# Fibonacci Gerbils

We started the evening math circle with this classic problem:

There are 4 apples in a basket. Can you divide them between 4 children in such a way that every child gets an apple, yet 1 apple is left in the basket?

Everyone was stumped. So we ended up role-playing it with an actual basket of actual apples.

Then I gave another classic puzzle:

Using just 6 toothpicks, build 4 triangles of the same size and shape.

I also gave two hints – I showed one way of arranging 9 toothpicks into 4 triangles of the same size and shape (and we briefly discussed what it means for the triangles to be exactly the same size and shape). I also told them that the solution might be easier to build if they work together (this was promptly ignored). Eventually, I had to point out that between the four of them, they produced lots and lots of non-working flat arrangements. After this, the problem was solved very quickly.

We did a few word problems of the type “53 books are placed on 2 shelves. One shelf has 3 more books than the other. How many books are there on each shelf?” The problems came straight from Anna Burago’s book. Anna’s approach is to “first determine what’s common in all unknowns, then… take away the elements that set these quantities apart.”

And then we moved on to the main event – the Fibonacci gerbils. I know, it’s usually the rabbits. And I was going to talk about rabbits. Except on Monday M read me a poem by Jack Prelutsky called “My Gerbil Seemed Bedgraggled”. It was perfect for the topic!

So A. read the poem. Then we discussed gerbils a bit – what kind of animals are they. None of the kids have gerbils. To my surprise, at least one thought that gerbils were a type of birds. Finally, it was time to move to counting the gerbils:

Gerbils mature very quickly – at just one month old they are ready to have babies. (I’m not sure if this is accurate). And each time they have babies, they have exactly two who are ready to have their own babies in one month’s time. So, if the boy from the poem started with one pair of gerbils, how many pairs of gerbils would he have a year later?

Note: we started with a situation in which in Month 1 we have 1 pair of adult gerbils. I explained that a month before that (Month 0), the boy went out to the store and bought a pair of baby gerbils (yes, it’s not in the original poem).

Together, we worked out how many pairs will there be in Month 2 – two pairs. At this point both A and M decided that it was a doubling patternÂ – 1, 1, 2, 4, 8, 16… But after working out the number of pairs in Month 3 – three pairs – this idea was abandoned. K drew a diagram that showed that in Month 4 there would be five pairs – we now had 1, 1, 2, 3, 5. From this K conjectured that the pattern is that of odd numbers. A pointed out that 2 is not an odd number. Back to diagramming.

M was busy drawing something. He said the whole thing reminded him of a tree fractal. And that’s what he was drawing. So we discussed whether his diagram accurately described our findings (it didn’t) and how it could be adjusted.

Eventually, somewhere around Month 6, the rule for the pattern of 1, 1, 2, 3, 5, 8, 13, … was discovered and the following Month’s gerbil count was accurately predicted – 21 pairs. From there, A and E quickly calculated the number of pairs in Month 12. Yep, a whole lot of gerbils!