Japanese puzzle expert and mathematics teacher Kobon Fujimura asked for the largest number
of nonoverlapping triangles that can be constructed using
lines (Fujimura 1978; Gardner 1983, p. 170). A Kobon
triangle is therefore defined as one of the triangles constructed in such a way.
While it appears to be very difficult to find an analytic expression for the general
th term, upper bounds are known. The first
few known terms for
,
3, ... are 1, 2, 5, 7, 11, 15, 21, 25, 32, 38, 47, ... (OEIS A006066).
A. Wajnberg (pers. comm., Nov. 18, 2005) found a configuration for 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).
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).
T. Suzuki (pers. comm., Oct. 2, 2005) found the above configuration for .
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 ,
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 ,
(1)
|
(Eppstein, Bartholdi et al. 2007, Clément and Bader 2007), where is the floor
function. Clément and Bader (2007) tightened the bound by one in the case
of
,
obtaining
(2)
| |||
(3)
|
(where the above expression was incorrectly written as
in the paper). The bound in the case of even
was tightened further by Bartholdi et
al. (2007) to
(4)
|
Finding an arrangement with equal to any (or in practice, the smallest) of these bounds
is therefore guaranteed to be an maximal configuration, meaning all the configurations
illustrated above are, in fact, maximal.
These upper bounds discussed above are summarized in the following table.
OEIS | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
Tamura bound | A032765 | 1 | 2 | 5 | 8 | 11 | 16 | 21 | 26 | 33 | 40 | 47 | 56 | 65 | 74 | 85 | 96 | 107 | 120 |
Clément and Bader bound | A000000 | 1 | 2 | 5 | 7 | 11 | 15 | 21 | 26 | 33 | 39 | 47 | 55 | 65 | 74 | 85 | 95 | 107 | 119 |
Bartholdi et al. bound | A000000 | 2 | 7 | 15 | 25 | 38 | 54 | 72 | 94 | 117 |
In addition, exact values are known for some special forms of as summarized in the following table (Bartholdi et al.
2007).
Known exact values of
for
are summarized in the following table.
comment | reference | ||
3 | 1 | ||
4 | 2 | upper limit construction | |
5 | 5 | ||
6 | 7 | upper limit construction | |
7 | 11 | ||
8 | 15 | upper limit construction | |
9 | 21 | ||
10 | 25 | upper limit construction | Kabanovitch (1999), Grabarchuk, Grünbaum (2003), Wajnberg (pers. comm., 2005) |
11 | 32 | proved maximal by SAT | Savchuk (2025) |
12 | 38 | upper limit construction | Kabanovitch (1999) |
13 | 47 | ||
15 | 65 | Suzuki (pers. comm., 2005) | |
16 | 72 | upper limit construction | Clément and Bader (2007) |
17 | 85 | Clément and Bader (2007) | |
19 | 107 | upper limit construction | Wood |
21 | 133 | upper limit construction | Savchuk (2025) |
23 | 161 | upper limit construction | Savchuk (2025) |
25 | 191 | ||
27 | 225 | upper limit construction | Savchuk (2025) |
29 | 261 | ||
31 | 299 | upper limit construction | Wood |
33 | 341 | ||
49 | 767 |