Problem of the Month(February 2013)

A cluster of triangles is a non-overlapping collection of triangles with the property that it is possible to get from any triangle to any other triangle by passing through edges that are completely shared by two triangles. Of all the clusters of the integer-sided triangles with perimeter equal to n, which cluster fits inside the smallest circle? How about clusters of integer-sided triangles with perimeter no more than n?

Smallest-Known Clusters of Integer-Sided Triangles With Perimeter n
 3r = 1/√3 = .5773+ 5r = 4/√15 = 1.0327+ 6r = 2/√3 = 1.1547+ 7r = 1.5302+
 8r = 9/4√2 = 1.5909+ 9r = 2.5468+ 10r = 12√(251-3√3)/37√5 = 2.2739+ 11r = 2.9634+
 12r = 3.4968+ 13r = 3.7801+(Jon Palin) 14r = 4.2752+(Jon Palin) 15r = 5.1607+(Maurizio Morandi)
 16r = 5.0334+(Maurizio Morandi) 17r = 5.9147+(Joe DeVincentis) 18r = 6.2967+(Joe DeVincentis) 19r = 6.8379+(Joe DeVincentis)
 20r = 7.3534+(Joe DeVincentis) 21r = 8.2535+(Joe DeVincentis) 22r = 8.5766+(Joe DeVincentis) 23r = 9.5137+(Joe DeVincentis)
 24r = 9.8888+(Joe DeVincentis) 25r = 11.3464+(Joe DeVincentis) 26r = 11.7197+(Joe DeVincentis) 27r = 13.2703+(Joe DeVincentis)
 28r = 13.3210+(Joe DeVincentis) 29r = 14.8887+(Joe DeVincentis) 30r = 15.5952+(Joe DeVincentis)

Smallest-Known Clusters of Integer-Sided Triangles With Perimeter No More Than n
 3r = 1/√3 = .5773+ 5r = (√15+√3)/4 = 1.4012+ 6r = 2√(52+9√5)/11 = 1.5441+ 7r = 1.9816+
 8r = 2.2516+(Jon Palin) 9r = 2.8745+(Joe DeVincentis) 10r = 3.2920+(Joe DeVincentis) 11r = 4.0568+(Joe DeVincentis)
 12r = 5.0037+(Joe DeVincentis) 13r = 5.9851+(Joe DeVincentis) 14r = 6.7984+(Joe DeVincentis) 15r = 8.0494+(Joe DeVincentis)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/13/13.