# Floor Functions

Every Sunday afternoon I spend a happy hour in a cafe with Desmond, a student preparing to study Maths at university. We struggle in the darkness, wading through difficult problems. It is beneficial for both of us. This week, a problem from UKMT (3.5 hours, 4 questions to think about).

Progression of the problem:

1. Guess it has something to do with odds and evens, and try that.
2. Give up
3. Calculate the first few values by hand
4. Create a graph of the function on Desmos
5. Realise that the number of factors of n is somehow important
6. Claim that  Twin Prime Conjecture and Mersenne Prime Conjecture are both true
7. Get sad when our phone tells us neither has been proved yet
8. Talk lots about numbers being dragged up or down

I really enjoyed applying the skills I had been trying to teach at school – chunking the problem, drawing a picture, making links. We started to understand the problem a bit, but definitely were not near to a solution. See strategies at end of this post.

Final stage: ask girlfriend, who happens to be doing a PHD in Maths, to solve it for you. In her words:

“I thought I would want to use numbers that I could understand the factors easily. I realized that if I understand the factors of n, then I don’t know anything about the factors of n+1. So I could do it in a straightforward way.

And your graph showed that it increased on a large scale, but not on a small scale. So I guessed, and knew I would want to approximate above and below”

Knowing to play around with powers of 2 shows great intuition, learnt from many years of practise.

Problem solving at School 21: