What makes a number a perfect square

0:00A corridor, a hundred lockers numbered one to a hundred, every one of them shut. A hundred people are going to walk down it, one after another. The first person opens every locker. The second visits every second locker - two, four, six - and changes what they find: open doors they shut, shut doors they open. The third person does the same to every third locker. And so on down to the hundredth person, who touches one door and goes home.

0:34When the last of them has gone, some are open and some shut. Which ones are open? You could find out by walking the corridor; by the end of this we will name them without walking it at all. Forget the other ninety-nine and watch locker six. Person one touches everything, so person one touches it. Person two does, because six is every second locker. Person three does. Person four does not - four, eight, twelve, and six is not on that list - and nor does person five.

1:11Person six does, and then nobody, because everyone left starts past it. Four people. Now the thing that decides its fate. It started shut and every touch flips it: shut, open, shut, open, shut. Four flips, and it is exactly where it began. A locker finishes open when the number of people who touched it is odd, and shut when it is even. Not who they were, not what order they came in. Just how many.

1:43Open doors will be gold from here on, shut ones grey. So it is a counting question, and we should work out what we are counting. When does person d touch locker n? Person d visits d, two d, three d - the multiples of d. So person d touches locker n exactly when d divides n, leaving nothing over. Back at locker six, the people who touched it were one, two, three and six.

2:14Those are precisely the numbers that divide six - all of them, and nothing else. So the corridor has stopped being about doors. The number of people who touch locker n is the number of divisors n has. And the real question is: for which numbers is that count odd? Divisors do not turn up one at a time. Three divides thirty-six - and thirty-six divided by three is twelve, so twelve divides thirty-six as well.

2:47They arrive together, as a couple, because three times twelve is thirty-six. I will draw that as an arc joining the two of them, and the arc will always mean the same thing: these two multiply to give the number. Take six: one and six, two and three. Two arcs, four divisors. Take ten: one and ten, two and five. Four again. Every arc brings in two divisors, so the count climbs two at a time from nothing, which makes it even.

3:20Every time? Try thirty-six. One and thirty-six. Two and eighteen. Three and twelve. Four and nine. And then six. Six times six is thirty-six, so the partner of six is six. There is nobody to join it to. The arc closes on itself, and I will draw that as a loop. Count them: one, two, three, four, six, nine, twelve, eighteen, thirty-six. Nine divisors. Odd. So locker thirty-six should end open - and walk the corridor, and it is.

4:00This is the point where it is easy to stop one step too soon. The tempting sentence is: thirty-six has six times six, so its count is odd, done. That is not done. Look at what the count actually is. Two for every arc, plus one for every loop. Four arcs is eight, plus one loop, is nine. It came out odd because the number of loops was odd. One is odd - but so is three.

4:29Three loops would give even plus three, still odd, and the argument would survive. Two loops would give even plus two, which is even, and the door shut. So knowing that thirty-six has a loop is not enough. We need to know how many loops a number can have. A loop is a divisor d with d times d equal to n. For a given n there is exactly one number that does that, or there is none at all.

4:59There is never a second one. That single sentence is what the whole argument is standing on. Now there are only two cases. No loop: two for each arc, so the count is even and the door is shut. One loop: even plus one, so it is odd, and the door is open. There is no third case. A locker ends open exactly when its number has a divisor that is its own partner - exactly when the number is d times d for some whole number d.

5:34Which lockers are those? One, four, nine, sixteen, twenty-five, thirty-six, forty-nine, sixty-four, eighty-one, a hundred. Ten doors out of a hundred, and we named every one of them without a single person walking anywhere. One is the number people expect the rule to break on, and it is the opposite of that. One has a single divisor, and that divisor is one. Its partner is one divided by one, which is one again.

6:08So it pairs with itself. No arcs, one loop, a count of one. One is odd, so the door is open - and locker one is open. And one is one times one. It is not an awkward case the rule has to survive. It is the cleanest example of the rule there is. A second question about the same corridor. Which lockers were touched exactly twice - two divisors and no more?

6:38Every number has one and itself on its list, so a number with exactly two divisors is a number with nothing in between. That is what a prime is. Two, three, five, seven, eleven, and on from there. And watch what happens to one. For one to be on the list its two divisors would have to be one and itself, and those are the same number. One is touched once, not twice.

7:06It falls out on its own, and nobody has to remember to keep it out. There is a word for the numbers that ended up open, and it is worth seeing where it comes from. Lay out tiles in a square array. One tile along each side: one in total. Two along each side: four. Three: nine. Four: sixteen. Five: twenty-five. Ten along each side: a hundred. Those areas are the open lockers, in order.

7:40The word square is not a metaphor here. A number is a square because it is the area of a square whose side is a whole number of units. And the side is the divisor with no partner - the one that opened the door. Writing n times n gets tiring, so it is written n with a small two raised beside it: n squared. Five squared is twenty-five. The small two counts how many n's are multiplied together.

8:13It is not something you multiply by, and that is worth saying, because squaring is not doubling. Two and a half doubled is five. Two and a half squared is six point two five. You can square anything, not only whole numbers. Three fifths squared is nine over twenty-five - top times top, bottom times bottom. Notice that one came out smaller than what we started with, which doubling never does.

8:43Nine over twenty-five is a square. It is not a perfect square. A perfect square is what you get when the thing squared was a natural number: one, four, nine, sixteen, twenty-five. Six point two five is a square too, and not a perfect one, because two and a half is not a natural number. So the doors that finished open are precisely the perfect squares from one to a hundred.

9:12That is a much stronger statement than a list of ten numbers: it tells you what a corridor of a thousand lockers would do, without your having to build one. Nobody walked the corridor. A question about doors became a question about counting, because a door's fate depends on nothing except how many times it was touched. Counting became pairing, because divisors arrive two at a time, each one with a partner.

9:45Pairing had exactly one thing that could go wrong: a divisor whose partner is itself. And that one exception turned out to be a shape. The doors left open are the areas of squares. Each step was small, and together they turned a hundred separate questions into one. That is the trade this subject keeps making.