Kobon Triangle
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).
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