Orchard-Planting Problem
Also known as the Tree-Planting Problem. Plant 
trees so that there will be 
straight rows with 
treesin each row. The following table gives max() for various . 
is Sloane's A003035 and 
is Sloane's A006065.
| | |  |
| 3 | 1 | -- | -- |
| 4 | 1 | 1 | -- |
| 5 | 2 | 1 | 1 |
| 6 | 4 | 1 | 1 |
| 7 | 6 | 2 | 1 |
| 8 | 7 | 2 | 1 |
| 9 | 10 | 3 | 2 |
| 10 | 12 | 5 | 2 |
| 11 | 16 | 6 | 2 |
| 12 | 19 | 7 | 3 |
| 13 |  |  | 3 |
| 14 |  |  | 4 |
| 15 |  |  |  |
| 16 | 37 |  | |
| 17 |  | |  |
| 18 |  |  | |
| 19 |  |  | |
| 20 |  |  |  |
| 21 |  | ||
| 22 |  | ||
| 23 |  | ||
| 24 |  | ||
| 25 |  |
Sylvester showed that
where
is the Floor Function (Ball and Coxeter 1987). Burr, Grünbaum and Sloane (1974) have shownusing cubic curves that
except for
, 11, 16, and 19, and conjecture that the inequality isan equality with the exception of the preceding cases. For 
,
where
is the Ceiling Function.See also Orchard Visibility Problem
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 104-105 and 129, 1987. Burr, S. A. ``Planting Trees.'' In The Mathematical Gardner (Ed. David Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 90-99, 1981. Dudeney, H. E. Problem 435 in 536 Puzzles & Curious Problems. New York: Scribner, 1967. Dudeney, H. E. The Canterbury Puzzles and Other Curious Problems, 7th ed. London: Thomas Nelson and Sons, p. 175, 1949. Dudeney, H. E. §213 in Amusements in Mathematics. New York: Dover, 1970. Gardner, M. Ch. 2 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, 1977. Gardner, M. ``Tree-Plant Problems.'' Ch. 22 in Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman, pp. 277-290, 1988. Grünbaum, B. ``New Views on Some Old Questions of Combinatorial Geometry.'' Teorie Combin. 1, 451-468, 1976. Grünbaum, B. and Sloane, N. J. A. ``The Orchard Problem.'' Geom. Dedic. 2, 397-424, 1974. Jackson, J. Rational Amusements for Winter Evenings. London, 1821. Macmillan, R. H. ``An Old Problem.'' Math. Gaz. 30, 109, 1946. Sloane, N. J. A. SequencesA006065/M0290and A003035/M0982in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and extended entry in Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.
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