5th Grade · Ages 10–11
The Odd-Factor Year
Ten is when a child can be shown that the right question is a different question — and can feel the click when the problem changes shape.
- 1 Hard Thing
- 5 days · 105 minutes
- 3 unmarkable questions
The Hard Thing · about a grade above
The Hundred Lockers
A hallway has 100 lockers, all shut. Student 1 walks past and opens every locker. Student 2 changes every 2nd locker — shutting the open ones. Student 3 changes every 3rd. It goes on all the way to student 100. When it is over, which lockers are open?
Stuck for ten minutes? Good. Open one hint, not three.
1Hint 1
Do not try to run all 100. Do the first ten by hand and write down which are open.
2Hint 2
Locker 12 gets touched by students 1, 2, 3, 4, 6 and 12. Where does that list come from?
3Hint 3
A locker ends up open if it was touched an odd number of times. So: which numbers have an odd number of factors?
The answer — only after you have written something down
Ten lockers — 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100. The perfect squares. Factors normally come in pairs (12 = 1×12 = 2×6 = 3×4, so six of them), which means an even number of touches and a shut locker. A square is the exception: 36 = 6×6, and that 6 only counts once. Odd number of touches, so it ends open.
The genius move
“Stop asking what happens. Start asking how many times.”
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
The Hundred Lockers. No spreadsheet.
25 min - TuesdayBuild
Draw 100 squares and actually run the first 25 lockers. Watching the pattern arrive is not the same as being told it.
25 min - WednesdayRead up
Find out what a prime number is, if you do not know — and what a perfect number is, if you do.
20 min - ThursdayTeach
Teach why 36 behaves differently from 12. Get that across and you have taught the whole thing.
15 min - FridayBreak
How many lockers would be open out of 1,000? Answer it without drawing 1,000 squares.
20 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
Why do squares behave differently from every other number?
- ?2
Is there a largest prime number?
- ?3
Could you tell whether a number is a square without working out its square root?
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
100 lockers, and factors counted.
Next year
Factor trees, and why every number breaks into primes in exactly one way.
The year after
The spacing of the squares — 1, 4, 9, 16 — and the odd numbers hiding in the gaps.
What lives on the desk this year
- Squared paper, a lot of it
- A hundred-square on the wall
- Coloured pencils, one colour per student in the puzzle
- A calculator you are only allowed to check with
- A list of the numbers you have decided are interesting
The rest of 5th Grade
One hard thing a day is the sharp end. It sits on top of an ordinary year.
Last updated: September 2026