Koenigsberg Bridge problem, and a problem of roots
Date: Around 1995
Can someone tell me about the Koenigsberg bridge problem in elementary
terms? Did Euler's solution lead to any generalizations? What broad area
of mathematics is this problem in? I would be grateful if someone could
tell me what the n roots of x^n - 1 are, and explain how they arrived at
that newer answer. I realize those are not similar in any way but I would
really like to know the answer to both.
Since I do not recall the details of the bridge problem I will leave that to
somebody else, except I think it is in the general area of topology (the
shape of arbitrary things) and there certainly are generalizations.
The n roots of x^n - 1 are obtained from the formula
e^(2 pi i) = 1
which is a very important formula in the theory of complex numbers.
Let z = e^(2 pi i/n). Then z^n = e^(2 pi i) = 1 satisfies the formula
x^n-1 = 0, and also z^2 has z^n = e^(4 pi i) = 1^2 = 1 satisfies it,
and in fact z^k for any k = 0, 1, 2, . . ., n - 1 gives a distinct
solution. The actual value is found using the cosine/sine decomposition of
exponentials with imaginary argument:
z^k = e^(2 k pi i/n) = cos (2 k pi/n) + i sin (2 k pi/n)
The Koenigsberg bridge problem dates from 1736 and deals with a puzzle
problem requiring creation of a walking route which would cross each of
seven bridges on the Pregel river exactly once and returns to the starting
point. The configuration involves a fork in the river and an island in the
river and would not be easy to reproduce here. It should be easy, however,
to find a diagram of this. The general area is Graph Theory and this
problem is described in most introductory discrete mathematics and combina-
torics textbooks. Look in the index under Euler circuits or just circuits.
The problem generalizes to a large area of study.
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Update: June 2012