Kobon Fujimura asked for the largest number 
of nonoverlapping triangles that can be constructed using
lines (Gardner 1983, p. 170). A
Kobon triangle is therefore defined as one of the triangles constructed in such a
way. The first few terms are 1, 2, 5, 7, 11, 15, 21, ... (OEIS A006066).
It appears to be very difficult to find an analytic expression for the 
th term, although Saburo Tamura has proved an upper bound on
of 
, where 
is the floor function
(Eppstein). For 
,
3, ..., the first few upper limits are therefore 2, 5, 8, 11, 16, 21, 26, 33, ...
(OEIS A032765).
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. The upper bound on 
means that the maximum must be either 25 or 26 (but it
is not known which). Two other distinct solutions were found in 1996 by Grabarchuk
and Kabanovitch (Kabanovitch 1999, Pegg 2006).
Honma illustrates an 11-line, 32-triangle configuration, where 33 triangles is the theoretical maximum possible. Another solution was found by Kabanovitch (1999; Pegg 2006), who also found a 12-line, 38-triangle configuration (upper bound is 40), and a 13-line 47-triangle configuration (which meets the upper bound of 47 triangles).
T. Suzuki (pers. comm., Oct. 2, 2005) found the above configuration for 
, which is maximal since it satisfies
the upper bound of 
.
Further study has found configurations for 14 lines and 53 triangles (upper bound is 56), 16 lines and 72 triangles (74), and 17 lines and 85 triangles, a new solution matching the upper bound (Clément and Bader 2007).
