Socrates and Shoelaces

I have quite some time to think these days. Thinking is a process that is frequently non-linear, arbitrary and sometimes causeless. I rely on tools, mostly mathematical and sometimes probabilistic which shape the next part of the thread. Thinking is exhausting, especially when a degree of conscientiousness is required. It has been suggested, erroneously in my view, that ten thousand hours of practice can turn a novice into an expert, irrespective of innate ability.  Building neural pathways to solve specific problems is not enough. Until we learn and appreciate the value of lateral observation, thinking becomes no more than a mechanistic, hence unrewarding activity, which, dopamine-addicted as we all are, is counter-productive. Great teachers are not those who can manage gigantic classes and convey some information. Great teachers ask individuals, who then deduce partial solutions from which greater understanding comes. It comes as no surprise to learn that tutored students do better by a measurably significant amount than those who sit in large classrooms and are lectured along with a large number of others. It takes intense concentration for individuals to engage appropriately with the material in such a setting with little or no feedback. ‘How am I doing?’ is the perennial question, the answer to which is what every student really wants to know.

For these questions, you need to ask someone else the how-am-I-doing part. There aren’t any right answers, just interesting lines of enquiry. For each of them, you have to ask subsidiary questions, the accuracy of which will determine the quality of your final answer.

First, Fermi estimation. This is the process of coming up with estimates of the correct order of magnitude for various real-world quantities often with little or no hard data. Here are a few examples.

1. How much does a cloud weigh?

2. How many people could fit into the Island of Manhattan?

3. How many piano tuners are there in Chicago? [a classic example- I’ll post the standard solution in a comment – you might like to try to compute the uncertainties in the calculation]

4. If the average temperature of the sea were to rise by a degree, then by how much would thermal expansion cause sea levels to rise?

5. How many molecules from Socrates’s last breath are in the room?

Now for some more difficult ones.

6. How do speed cameras work? How accurate are they likely to be? (The basic technique I’m talking about is taking two photos in quick succession.)

7. Why does a mouse survive a big fall when a human doesn’t? (There are many questions similar to this, such as why elephants have thick legs, ants can carry several times their body weight, etc.)

8. Somebody pours you a cup of coffee but you aren’t yet in a position to drink it. You take milk, and the milk provided is cold. You want your coffee as warm as possible. When should you put in the milk: now, or just before you drink it, or some time in between?

9. You are walking from one end of an airport terminal to the other. The airport has several moving walkways, and you need to stop to tie your shoelace. Assuming you want to get to the other end as quickly as possible, is it better to tie your shoelace while you are on a moving walkway or while you are between walkways?

This question comes from a blog post ofTerence Tao

10. You have a collection of suitcases, boxes and bags of various sizes, shapes and degrees of squashiness. You want to pack them all into the boot of a car and it’s not obvious whether you can. What is the best method to use? If you don’t like the idea of suitcases, try thinking about plastic containers of varying sizes and robustness, some with lids and some without. How can they be fitted into the least and most accessible space?

If these kinds of exercises bewilder you, that's fine. A significant part of our time is spent wondering what to do next. Wondering is what we do best. I sometimes wonder in common with the comedian Mark Russell whether  the rings of Saturn are entirely composed of lost airline luggage.