9th Grade · Ages 14–15
The Contradiction Year
Fourteen is the year to learn the difference between being confident and being certain — and it takes exactly one proof to feel it.
- 1 Hard Thing
- 5 days · 120 minutes
- 3 unmarkable questions
The Hard Thing · about a grade above
Prove That √2 Is Not a Fraction
The square root of 2 is about 1.41421356… Somebody claims it can be written as a fraction — one whole number over another. Prove them wrong. Not 'I checked a lot of fractions'. Prove it.
Stuck for ten minutes? Good. Open one hint, not three.
1Hint 1
You cannot test every fraction, so do not try. Assume they are right and see what breaks.
2Hint 2
Write √2 = a/b, already cancelled as far as it will go. Square both sides.
3Hint 3
You will find that a is even. Write a = 2k and keep going. Watch what happens to b.
The answer — only after you have written something down
Assume √2 = a/b in lowest terms. Then a² = 2b², so a² is even, so a is even — because an odd number squared is always odd. Write a = 2k: then 4k² = 2b², so b² = 2k², so b is even as well. But a and b were in lowest terms and cannot both be even. The assumption breaks, so no such fraction exists.
The genius move
“Assume the opposite and break it. You cannot check infinitely many fractions — but you only have to kill one assumption.”
Five days, five muscles
The week
Never the same muscle two days running. The weekend is not on this list on purpose — being bored is the sixth muscle and it needs two clear days.
- MondayThe Hard Thing
Prove it. Twenty-five minutes of being stuck counts as the work.
30 min - TuesdayBuild
Write the proof out from memory with no notes. If you cannot, you have read it, not learned it.
25 min - WednesdayRead up
Read about the Pythagoreans and what is said to have happened when this got out.
20 min - ThursdayTeach
Teach it to somebody who has never seen a proof. The hard part is explaining why testing fractions is not enough.
20 min - FridayBreak
Run the same proof on √4 and find the exact line where it fails. That line is why 4 is different.
25 min
Start Monday now
Read the problem once more, out loud, and press start. Solving it is not the measurement.
Nothing rung today yet
The twenty-five minutes
Four phases. The middle one is most of it, and it is meant to feel slow.
- 1SitRead the problem again, out loud. Write nothing yet.
- 2AttackTry the wrong thing. It is faster than thinking about the right one.
- 3StuckWrite down the exact place you are stuck. “I don’t get it” is not a place.
- 4LogOne sentence: what you will try tomorrow. Then close the book.
No answer key
Three questions you cannot mark
Ask one at dinner. Do not resolve it. A question that survives the washing-up is doing its job.
- ?1
Was mathematics discovered or invented?
- ?2
Can something be true that nobody can ever prove?
- ?3
Why is 'I checked a thousand cases' not a proof?
Where this goes
The ladder
The same idea, three times, roughly a year apart. This is what depth looks like from the outside.
This year
√2, by contradiction.
Next year
√3 and √5 by the same argument — then the general statement about any number that is not a perfect square.
The year after
Gödel: that there are true statements no proof can reach.
What lives on the desk this year
- Something you actually enjoy writing with
- A proof notebook, separate from everything else
- A wall list of theorems you can prove from memory
- Somebody to be wrong in front of
- One problem you have not solved for six months
The rest of 9th Grade
One hard thing a day is the sharp end. It sits on top of an ordinary year.
Last updated: September 2026