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.