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 $\text{B}(L)$ 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 $\text{B}(L) = 450$.
To provide a detailed answer to this problem, we first need to have a clear understanding of the geometry of a honeycomb (or hexagonal) lattice.
The hexagonal lattice can be drawn as a grid of equilateral triangles, and we can see that each “layer” around the queen bee forms a hexagon, with each side of the hexagon having a certain number of cells. Thus, finding B(L) which gives the total number of cells at a certain distance, L, from the queen corresponds to determining the number of sides of these hexagons.
The relevant fact here is that the number of cells in a “layer” (which is essentially a hexagon) formed around the queen bee is 6*(n-1) when the distance to any hexagon of this layer from the queen is n (which is an integer). Therefore, B(n) = 6*(n-1).
If B(L) = 450, then according to the formula, we have 6*(n-1) = 450, which gives n = 76, which means if the distance is measured in whole number of cells from the queen, it needs to span 76 cells to create a layer with 450 cells.
However, we need to handle the fact that L is a real number, not necessarily an integer, and find how many such L’s are there, L <= 5*10^11. Given the geometry of a honeycomb, for each disrete 'n', the continuous length from centre to centre, 'L' could be in a range such that [(n-1)sqrt(3), nsqrt(3)] (for n>=2) because a side of an equilateral triangle*sqrt(3) gives the distance from center to center for two consecutive layers.
Thus we want to find all ‘n’ such that (n-1)sqrt(3) <= L <= nsqrt(3), for L <= 5*10^11.
Given 450=6*(n-1), we found n=76, thus the range of L that statisfies the condition would be (75sqrt(3), 76sqrt(3)].
Given L <= 5*10^11, we need to find how many such pairs of n exists.
We need to find 'm' and 'n' (both integers) such that m
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