## Abstract

In 1821, John Jackson published this mathematical conundrum in a book of problems called *Rational Amusement for Winter Evenings* [4]. These days, verse is not as popular, and a modern-day puzzle poser might even dispense with the trees, saying: Arrange nine points on a plane so that there occur ten rows of three points. When a mathematician encounters such a problem, he feels a natural urge to generalize it and then wants to make it more precise. This leads to the following version: Given a positive integer *p*, how can *p* points (*p ≥* 3) be arranged on a plane, no four in a straight line, so that the number of straight lines with three points on them is maximized? We will call this maximal number of lines *l*(*p*).

## Keywords

Parametric Representation Projective Transformation Artful Computer Program Winter Evening Natural Urge## Preview

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## References

- 1.Burr, S. A.; Grünbaum, B.; and Sloane, N. J. A. 1974. The Orchard Problem.
*Geometriae Dedicata*2: 397–424.CrossRefGoogle Scholar - 2.Gardner, M. 1976. Mathematical Games.
*Scientific American*, 102–109.Google Scholar - 3.Grünbaum, B. 1972.
*Arrangements and Spreads*. Providence, R.I.: Amer. Math. Soc.Google Scholar - 4.Jackson, J. 1821.
*Rational Amusement for Winter Evenings*. London: Longman, Hurst, Rees, Orme, and Brown.Google Scholar - 5.Kelley, L. M., and Moser, W. O.J. 1958. On the Number of Ordinary Lines Determined by
*n*Points,*Canad. J. Math.*10: 210–219.CrossRefGoogle Scholar - 6.Sylvester, J. J. 1886. Problem 2572.
*Math Questions from the Educational Times*45: 127–128.Google Scholar