A hundred lockers in a row, all shut. Student 1 opens every one, student 2 changes every second, student 3 every third, and so on to student 100. When the corridor goes quiet, only a surprising few are open — which ones, and how many?
Every other counting sheet here asks for a number you could, in principle, reach by listing. This one asks for one you reach by NOT listing. Locker L is touched once by every student whose number divides L, so it ends open exactly when L has an odd number of divisors. Divisors come in pairs — d with L ÷ d — so the count is even, unless a divisor pairs with itself. That happens only when L is a perfect square. The open lockers are 1, 4, 9, 16, …, and the answer is however many squares fit.
Why this sheet works
The first answer almost everyone gives is zero, and it is worth letting happen. The pairing argument is real and persuasive — divisors come in pairs, so surely every locker is flipped an even number of times and they all end shut — and it is exactly right except for the one case it forgets. Committing to "all shut" and then finding the single locker that breaks it is most of the lesson: it is the difference between a rule and a rule whose exception is understood.
A row of lockers, and every student flips a share of them. Say how many end open, and why.
1A school corridor has 68 lockers in a row, numbered 1 to 68, all shut. Student 1 opens every locker. Student 2 changes every second locker (2, 4, 6, …). Student 3 changes every third, and so on up to student 68 — each student k changes every locker whose number k divides.In the end, how many lockers are left open?Why
2A hall has 55 lamps in a line, numbered 1 to 55, all off. Switch 1 flips every lamp. Switch 2 flips every second lamp, switch 3 every third, and so on up to switch 55 — switch k flips every lamp whose number k divides.In the end, how many lamps are left on?Why
Locker doorsUp to 100 lockers
Name: Date:
A row of lockers, and every student flips a share of them. Say how many end open, and why.
3A hall has 87 lamps in a line, numbered 1 to 87, all off. Switch 1 flips every lamp. Switch 2 flips every second lamp, switch 3 every third, and so on up to switch 87 — switch k flips every lamp whose number k divides.In the end, how many lamps are left on?Why
4A long hallway has 56 doors, numbered 1 to 56, all closed. Person 1 opens every door. Person 2 changes every second door, person 3 every third, and so on to person 56 — person k changes every door whose number k divides.In the end, how many doors are left open?Why
Locker doors · Up to 100 lockersAnswer key
1. Each locker numbered L is changed once for each divisor of L. Divisors come in pairs (d with L ÷ d), an even count, so a locker ends shut unless a divisor pairs with itself — that is, unless L is a perfect square. The lockers left open are the perfect squares up to 68: 1, 4, 9, 16, …, 64. The largest is 8 × 8 = 64, so there are 8 of them and 8 lockers stay open.2. Each lamp numbered L is changed once for each divisor of L. Divisors come in pairs (d with L ÷ d), an even count, so a lamp ends off unless a divisor pairs with itself — that is, unless L is a perfect square. The lamps left on are the perfect squares up to 55: 1, 4, 9, 16, …, 49. The largest is 7 × 7 = 49, so there are 7 of them and 7 lamps stay on.3. Each lamp numbered L is changed once for each divisor of L. Divisors come in pairs (d with L ÷ d), an even count, so a lamp ends off unless a divisor pairs with itself — that is, unless L is a perfect square. The lamps left on are the perfect squares up to 87: 1, 4, 9, 16, …, 81. The largest is 9 × 9 = 81, so there are 9 of them and 9 lamps stay on.4. Each door numbered L is changed once for each divisor of L. Divisors come in pairs (d with L ÷ d), an even count, so a door ends closed unless a divisor pairs with itself — that is, unless L is a perfect square. The doors left open are the perfect squares up to 56: 1, 4, 9, 16, …, 49. The largest is 7 × 7 = 49, so there are 7 of them and 7 doors stay open.
