Honeybee Family Tree

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Honeybees Solution

Here is what we might get if we label the bees and then continue the pattern. (Of course this is a vastly simplified model of the real world. )

What to notice:

If we keep track of how many bees are in each generation, as well as how many males and females are in each generation, we notice something familiar and surprising.

Just for convenience, let's say that the single bee at the bottom of the diagram is in "Generation 1." His mother, just above him in the diagram, is in Generation 2. And so forth.

Notice that each column contains the Fibonacci sequence! Each number in the Fibonacci sequence is the sum of the two prior numbers in the sequence. The number of bees in any generation is equal to the sum of the number of bees in the two prior generations.

Why is it true?

Every bee has one female parent, so the total number of bees in one generation is equal to the number of female bees in their parents' generation. 

Every one of those female bees also has a male parent, so:

Since the total number of bees in any generation is equal to the number of males plus the number of females, we also have:

Substituting the first two equations into the third gives us:

which is the recursive formula for the Fibonacci sequence.

Thank you to Arushi G. for your help with this solution!