Kobon Triangle


Kobon Fujimura asked for the largest number N(n) of nonoverlapping triangles that can be constructed using n lines (Gardner 1983, p. 170). A Kobon triangle is therefore defined as one of the triangles constructed in such a way. The first few terms are 1, 2, 5, 7, 11, 15, 21, ... (OEIS A006066).

It appears to be very difficult to find an analytic expression for the nth term, although Saburo Tamura has proved an upper bound on N(n) of |_n(n-2)/3_|, where |_x_| is the floor function (Eppstein). For n=2, 3, ..., the first few upper limits are therefore 2, 5, 8, 11, 16, 21, 26, 33, ... (OEIS A032765).


A. Wajnberg (pers. comm., Nov. 18, 2005) found a configuration for n=10 containing 25 triangles (left figure). A different 10-line, 25-triangle construction was found by Grünbaum (2003, p. 400), and a third configuration is referenced by Honma. The upper bound on n=10 means that the maximum must be either 25 or 26 (but it is not known which). Two other distinct solutions were found in 1996 by Grabarchuk and Kabanovitch (Kabanovitch 1999, Pegg 2006).


Honma illustrates an 11-line, 32-triangle configuration, where 33 triangles is the theoretical maximum possible. Another solution was found by Kabanovitch (1999; Pegg 2006), who also found a 12-line, 38-triangle configuration (upper bound is 40), and a 13-line 47-triangle configuration (which meets the upper bound of 47 triangles).


T. Suzuki (pers. comm., Oct. 2, 2005) found the above configuration for n=15, which is maximal since it satisfies the upper bound of N(15)=65.

Further study has found configurations for 14 lines and 53 triangles (upper bound is 56), 16 lines and 72 triangles (74), and 17 lines and 85 triangles, a new solution matching the upper bound (Clément and Bader 2007).

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Clément, G. and Bader, J. "Tighter Upper Bound for the Number of Kobon Triangles." Dec. 21, 2007., D. "Kabon Triangles.", M. Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 170-171 and 178, 1983.Grünbaum, B. Convex Polytopes, 2nd ed. New York: Springer-Verlag, p. 400, 2003.Honma, S. "Kobon Triangles.", V. "Kobon Triangle Solutions." Sharada (Charade, publication of the Russian puzzle club Diogen) 6, 1-2, June 1999.Pegg, E. Jr. "Math Games: Kobon Triangles." Feb. 8, 2006., N. J. A. Sequences A006066/M1334 and A032765 in "The On-Line Encyclopedia of Integer Sequences."

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Kobon Triangle

Cite this as:

Weisstein, Eric W. "Kobon Triangle." From MathWorld--A Wolfram Web Resource.

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