Problem 354
Distances in a bee’s honeycomb
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.
蜜蜂蜂巢距离
蜜蜂蜂巢的每个单元都是边长为1的正六边形。
其中一个单元被蜂后占据。
对于正实数L，记B(L)为距离蜂后所在单元的距离为L的单元总数（单元的距离指单元中心之间的距离）；你可以假定蜂巢足够大，允许我们在本题中考虑任意远的距离。
例如，B(√3) = 6，B(√21) = 12以及B(111 111 111) = 54。
有些L ≤ 5·1011满足B(L) = 450，求所有这样的L的数目。