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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 whose sides lie on an arrangement of ''k'' lines. Variations of the problem consider the projective plane rather than the Euclidean plane, and require that the triangles not be crossed by any other lines of the arrangement.〔.〕 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. 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).〔(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.〔(Ed Pegg Jr. on Math Games )〕 For ''k'' = 10, 11 and 12, the best solutions known reach a number of triangles one less than the upper bound. Given a perfect solution with ''k0'' lines, other Kobon triangle solution numbers can be found for all ''ki''-values where : by using the procedure by D. Forge and J. L. Ramirez Alfonsin.〔〔("Matlab code illustrating the procedure of D. Forge and J. L. Ramirez Alfonsin", Retrieved on 9 May 2012. )〕 For example, the solution for ''k0'' = 3 leads to the maximal number of nonoverlapping triangles for ''k'' = 3,5,9,17,33,65,... == Examples == Image:KobonTriangle_3.svg|3 straight lines result in one triangle Image:KobonTriangle_4.svg|4 straight lines Image:KobonTriangle_5.svg|5 straight lines Image:KobonTriangle_6.svg|6 straight lines Image:KobonTriangle_7.svg|7 straight lines 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Kobon triangle problem」の詳細全文を読む スポンサード リンク
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