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Encounter at far point. (English) Zbl 1227.00008

Summary: I doubt I will ever tire of browsing through R. K. Guy “Unsolved problems in number theory” [Springer-Verlag, New York (1981; Zbl 0474.10001)]. I can still vividly recall my shock the first time I read Problem D19, “Is there a point all of whose distances from the corners of the unit square are rational?” How is it possible that this is not known! But so it goes when you meet \(\mathbb{Q}\). Here is a grab-bag of results involving configurations of points in the plane separated by rational distances

MSC:

00A08 Recreational mathematics
11D09 Quadratic and bilinear Diophantine equations
51M20 Polyhedra and polytopes; regular figures, division of spaces
97F60 Number theory (educational aspects)

Citations:

Zbl 0474.10001
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References:

[1] Guy, Richard.Unsolved Problems in Number Theory, 3rd ed. Springer, New York, 2004. · Zbl 1058.11001
[2] Kreisel, Tobias and Kurz, Sascha. There are integral heptagons, no three points on a line, no four on a circle.Discrete and Computational Geometry, to appear. DOI 10.1007/S00454-007-9038-6. · Zbl 1145.52010
[3] Kemnitz, A.Punktmengen mit ganzzahligen Abständen. Habilitationsschrift, TU Braunschweig, 1988. · Zbl 1109.52301
[4] Geelen, Jim, Guo, Anjie, and McKinnon, David. Straight line embeddings of cubic planar graphs with integer edge lengths.J. Graph Theory, to appear. · Zbl 1152.05023
[5] Berry, T. G. Points at rational distance from the vertices of a triangle.Acta Arith. 62 (1992)391–398. · Zbl 0758.11019
[6] Skiena, Steven, Smith, Warren, and Lemke, Paul.Reconstructing Sets From Interpoint Distances. Sixth ACM Symposium on Computational Geometry, June 1990 (final version, 1995). · Zbl 1104.68803
[7] Pegg, Ed. Web site http://www.mat.hpuzzle.com, material added 13 November 2006.
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