A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane.

0, 0, 1, 2, 5, 7, 11, 15, 21, 25, 32, 38, 47

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• hard
• more
• changed

Comments

The known values a = a(n) and upper bounds U (usually A032765(n)) with name of discoverer of the arrangement when known are as follows:

n a U [Found by]

------------------------------

1 0 0

2 0 0

3 1 1

4 2 2

5 5 5

6 7 7

7 11 11

8 15 16

9 21 21

10 25 25 [Grünbaum]

11 32 33 [32-triangle solutions found by Honma and Kabanovitch; proved maximal by Savchuk 2025]

12 38 38 [Kabanovitch]

13 47 47 [Kabanovitch]

14 >= 53 54 [Bader]

15 65 65 [Suzuki]

16 72 72 [Bader]

17 85 85 [Bader]

18 >= 93 94 [Bader]

19 107 107 [Wood]

20 >= 116 117 [Wood]

21 133 133 [Savchuk]

22 >= 143 144 [Savchuk]

23 161 161 [Savchuk]

24 >= 172 173 [Savchuk]

25 191 191 [Bartholdi]

26 ? 205

27 225 225 [Savchuk]

28 ? 239

29 261 261 [Bartholdi]

30 ? 276

31 299 299 [Wood]

32 ? 316

33 341 341 [Bartholdi]

Ed Pegg's web page gives the upper bound for a(6) as 8. But by considering all possible arrangements of 6 lines - the sixth term of A048872 - one can see that 8 is impossible. - N. J. A. Sloane, Nov 11 2007

Although they are somewhat similar, this sequence is strictly different from A084935, since A084935(12) = 48 exceeds the upper bound on a(12) from A032765. - Floor van Lamoen, Nov 16 2005

The name is sometimes incorrectly entered as "Kodon" triangles.

Named after the Japanese puzzle expert and mathematics teacher Kobon Fujimura (1903-1983). - Amiram Eldar, Jun 19 2021

Examples

			a(17) = 85 because the a configuration with 85 exists meeting the upper bound.
		

References

• Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, "On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles", 2008, in Goodman, Jacob E.; Pach, János; Pollack, Richard (eds.), Surveys on Discrete and Computational Geometry: Proceedings of the 3rd AMS-IMS-SIAM Joint Summer Research Conference "Discrete and Computational Geometry—Twenty Years Later" held in Snowbird, UT, June 18-22, 2006, Contemporary Mathematics, vol. 453, Providence, Rhode Island: American Mathematical Society, pp. 105-116, doi:10.1090/conm/453/08797, ISBN 978-0-8218-4239-3, MR 2405679
• Martin Gardner, Wheels, Life and Other Mathematical Amusements, Freeman, NY, 1983, pp. 170, 171, 178. Mentions that the problem was invented by Kobon Fujimura.
• Branko Grünbaum, Convex Polytopes, Wiley, NY, 1967; p. 400 shows that a(10) >= 25.
• Viatcheslav Kabanovitch, Kobon Triangle Solutions, Sharada (Charade, by the Russian puzzle club Diogen), pp. 1-2, June 1999.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Johannes Bader, Kobon Triangles.
• Johannes Bader, Kobon Triangles. [Cached copy, with permission, pdf format]
• Johannes Bader, Illustration showing a(17)=85, Nov 28 2007.
• Johannes Bader, Illustration showing a(17)=85, Nov 28 2007. [Cached copy, with permission]
• Nicolas Bartholdi, Jérémy Blanc, and Sébastien Loisel, On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles, arXiv:0706.0723 [math.CO], 2007.
• Nicolas Bartholdi, Jérémy Blanc, Sébastien Loisel, and Pavlo Savchuk, Illustration showing a(33) = 341, 2008.
• Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007.
• Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007. [Cached copy, with permission]
• Martin Gardner, Letter to N. J. A. Sloane, Jun 20 1991.
• S. Honma, 三角形の最大数
• S. Honma, Illustration showing a(10)>=25
• S. Honma, Illustration showing a(11)>=32
• S. Honma, n本の直線でn(n-2)/3個の三角形が出来る条件についての考察: part 1, part 2, part 3.
• Ed Pegg, Jr., Kobon triangles, 2006.
• Ed Pegg, Jr., Kobon Triangles, 2006. [Cached copy, with permission, pdf format]
• Luis Felipe Prieto-Martínez, A list of problems in Plane Geometry with simple statement that remain unsolved, arXiv:2104.09324 [math.HO], 2021.
• Pavlo Savchuk, Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straightening, arXiv:2507.07951 [math.CO], 2025. See pp. 1, 17.
• Pavlo Savchuk, Illustration showing a(21) = 133
• Pavlo Savchuk, Illustration showing a(23) = 161
• Pavlo Savchuk, Corresponding pseudo-line arrangement for the optimal n=23 solution
• Pavlo Savchuk, Illustration showing a(27) = 225
• Pavlo Savchuk, Illustration showing a(22) >= 143
• Pavlo Savchuk, Illustration showing a(24) >= 172
• N. J. A. Sloane, Illustration for a(5) = 5 (a pentagram)
• Alexandre Wajnberg, Illustration showing a(10) >= 25. [A different construction from Grünbaum's]
• Eric Weisstein's World of Mathematics, Kobon Triangle.
• Kyle Wood, Illustration showing a(19) = 107
• Kyle Wood, Illustration showing a(20) >= 116
• Kyle Wood, Illustration showing a(31) = 299

Formula

An upper bound on this sequence is given by A032765.

For any odd n > 1, if n == 1 (mod 6), a(n) <= (n^2 - (2n + 2))/3; in other odd cases, a(n) <= (n^2 - 2n)/3. For any even n > 0, if n == 4 (mod 6), a(n) <= (n^2 - (2n + 2))/3, otherwise a(n) <= (n^2 - 2n)/3. - Sergey Pavlov, Feb 11 2017

The upper bound for even n can be improved: floor(n(n-7/3)/3), proven by Bartholdi et. al.

Extensions

a(15) = 65 found by Toshitaka Suzuki on Oct 02 2005. - Eric W. Weisstein, Oct 04 2005

Grünbaum reference from Anthony Labarre, Dec 19 2005

Additional links to Japanese web sites from Alexandre Wajnberg, Dec 29 2005 and Anthony Labarre, Dec 30 2005

A perfect solution for 13 lines was found in 1999 by Kabanovitch. - Ed Pegg Jr, Feb 08 2006

Updated with results from Johannes Bader (johannes.bader(AT)tik.ee.ethz.ch), Dec 06 2007, who says "Acknowledgments and dedication to Corinne Thomet".

a(11)-a(13) from Eric W. Weisstein, Jul 26 2025