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. The next screens hold one hint each.

Hint 1 of 3

Only open it if you need it

1Show hint 1

Do not try to run all 100. Do the first ten by hand and write down which are open.

Hint 2 of 3

Only open it if you need it

2Show hint 2

Locker 12 gets touched by students 1, 2, 3, 4, 6 and 12. Where does that list come from?

Hint 3 of 3

Only open it if you need it

3Show hint 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

Show the answer

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.”

What it is training: Re-describing a problem until it turns into a different, easier problem.

Five days, five muscles

The week

MondayThe Hard Thing25 min
The Hundred Lockers. No spreadsheet.
TuesdayBuild25 min
Draw 100 squares and actually run the first 25 lockers. Watching the pattern arrive is not the same as being told it.
WednesdayRead up20 min
Find out what a prime number is, if you do not know — and what a perfect number is, if you do.
ThursdayTeach15 min
Teach why 36 behaves differently from 12. Get that across and you have taught the whole thing.
FridayBreak20 min
How many lockers would be open out of 1,000? Answer it without drawing 1,000 squares.

Start Monday now

Read the problem once more, out loud, and press start. Solving it is not the measurement.

25:00Sit

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.

?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?

That is the year’s Hard Thing.

Twenty-five minutes a day at the edge of what you cannot do yet.

🎮 Play it — 100 challenges Next year: 6th Grade →The Genius Room

The Odd-Factor Year

The Hundred Lockers

Only open it if you need it

Only open it if you need it

Only open it if you need it

Only after you have written something down

The week

The twenty-five minutes

Three questions you cannot mark

5th Grade · The Hundred Lockers

The ladder

This year

Next year

The year after

What lives on the desk this year

The rest of 5th Grade

All twelve years

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