Tuesday, June 9, 2015

Euler and the 7 Bridges Puzzle Series #1

Leonhard Euler (/OY-lər)15 April 1707– 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in many fields. He also introduced much of the modern mathematical terminology and notation. He is also renowned for his work in mechanics, fluid dynamics, optics, astronomy, and music theory.

Euler is considered to be the most famous mathematician of the 18th century and one of the greatest mathematicians to have ever lived. He is also one of the most prolific mathematicians; his books fill 60–80 volumes. He spent most of his adult life in St. Petersburg, Russia and Berlin, Prussia.

The Seven Bridges of Königsberg is a historically notable problem in mathematics. Leonhard solved this problem in 1735.

The problem was to find a way to walk through the city that would cross each of 7 bridges once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time; one could not walk halfway onto the bridge and then turn around and later cross the other half from the other side (the walk does not need to start and end at the same spot). Euler didn't show a way to walk the seven bridges, but he proved that the problem has no solution- there was no way to do it.

To solve this problem, Euler realized that the shape of the island and city do not matter. The only thing that matters is the number of bridges and where they connect, like so:
Konigsberg bridges.png7 bridges.svgKönigsberg graph.svg
In modern mathematical terms, each land mass becomes a "vertex" (blue dot), and each bridge becomes an "edge” (line), the resulting mathematical figure is called a “graph”.

If you can draw a line that starts on a vertex and covers each edge once and only once without lifting the pencil, you have solved the problem by proving it is possible. Remember that Euler proved that this graph is impossible, which means you cannot draw the graph without lifting the pencil and starting at a new vertex.

Can you guess how Euler knew? Look for a new blog showing you how to tell whether you can draw a figure without lifting the pencil.

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