"The particular problem of the seven bridges of Königsberg could be solved by carefully tabulating all possible paths, thereby ascertaining by inspection which of them, if any, met the requirement. This method of solution, however, is too tedious and too difficult because of the large number of possible combinations, and in other problems where many more bridges are involved it could not be used at all." [1].So, Euler decided to think about the problem, instead. This points out one difference between Euler and me. I would have jumped to my keyboard and started writing a program, since I find thinking to be very difficult. His first observation was that every time you reach a vertex by a bridge, you need to exit the vertex by another bridge. After a little thought, you realize that the number of connections to each vertex must be even if no bridge is traversed twice. You see that there are four land masses (four vertices), one of which has five connections, and the other three have three connections. So, Euler's 1736 resolution of this problem is that such a path (now called an Eulerian circuit) does not exist. In contemporary mathspeak,

A connected graph G = (V,E) contains an Eulerian circuit if and only if the degree of every vertex v ∈ V is even.[2]where the graph G is specified by its set of vertices (V) and set of edges (E), and each vertex v is a member of V There's another piece of mathematics, called the St. Petersburg paradox, that has as much to do with economics and psychology as it does with math. The paradox was named by Daniel Bernoulli, who published it and his solution in 1738; but he had heard it many years earlier from a cousin, Nicolas Bernoulli. Anyone in math or physics knows that there were many important Bernoullis, and they're interchangeable in most discussions, since no one can keep track of who is who. The paradox relates to a particular lottery and how much you would be willing to pay for a ticket. This lottery is quite different from the usual games of chance, so the odds calculation is, to say the least, unusual. Here's the way the lottery is played.

1) at each stage of game play, a fair coin is tossed.At first glance, I might be willing to pay two dollars for a lottery ticket, since it seems as if this process is a little better than a simple coin toss with 50:50 odds when taken to the second step; that is, I could loss a dollar, but I might win two dollars. Bernoulli showed how wrong my approximation of the odds can be. The expected value of the payout is easy to calculate by multiplying the winnings by the probability of winning at each stage and summing; viz., In short, this is a lottery you can't lose if you play an infinite number of rounds; but what if you're not that patient, or don't have a lot of money? Does it still make sense to play? As would any computer person, I decided to simulate this game with a simple program (C source code here). The figure below shows the results of 10,000 trials of 10,000 games each. This would approximate the payout if you bought a ticket each day for twenty-seven years.

2) On heads, the lottery pays $1, and the game ends. On tails, the coin is tossed again.

3) On heads, the lottery pays $2, and the game ends. On tails, the coin is tossed again.

...

n) On heads, the lottery pays $2^{(n-1)}, and the game ends. On tails, the coin is tossed again.

- L. Euler, "Solutio problematis ad geometrian situs pertinentis," Comm. Acad. Sci. Imper. Petropol.," vol. 8, pp. 128-140, 1736 (as cited in Ref. 3).
- Stephan Mertens, "Computational Complexity for Physicists," arXiv Preprint, April 7, 2002.
- Ole Peters, "The time resolution of the St. Petersburg paradox," arXiv Preprint, November 19, 2010.
- St. Petersburg Paradox Page on Wikipedia.
- Anders Martin-Löf, "An analysis of two modifications of the Petersburg game," arXiv Preprint, December 27, 2007.