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 25 [Grünbaum]
11 32? 33 [See link below]
12 38 38
13 47 47 [Kabanovitch]
14 >= 53 56 [Bader]
15 65 65 [Suzuki]
16 72 72 [Bader]
17 85 85 [Bader]
18 >= 93 96 [Bader]
19 107 107 [Wood]
20 >= 115 120 [Bader]
21 133 133 [Savchuk]
22 ? 146
23 ? 161
24 ? 176
25 191 191 [Bartholdi]
26 ? 208
27 ? 225
28 ? 242
29 261 261 [Bartholdi]
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).
Bartholdi, Nicolas; Blanc, Jérémy; Loisel, Sébastien (2008), "On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of triangles", 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, arXiv:0706.0723, doi:10.1090/conm/453/08797, ISBN 978-0-8218-4239-3, MR 2405679
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]
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.
Pavlo Savchuk, Illustration showing a(21) = 133
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
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.
EXAMPLE
a(17) = 85 because the a configuration with 85 exists meeting the upper bound.
CROSSREFS
KEYWORD
nonn,hard,more,changed
AUTHOR
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