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:
- Guess it has something to do with odds and evens, and try that.
- Give up
- Calculate the first few values by hand
- Create a graph of the function on Desmos
- Realise that the number of factors of n is somehow important
- Claim that Twin Prime Conjecture and Mersenne Prime Conjecture are both true
- Get sad when our phone tells us neither has been proved yet
- 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:
Glimpses of Infinity, Reminders of Reality 