{"app":{"locale":"en-GB"},"problem":{"content":"

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

$\"p354_bee_honeycomb.png\"$

One particular cell is occupied by the queen bee.
For a positive real number $L$, let $\\text{B}(L)$ 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, $\\text{B}(\\sqrt 3)=6$, $\\text{B}(\\sqrt {21}) = 12$ and $\\text{B}(111\\,111\\,111) = 54$.

Find the number of $L \\le 5 \\times 10^{11}$ such that $\\text{B}(L) = 450$.

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