Solved

# Calculating the maximum

Posted on 2014-12-30

Ok, a silly meme appeared on a well known social network. It seemed so ridiculous that I chimed in and derided it quickly with a throw-away comment. Then my problem solving brain woke up and thought, "hang on, could that figure be correct?", so I checked my thinking again. I immediately noticed my initial scoffing of the meme was incorrect based on the fact that I had misread it, but then I wondered what the real figure was and not being the best mathematician in the world, I decided to see if I could work it out. Then I got stuck, because other than doing a brute force attempt, I couldn't see a simple way of working it out. So, I thought I'd post here. The distilled meme is along the lines of:

"In six years with 2 litters of 6 - 10 puppies a year, one un-spayed female dog and her un-spayed offspring can theoretically produce 67,000 dogs."

There are some serious assumptions here which are:

She only produces females

All females reproduce at the same rate

All puppies survive

If we assume the maximum litter is 10 each, what is the final figure after a 6 year time span, starting with the first birth of 10 puppies from a 2 year old female? We will assume that the first litter arrives dead on the 2 year age for each dog and they reproduce at the stated rate.

My feeling is that there must be some sort of formula that can be used to work this out. It feels somewhat recursive but I got stuck and it's bugging me.

Anyone? If you get a figure, can you show how you worked it out? I'm stumped.