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The Three Houses Problem

Published online by Cambridge University Press:  03 November 2016

G. T. Kneebone
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Extract

I Give below a topological proof of the following theorem, based on a well-known conundrum:

A, B, C are three houses, and A′, B′, C′ are the gas, electric and water companies, arranged as shown (Fig. 2). It is impossible to join each house to each company in such a way that no two lines cross and no line passes through a third building.

Type
Research Article
Information
The Mathematical Gazette , Volume 25 , Issue 264 , May 1941 , pp. 78 - 81
Copyright
Copyright © Mathematical Association 1941

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References

* A Jordan curve is a curve having parametric equations x=f(t), y=g(t) where f, g are one-valued, continuoua functions in the range 0 ≤ t ≤ 1. Either no two values of t give the same point (x, y), when the curve is open and is called a Jordan arc, or there is just one pair of values, 0 and 1, of t giving the same point and the curve is closed. Curve, throughout this paper, means Jordan curve.

† For the proofs, which are long and rather delicate, see e.g. von Kerékjártó, Vorlewmgen über Topologie, pp. 59-67.