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 first. And once you can find an answer, /guides/what-a-proof-is is how you make it certain.
The phases
- The loop: understand, plan, do, look back - the four-step frame every solver runs, named.
- The moves: how to actually get an idea - small cases, patterns, working backwards, invariants.
- Stuck is the job: getting unstuck - what to do when nothing comes, and why struggle beats talent.