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).
Your aid I want, nine trees to plant
In rows just half a score;
And let there be in each row three.
Solve this: I ask no more.
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References
Burr, S. A.; Grünbaum, B.; and Sloane, N. J. A. 1974. The Orchard Problem. Geometriae Dedicata 2: 397–424.
Gardner, M. 1976. Mathematical Games. Scientific American, 102–109.
Grünbaum, B. 1972. Arrangements and Spreads. Providence, R.I.: Amer. Math. Soc.
Jackson, J. 1821. Rational Amusement for Winter Evenings. London: Longman, Hurst, Rees, Orme, and Brown.
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.
Sylvester, J. J. 1886. Problem 2572. Math Questions from the Educational Times 45: 127–128.
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© 1981 Wadsworth International
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Burr, S. (1981). Planting Trees. In: Klarner, D.A. (eds) The Mathematical Gardner. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-6686-7_11
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DOI: https://doi.org/10.1007/978-1-4684-6686-7_11
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