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Distances in a bee's honeycomb

Problem 354

Published on Sunday, 16th October 2011, 07:00 am; Solved by 263; Difficulty rating: 65%

Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length 1.


One particular cell is occupied by the queen bee.
For a positive real number L, let B(L) count the cells with distance L from the queen bee cell (all distances are measured from centre to centre); you may assume that the honeycomb is large enough to accommodate for any distance we wish to consider.
For example, B(√3) = 6, B(√21) = 12 and B(111 111 111) = 54.

Find the number of L ≤ 5·1011 such that B(L) = 450.