Kobon triangle problem

Kobon triangle problem

The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura. The problem asks for the largest number "N"("k") of nonoverlapping triangles that can be produced by "k" straight line segments.

Saburo Tamura proved that the largest integer not exceeding "k"("k" − 2)/3 provides an upper bound on the maximal number of nonoverlapping triangles realizable by "k" lines. [MathWorld|urlname=KobonTriangle|title=Kobon Triangle] This sequence is captured in the On-Line Encyclopedia of Integer Sequences as OEIS2C|id=A032765. In 2007, a tighter upper bound was found by Johannes Bader and Gilles Clément, by proving that Tamura's upper bound couldn't be reached for any "k" congruent to 0 or 2 (mod 6). [ [http://www.tik.ee.ethz.ch/sop/publicationListFiles/cb2007a.pdf G. Clément and J. Bader. Tighter Upper Bound for the Number of Kobon Triangles. Draft Version, 2007.] ] The maximum number of triangles is therefore one less than Tamura's bound in these cases. Perfect solutions (Kobon triangle solutions yielding the maximum number of triangles) are known for "k" = 3, 4, 5, 6, 7, 8, 9, 13, 15 and 17. [ [http://www.maa.org/editorial/mathgames/mathgames_02_08_06.html Ed Pegg Jr. on Math Games] ] For other "k"-values the Kobon triangle solution numbers are not known. For "k" = 10, 11 and 12, the best solutions known reach a number of triangles one less than the upper bound.

The known Kobon triangle solution numbers are captured in the On-Line Encyclopedia of Integer Sequences as OEIS2C|id=A006066.

Examples

References


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