere and end up back there, but if it has a pair of odd nodes then you must start at one and end at the other.* If there are two pairs of odd nodes or more, we cannot draw the graph without repeating a line; the graph is non-Eulerian. Because odd nodes are essentially ‘unstable’ and have more ways in than out (or vice versa) they can make up only the start or end of your route, never a middle part. No route can have more than one start and end, so more than a pair of odds is an insurmountable problem.

13 July

Yesterday’s discovery about which graphs can and can’t be drawn without repeating an edge came as a result of the Bridges of Königsberg problem, in which residents of the town of Königsberg (now Kaliningrad, in that strange bit of Russia sandwiched between Poland and Lithuania but not connected to the rest of Russia) tried to walk over all seven bridges in the town without going over a bridge twice. When this problem is reconfigured as a graph, in which each bridge is an edge and the four main land masses connected by the bridges are represented as nodes, it becomes clear that every ‘island’ of land is an odd node, so it cannot be done.



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