This page explains the exercise template used on this website.
Exercise statement
This section states the exercise/theorem/proposition/whatever.
Hints
This section contains hints for the exercise, so that you can be pointed in the right direction without being told the entire answer. If the book contains hints, I will list them here verbatim, but I may also list some hints of my own.
How to think about the exercise
This section gives you an idea of how one might arrive at a solution. It will contain analogies, simple examples, scratch work, setup, stream of consciousness, etc. These are the kinds of thoughts you want to have when attacking the problem, but you don’t want to write any of these things up when you write out your solution.
I will also sometimes include background context, applications, and further reading in this section.
Model solution
This section contains a “good” solution to the exercise—one that is written in proper mathematical prose, is rigorous, and so forth. This is the sort of thing you would want to turn in, if you were solving the exercise for a class (homework or exam). If you are just learning how to write proofs, this is the sort of output you should aspire toward. See also my Ethics statement.