A006066 Kobon triangles: maximal number of nonoverlapping triangles that can be formed from n lines drawn in the plane. (Formerly M1334)

0, 0, 1, 2, 5, 7, 11, 15, 21

Offset

1,4

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? 26 [Grünbaum]

11 32? 33 [See link below]

12 ? 40

13 47 47 [Kabanovitch]

14 >= 53 56 [Bader]

15 65 65 [Suzuki]

16 >=72 74 [Bader]

17 85 85 [Bader]

18 >= 93 96 [Bader]

19 >= 104 107 [Bader]

20 >= 115 120 [Bader]

21 >= 130 133 [Bader]

22 ? 146

23 ? 161

24 ? 176

25 ? 191

26 ? 208

27 ? 225

28 ? 242

29 ? 261

30 ? 280

31 ? 299

32 ? 320

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

References

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

Table of n, a(n) for n=1..9.

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]

Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007.

Gilles Clement and Johannes Bader, Tighter Upper Bound for the Number of Kobon Triangles, Unpublished, 2007.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, Title? (A related site)

S. Honma, Title? (A related site)

S. Honma, Illustration showing a(11)>=32

S. Honma, Title? (A related site)

S. Honma, Title? (A related site)

S. Honma, Title? (A related site)

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.

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.

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

Example

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

Crossrefs

Sequence in context: A216094 A184857 A032616 * A084935 A239072 A317242

Adjacent sequences: A006063 A006064 A006065 * A006067 A006068 A006069

Keyword

nonn,hard,more

Author

N. J. A. Sloane

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".

Status

approved