Problem
Lockers left open by divisor-count parity
Ten lockers numbered 1 to 10 all start closed. Student n flips every locker whose number is a multiple of n. Ten students walk past in turn. Find which lockers end up open.
Reasoning · Grade 5-1 Applications of Prime Factorization
Your answer
Watch out Divisors seem to come in pairs, but a square repeats one and lands on an odd count.
How to solve
Strategy Make a Systematic List — Chasing all ten students one after another means keeping track of a hundred visits, which is easy to lose count of. Instead I follow one locker at a time and ask a much smaller question: which students touch this particular locker? Student k touches locker n exactly when k is a divisor of n, so the number of flips a locker receives is simply its number of divisors. Once that is set up, a systematic list of divisor counts for 1 to 10 answers everything, and looking at which counts come out odd reveals the pattern — the open lockers are precisely the square numbers, which I can then explain by pairing divisors up.
1STEP 1
Work out who touches a given locker
The students who visit are the divisors of that number.
6 = 1 × 6 = 2 × 3 → students 1, 2, 3, 6
Following one locker instead of all ten students turns a long simulation into a divisor question, and finding the divisors of a number under 10 is something a fourth grader can do by trying 1, 2, 3, ... in order.
4.OA.B.4Solve An Easier Related Problem2STEP 2
Count the flips for every locker
Count the flips for each locker.
1{:}1, 2{:}2, 3{:}2, 4{:}3, 5{:}2, 6{:}4, 7{:}2, 8{:}4, 9{:}3, 10{:}4
Ten short divisor lists replace a hundred separate flips, and writing them all out in order means no locker is forgotten and none is counted twice.
4.OA.B.4Make A Systematic List3STEP 3
Decide open or closed from the flip count
Starting closed, it needs an odd number of flips.
closed → open → closed → open …
Flips come in pairs that cancel each other, so only whether the count is odd or even matters — the exact number of flips is irrelevant.
2.OA.C.3Look For A Pattern4STEP 4
Pick out the odd counts
The odd counts are lockers 1, 4 and 9.
1, 3, 3 are odd → lockers 1, 4, 9
Reading a finished list for odd numbers is a simple sweep, and it is far safer than trying to remember the state of ten doors while ten students go past.
2.OA.C.3Make A Systematic List5STEP 5
See why 1, 4 and 9 are special
Only these are squares, so only they have odd divisor counts.
1 = 1 × 1, 4 = 2 × 2, 9 = 3 × 3
Pairing each divisor with its partner is the same trick as pairing up socks: everything matches up two by two unless one item is its own match, and that lone unmatched divisor is what leaves the door open.
4.OA.B.4Look For A PatternLockers 1, 4 and 9 stay open because a square number is the only kind with an odd number of divisors.
Why?
Divisors pair off two by two, and the only way one is left unmatched is when a number is its own partner.
🧱Divisors come in pairsEach divisor hands over its partner for free, so divisors pair off two by two unless one is its own partner.
Why?
Each flip swaps the door between open and closed, so only whether the count of flips is odd or even decides the end state.
🧱Even and odd behave predictablyTwice any whole number is even and adding an odd amount always swaps even with odd, so a total's parity is settled by its parts before anything is computed.
6STEP 6
Spot-check by replaying the students
Replaying locker 4 shows it really is open.
locker 4: open → closed → open
Replaying one open locker and one closed locker by hand tests both sides of the rule, which is enough to trust the divisor-count shortcut for all ten.
4.OA.C.5Solve An Easier Related ProblemAnswer
1, 4, 9 lockers
1 × 1, 2 × 2, 3 × 3
The answer is a set of locker numbers, all of them between 1 and 10 as they must be, and only 3 of the 10 lockers finish open — a modest number, which fits the fact that most numbers have an even count of divisors that cancels out. Locker 1 is open because only one student ever touches it, which is obviously right. The result matches the square-number rule exactly, and a direct replay of lockers 4 and 6 confirmed both an open case and a closed case. Nothing here suggests an off-by-one slip.
Takeaway
A door ends up open only if it was flipped an odd number of times, and that happens only for square numbers — 1, 4 and 9 — because every other number's divisors pair up perfectly.
- Work out who touches a given locker
- Count the flips for every locker
- Decide open or closed from the flip count
- Pick out the odd counts
- See why 1, 4 and 9 are special
- Spot-check by replaying the students