The Honeycomb/Rabbit Problem: Enter Fibonacci

The Rabbit Problem is a complicated one. The problem is set up by staring with two rabbits: one male and one female. At the end of every month, they can reproduce: one male and one female. The reproduction starts happening to this pattern:

After a year, the amount of bunnies that is produced is 233.

WAIT, HEY LOOK! IT’S THE FIBONACCI SEQUENCE!

The Fibonacci Sequence is the sequence of numbers where the sum of the previous two numbers is the next number in the series. So, 0+1 = 1, 1+1 = 2, 1+2 = 3, 2+3 = 5, and so on. This sequence goes on forever and even occurs in nature! Cool right?

Now what does this have to do with honeycombs?

As the question states, you can either start at cell 1 or cell 2. To get to cell 1, there is only one way. To get to cell 2, there are two ways, to get to cell three, there are 3 ways. Guess what, to get to cell 4, THERE WILL BE 5 WAYS. Freaky stuff right? I thought so too.

So the question asks, how many paths are there to cell n? The number of paths to cell n is equal to the sum of paths to cell n-1 and n-2. Thanks to this website, we can find an actual equation for this nth path, which is:

fb02

This formula is known as Binet’s Formula. 

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