# How to Think Like a Mathematician

You stare at a problem. Nothing happens. The page feels like a locked door with no
handle, and the longer you look the more certain you become that real
problem-solvers would have cracked it by now. That feeling is the lie this guide
takes apart.

Mathematicians are not faster than you. They are not staring at the locked door
either - they are walking around the building trying every window, and they have a
short list of windows worth trying. That list is a craft, not a gift. You can learn
it, the same way you learned to debug code: by having a few reliable moves and the
patience to run them.

## How to read this

Read the phases in order; each one builds on the move before it. Don't rush to the
heuristics in phase 2 before you've sat with phase 1 - the four-step loop is the
frame that makes every other trick land. Try the examples on paper as you go. The
whole point is that these moves only become yours once your own hand has run them on
a problem that was actually stuck.

If the word "math" still makes your shoulders tense, start with
[/guides/why-math-isnt-your-enemy](/guides/why-math-isnt-your-enemy) first. And once
you can find an answer, [/guides/what-a-proof-is](/guides/what-a-proof-is) is how you
make it certain.

## The phases

1. [The loop: understand, plan, do, look back](01-the-loop.md) - the four-step frame every solver runs, named.
2. [The moves: how to actually get an idea](02-the-moves.md) - small cases, patterns, working backwards, invariants.
3. [Stuck is the job: getting unstuck](03-stuck-is-the-job.md) - what to do when nothing comes, and why struggle beats talent.