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The No-Three-In-Line Problem

Published online by Cambridge University Press:  20 November 2018

Richard K. Guy  and
Patrick A. Kelly
Richard K. Guy
Affiliation:
University of Calgary Alberta
Patrick A. Kelly
Affiliation:
University of Calgary Alberta
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Let Sn be the set of n2 points with integer coordinates n (x, y), 1 ≤ x, y <n. Let fn be the maximum cardinal of a subset T of Sn such that no three points of T are collinear. Clearly fn < 2n.

For 2 ≤ n ≤ 10 it is known ([2], [3] for n = 8, [ 1] for n = 10, also [4], [6]) that fn = 2n, and that this bound is attained in 1, 1, 4, 5, 11, 22, 57, 51 and 156 distinct configurations for these nine values of n. On the other hand, P. Erdös [7] has pointed out that if n is prime, fn ≥ n, since the n points (x, x2) reduced modulo n have no three collinear. We give a probabilistic argument to support the conjecture that there is only a finite number of solutions to the no-three-in-line problem. More specifically, we conjecture that

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 4 , October 1968 , pp. 527 - 531
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Acland-Hood, , Bull. Malayan Math. Soc, 0(1952–53) E 11–12.Google Scholar
2. Ball, , Rouse, W.W., Mathematical recreations and essays (11th edition, revised by Coxeter, H.S.M., Macmillan, 1939) p. 189.Google Scholar
3. Dudeney, H.E., Amusements in mathematics (Nelson, 1917) pp. 94, 222.Google Scholar
4. Guy, Richard K., Bull. Malayan Math. Soc., 0 (1952–53) E 22.Google Scholar
5. Hardy, G.H. and Wright, E.M., Introduction to the theory of numbers (4th edition, Oxford, 1960).Google Scholar
6. Kelly, P. A., The use of the computer in game theory (Master's Thesis, University of Calgary, 1967).Google Scholar
7. Roth, K.F., On a problem of Heilbronn, J. Lond. Math. Soc, 26 (1951) 198–204, esp. p. 204.Google Scholar