

Koenigsberg Bridge problem, and a problem of roots
Name: yendor
Status: N/A
Age: N/A
Location: N/A
Country: N/A
Date: Around 1995
Question:
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.
Replies:
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^n1 = 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)
asmith
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.
chaffer
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Update: June 2012

