What a perfect square's last digits can and cannot be

0:00Square the whole numbers, one after another, and write down what you get. One, four, nine, sixteen, twenty-five, thirty-six. Keep going: forty-nine, sixty-four, eighty-one, a hundred. Past a hundred it slows down - a hundred and twenty-one, a hundred and forty-four. Thirty rows of this looks like busywork. It is not. It is data, and something is hiding in its right-hand edge. Cover everything except the last digit of each answer.

0:33One, four, nine, six, five, six, nine, four, one, nought. Some digits are showing up. And some are not showing up at all. Before reading anything off a table, ask where a last digit comes from at all. Multiply thirty-seven by eighty-four. The answer is three thousand one hundred and eight, and it ends in eight. Now multiply just the last digits: seven times four is twenty-eight. That ends in eight too.

1:06That is not a coincidence, and it is not about these two numbers. Split thirty-seven into thirty and seven, eighty-four into eighty and four, and multiply out. Four pieces. Three of them carry a thirty or an eighty, so all three are multiples of ten. A multiple of ten ends in nought, so it cannot put anything into the units place, however large it is. Only the fourth piece, seven times four, ever reaches that far.

1:41So here is the rule, and it is about multiplying, not about squares. The last digit of a product is decided by the last digits of the two things you multiplied. Nothing else in either number can touch it. Squaring is just the case where the two things happen to be the same. So the last digit of n squared is decided by the last digit of n, and by nothing else about n.

2:09So we need not square thirty numbers, or a thousand. There are only ten digits a number can end in. Ten cases, and we are finished for ever. Nought squared is nought, ending in nought. One squared is one. Two squared is four. Three squared is nine. Four squared is sixteen, which ends in six. Five squared is twenty-five, ending in five. Six squared is thirty-six - six again. Seven squared is forty-nine, so nine again.

2:44Eight squared is sixty-four, four again. And nine squared is eighty-one, which ends in one. Ten cases done. Now count the answers. Nought, one, four, five, six, nine. Ten digits went in and only six came out. That collapse is worth understanding, not just noticing. Look at one and nine. One squared ends in one; nine squared is eighty-one, which also ends in one. Two and eight: four, and sixty-four.

3:18Three and seven: nine, and forty-nine. Four and six: sixteen, and thirty-six. Every time, a digit and ten minus that digit land on the same ending. So the digits pair off, and each pair costs one ending instead of two. Four pairs. But two digits have nobody else to pair with: five would pair with five, and nought with nought. Those two stand alone. Four pairs and two on their own is six groups, and six groups is six endings.

3:53That is where the six comes from. Turn it round. Six endings occur, so four do not. Two, three, seven and eight. No perfect square, of any size at all, ends in any of those. Not a big one, not a clever one, not one nobody has checked. The ten cases were the whole of it, and none of them produced a two. So you can look at a number and reject it on sight.

4:22Anything ending in two: not a square. Ending in three, seven or eight: not a square. You have done no arithmetic and you are certain. Now the part that gets misremembered. Sixteen ends in six and is a square. Thirty-six ends in six and is a square. Twenty-six ends in six and is not. No whole number squares to twenty-six. So an ending of six tells you nothing at all - and it is not just six.

4:54Ten is not a square, and it ends in nought. Eleven ends in one and is not a square. Fourteen ends in four. Five ends in five. Nineteen ends in nine. Every one of the six permitted endings has non-squares wearing it, and not one apiece - hundreds. The test throws numbers out. It never lets one in. So use it for what it does. Two thousand and thirty-two, two thousand and forty-eight, one thousand and twenty-seven, one thousand and eighty-nine.

5:31Which are not perfect squares? The first ends in two, so it is out. The second ends in eight - out. The third ends in seven - out. Three of them settled with a glance and no calculation. The fourth ends in nine, which is allowed. And that is where the test stops and hands the problem back. Allowed does not mean square. Go and check, and thirty-three squared is one thousand and eighty-nine.

6:01It was a square - but the digit never told you so. The pairs are useful in their own right. Which numbers have squares ending in one? From the ten cases, exactly two digits do it: one and nine. So the family is one, nine, eleven, nineteen, twenty-one, twenty-nine, and on it goes. That list is not every other number and not a fixed step. It runs in pairs hugging each multiple of ten - one either side of twenty, one either side of thirty.

6:35That is the pairing showing through: one and nine are what is left when you count in from both ends. Now squares ending in six, and this is the same shape again. Two digits do it: four and six. Sixteen is four squared, thirty-six is six squared, a hundred and ninety-six is fourteen squared, two hundred and fifty-six is sixteen squared. Look at the numbers being squared: four, six, fourteen, sixteen.

7:08Four, six, four, six. Now a question. Of thirty-eight, thirty-four, forty-six, fifty-six, seventy-four and eighty-two, which have squares ending in six? Read the last digits: eight, four, six, six, four, two. Four and six are in the family, eight and two are not. So four of the six qualify, and you never squared anything. And notice the two that failed both landed on four instead - eight and two, two different digits, one ending.

7:43Zeros deserve their own rule. Ten squared is a hundred: one zero became two. Twenty squared is four hundred, forty squared is one thousand six hundred - one zero in, two out, every time. A hundred squared is ten thousand; two hundred squared is forty thousand; seven hundred squared is four hundred and ninety thousand. Two zeros became four. The reason is short: if ten divides the number then a hundred divides its square, because there is a ten from each factor.

8:21So the count doubles. Three zeros give six, not five. And say it exactly: exactly three zeros give exactly six. Glance at a thousand and you might call it two zeros. It has three, and its square has six. One thing falls straight out of doubling: a square can never end in an odd number of zeros. One more, and it is the shortest. An odd number times an odd number is odd.

8:52An even number times anything is even. A square is a number times itself, so it inherits. Odd numbers have odd squares, even numbers have even squares, every time. Run down the table and it alternates without a single break: one odd, four even, nine odd, sixteen even. Squares are not usually odd or usually even. They are exactly as odd or even as the thing you squared. None of this was really a fact about squares.

9:27It was a fact about multiplying: only the units places of what went in can reach the units place of the answer, because everything else is a multiple of ten. Squares are the case where the two are equal, which is why ten digits was the whole search. Ten possibilities collapsed onto six because they paired off, with two left standing alone. That is what makes the test one-way: two different digits arriving at the same ending means the ending cannot tell you which digit you came from.

10:03A test that throws things out is not a weaker version of one that lets things in. It is a different kind of thing, and knowing which you hold is most of using it properly.