 
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, ... (Sloane's 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, ... (Sloane's 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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