10th Grade · Ages 15–16
The Diagonal Year
Fifteen is old enough for the argument that broke mathematics open — and young enough to be annoyed that nobody mentioned it sooner.
- 1 Hard Thing
- 5 days · 125 minutes
- 3 unmarkable questions
The Hard Thing · about a grade above
Count the Infinite
You can list the whole numbers: 1, 2, 3, … and any one of them turns up eventually. Now try to list every number between 0 and 1. Somebody hands you a list and swears every one of them is on it. Prove they are wrong — and prove it for any list they could possibly hand you.
Stuck for ten minutes? Good. Open one hint, not three.
1Hint 1
You cannot read an infinite list. So do not. Build a number that cannot be on it.
2Hint 2
Write their list as a grid of decimals, one number per row. Look at the diagonal: the 1st digit of the 1st number, the 2nd of the 2nd, and so on down.
3Hint 3
Make a new number by changing every digit on that diagonal.
The answer — only after you have written something down
Take their list and build a number d whose nth digit differs from the nth digit of their nth number. Then d is different from row 1 in the first place, from row 2 in the second, from row n in the nth — so d is on no row at all. But d is a perfectly good number between 0 and 1. Their list was incomplete, and so is every possible list. There are strictly more real numbers than whole numbers. Cantor proved it in 1891 and the argument has never needed fixing.
The genius move
“When you cannot search the list, construct the thing the list must have missed.”
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
Count the Infinite. Sit with it before reading anything.
30 min - TuesdayBuild
Write out a made-up list of five decimals and physically build the diagonal number that defeats it.
25 min - WednesdayRead up
Read what the same argument does to the halting problem. It is the identical move.
25 min - ThursdayTeach
Teach it in five minutes with a grid on paper. If it takes longer, you have not got it yet.
20 min - FridayBreak
Why does the trick fail on the fractions? They CAN be listed — find the listing that does it.
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
Are some infinities bigger than others?
- ?2
Is there a question no computer can ever answer?
- ?3
What would it mean for something to be true but unprovable?
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
The diagonal, drawn by hand.
Next year
The halting problem — the same proof wearing different clothes.
The year after
Gödel's incompleteness: the third outing of the same trick, and the one that ended a century of certainty.
What lives on the desk this year
- Graph paper for the grid
- A reading list you wrote yourself
- One paper you are trying to read that is too hard for you
- A person a decade older who takes you seriously
- A five-year question
The rest of 10th Grade
One hard thing a day is the sharp end. It sits on top of an ordinary year.
Last updated: September 2026