## Distances in a bee's honeycomb

### Problem 354

Published on Sunday, 16th October 2011, 07:00 am; Solved by 249Consider 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` ≤ 5·10^{11} such that B(`L`) = 450.