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Kobon Triangle

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KobonTriangles

A Kobon triangle is a triangular region in an arrangement of straight lines. The Kobon triangle problem asks for the largest number N(n) of nonoverlapping triangles that can be constructed using n lines (Fujimura 1978; Gardner 1983, p. 170). While it appears to be very difficult to find an analytic expression for the general nth term, upper bounds are known. The first few known terms for n=3, 4, ... are 1, 2, 5, 7, 11, 15, 21, 25, 32, 38, 47, ... (OEIS A006066).

KobonTriangle10

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. Two other distinct solutions were found in 1996 by Grabarchuk and Kabanovitch (Kabanovitch 1999, Pegg 2006).

Kobon11-12-13

Honma illustrated an 11-line, 32-triangle configuration and another solution was found by Kabanovitch (1999; Pegg 2006). While 32-triangle solutions (one less than known upper bound as discussed below) had long been known, maximality was finally proved by Savchuk (2025).

Kabanovitch (1999) also found a 12-line, 38-triangle configuration, and a 13-line 47-triangle configuration (Pegg 2006).

KobonTriangle15

T. Suzuki (pers. comm., Oct. 2, 2005) found the above configuration for n=15.

A number of other configurations that are either maximal or the best known are summarized and linked to on the page for the sequence OEIS A006066 in the Online Encyclopedia of Integer Sequences. Savchuk (2025) enumerated all possible Kobon arrangements for n=3, 5, 9, 15, and 17 and also found new optimal Kobon arrangements for 23 and 27 lines.

The simplest bound was found by Saburo Tamura, who showed that for n>=4,

N(n)<=|_(n(n-2))/3_|
(1)

(Eppstein, Bartholdi et al. 2007, Clément and Bader 2007), where |_x_| is the floor function. Clément and Bader (2007) tightened the bound by one in the case of n=0,2 (mod 6), obtaining

N(n)<={|_(n(n-2))/3_|-1 for n=0,2 (mod 6); |_(n(n-2))/3_| otherwise
(2)
={(n(n-2))/3-1 for n=0,2 (mod 6); (n^2-2n-2)/3 for n=1,4 (mod 6); (n(n-2))/3 for n=3,5 (mod 6)
(3)

(where the above expression (n^2-2n-2)/3 was incorrectly written as (n-1)^2/3 in the paper). For simple arrangements, in which every pair of lines intersects and no three meet at one point, Bartholdi et al. (2007) proved that the maximum number N_s(n) of triangular regions for even n satisfies

N_s(n)<=|_n/3(n-7/3)_|.
(4)

An arrangement attaining an upper bound valid for its class is maximal within that class. The 11-line configuration does not attain the displayed bound of 33, and its optimality instead follows from Savchuk (2025).

These upper bounds for N(n) and, in the Bartholdi et al. row, N_s(n) are summarized in the following table.

nOEIS34567891011121314151617181920
Tamura boundA0327651258111621263340475665748596107120
Clément and Bader bound1257111521263339475565748595107119
Bartholdi et al. bound27152538547294117

In addition, exact values are known for some special forms of n, with t a nonnegative integer, as summarized in the following table (Bartholdi et al. 2007).

nN(n)
2·2^t+1(n(n-2))/3
14·2^t+1(n(n-2))/3
6·2^t+1(n(n-2)-2)/3

Known exact values of N(n) for n<50 are summarized in the following table.

nN(n)commentreference
312·2^t+1
42upper limit construction
552·2^t+1
67upper limit construction
7116·2^t+1
815upper limit construction
9212·2^t+1
1025upper limit constructionKabanovitch (1999), Grabarchuk, Grünbaum (2003), Wajnberg (pers. comm., 2005)
1132proved maximal by SATSavchuk (2025)
1238upper limit constructionKabanovitch (1999)
13476·2^t+1
156514·2^t+1Suzuki (pers. comm., 2005)
1672upper limit constructionClément and Bader (2007)
17852·2^t+1Clément and Bader (2007)
19107upper limit constructionWood
21133upper limit constructionSavchuk (2025)
23161upper limit constructionSavchuk (2025)
251916·2^t+1
27225upper limit constructionSavchuk (2025)
2926114·2^t+1
31299upper limit constructionWood
333412·2^t+1
497676·2^t+1

See also

Configuration, Projective Plane, Triangle

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References

Bartholdi, N.; Blanc, J.; and Loisel, S. "On Simple Arrangements of Lines and Pseudo-Lines in P^2 and R^2 with the Maximum Number of Triangles." 5 Jun 2007. https://arxiv.org/abs/0706.0723.Clément, G. and Bader, J. "Tighter Upper Bound for the Number of Kobon Triangles." Dec. 21, 2007. https://oeis.org/A006066/a006066.pdf.Eppstein, D. "Kabon Triangles." https://ics.uci.edu/~eppstein/junkyard/triangulation.html.Fujimura, K. "The Tokyo Puzzle Charles." New York: Scribner's Sons, 1978.Gardner, M. Wheels, Life, and Other Mathematical Amusements. New York: W. H. Freeman, pp. 170-171 and 178, 1983.Grünbaum, B. Arrangements and Spreads. Providence, RI: Amer. Math. Soc., 1972.Grünbaum, B. Convex Polytopes, 2nd ed. New York: Springer-Verlag, p. 400, 2003.Honma, S. "Kobon Triangles." http://www004.upp.so-net.ne.jp/s_honma/triangle/triangle2.htm.Kabanovitch, 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. https://www.mathpuzzle.com/MAA/45-Kobon/mathgames_02_08_06.html.Prieto-Martínez, L. F. "A List of Problems in Plane Geometry with Simple Statement That Remain Unsolved." 16 Apr 2021. https://arxiv.org/abs/2104.09324.Savchuk, P. "Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening." 10 Jul 2025. https://arxiv.org/abs/2507.07951.Sloane, 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 Resource. https://mathworld.wolfram.com/KobonTriangle.html

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