Mathematical Scales

Thanks to the move to the US, my son has a new piano teacher. He is playing at an advanced level, beyond grade 8 (for the UK audience), with pieces by Bach, Mozart and Chopin often ringing out. Yet for the last couple of months he has been taken right back to the basics. Looking again at simple techniques on how fingers hit the keys and going over scales.

I am in love with this idea of training, taking someone who has proved incredibly able in an area and taking them back to the most basic ideas. I started to wonder what the equivalent might be for mathematics. What exercises should we be giving to starting PhD students?What exercises could we ourselves try in order to gain intuition and insight into the basic workings of our subject. I have a first proposal, but am sure there are others? What do you think? Of the idea itself, or of suggestions of possible exercises?

Multiplication Exercise

Multiply all possible pairs of numbers from 1 to 99, that is 4950 different calculations. At a conservative estimate of 120 per hour (most will be a lot quicker than 30s, some will be longer!) that is just over 40 hours work. That could spread quite nicely over a month, maybe two along with other activities. It would be 40 hours of meditation on the most fundamental of mathematical operations, what might come from that?
Other suggestions

A couple of excellent suggestions from commentors in a lively debate on reddit:

1) Teaching, which of course is already a significant part of graduate training in the US, unfortunately less so in the UK (those being the two systems I have worked in).

2) Deep study of proofs, with mention of this beautiful paper of Dykstra.

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