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果园种植问题

OrchardProblem

果园种植问题（也称为果园问题或植树问题）要求种植 n 棵树，使得存在 r(n,k) 行，每行有 k 棵树。对于 n 个点，寻找三点共线的最大线数的问题归功于 Sylvester（Croft 等人，1991 年，第 159 页）。下表给出了各种 k 和 n 的 max_(k)(r(n,k))。

n\k345
OEISA003035A006065A008997
31----
411--
5211
6411
7621
8721
91032
101252
111662
121973
13[22,24]>=93
14[26,27]>=104
15[31,32]>=12>=6
1637>=15>=6
17[40,42]>=15>=7
18[46,48]>=18>=9
19[52,54]>=19>=10
20[57,60]>=23>=11
21[64,67]
22[70,73]
23[77,81]
24[85,88]
25[92,96]

Sylvester 证明了

r(k=3)>=|_1/6(n-1)(n-2)_|,
(1)

其中 |_x_| 是 向下取整函数（Ball 和 Coxeter，1987 年）。Burr 等人（1974 年）使用三次曲线证明了

r(k=3)>=1+|_1/6n(n-3)_|,
(2)

除了 n=7、11、16 和 19 之外，并推测除了上述情况外，该不等式都成立。对于 n>=4，

r(k=3)<=|_1/3[1/2n(n-1)-[3/7n]]_|,
(3)

其中 [x] 是 向上取整函数。

另请参阅

构型, 欧几里得果园, Grünbaum-Rigby 构型, 果园可见性问题, Sylvester 直线问题, 可见点

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参考文献

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 Gardener (Ed. David Klarner). Boston, MA: Prindle, Weber, and Schmidt, pp. 90-99, 1981.Burr, S. A.; Grünbaum, B.; and Sloane, N. J. A. "The Orchard Problem." Geom. Dedicata. 2, 397-424, 1974.Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 159, 1991.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.Friedman, E. "Tree Planting Problems." http://www.stetson.edu/~efriedma/trees/.Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, pp. 18-20 and 26, 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, Or, A Collection of Above 200 Curious and Interesting Puzzles and Paradoxes Relating to Arithmetic, Geometry, Geography, &c. with Their Solutions, and Four Plates, Designed Chiefly for Young Persons. London: J. and A. Arch, 1821.Macmillan, R. H. "An Old Problem." Math. Gaz. 30, 109, 1946.Sloane, N. J. A. Sequences A003035/M0982, A006065/M0290, and A008997 in "The On-Line Encyclopedia of Integer Sequences."Sloane, N. J. A. and Plouffe, S. Figure M0982 in The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

在 Wolfram|Alpha 中被引用

果园种植问题

请引用为

Weisstein, Eric W. "Orchard-Planting Problem." 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/Orchard-PlantingProblem.html

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