Decoding a divisibility trick with algebra

0:00Here is a trick you can do on anyone. Ask them for a two-digit number with two different digits, kept to themselves. Write the same digits the other way round, take the smaller from the larger, and divide what is left by nine. Then tell them, before they finish, that it came out exactly. Say they picked forty-seven. Backwards that is seventy-four; take one from the other and you get twenty-seven; and twenty-seven divided by nine is three, exactly.

0:34One example proves nothing. There are eighty-one numbers they could have picked, and checking every one would still not tell you why. So instead of checking, we are going to find out where the nine comes from. The first move is to write the number so its digits show. Forty-seven is four tens and seven ones. Ten times four, plus seven. Written that way you can see the two digits, and something else: the number each one is being multiplied by. Four by ten, seven by one.

1:10Call those the weights. A digit is worth its own value times the weight of the place it sits in, and that is the only thing the position of a digit ever does. Now reverse it, and watch what actually changes. Forty-seven becomes seventy-four. The digits are still a four and a seven. Not one of them has become a different digit. What has changed is which weight each of them is carrying.

1:41In forty-seven, the four carries ten and the seven carries one. In seventy-four, the four carries one and the seven carries ten. They have traded. One gives up ten and takes one; the other does the opposite. That is the whole of what reversing does, and once you see it that way the nine is almost already there. Take one from the other and let the digits stay as letters.

2:11Seventy-four is ten times seven, plus four. Forty-seven is ten times four, plus seven. Subtract. The tens parts leave ten sevens minus ten fours. The ones parts leave four minus seven. Collect them, and you have ten of one difference and one of the opposite difference: nine times the gap between the digits. Nothing about forty-seven survived that. The letters could have been any two digits at all. And there is the nine. It is ten, minus one. The difference between the two weights that swapped places.

2:52What we proved says more than the trick claimed. The difference is not merely a multiple of nine. It is nine times the gap between the digits. So the quotient is not a leftover number - it is the gap. Four and seven differ by three, and the answer was three. So the trick has a second half: you can announce the quotient too. It runs from one to nine. Twelve and twenty-one differ by nine itself. And ninety, reversed, is just nine; ninety take nine is eighty-one, which is nine nines.

3:30The gap does the work. The nine was never the interesting part. There is a case we skipped. We assumed the ones digit was the larger. What if the first one is? Run the same subtraction the other way round and you get nine times the gap the other way round. But the instruction was to take the smaller from the larger, so the gap is positive either way. Which is why forty-seven and seventy-four both give twenty-seven. Same pair of digits, and the trick cannot tell which you started from.

4:09Now change one word in the instructions. Instead of subtracting, add. Thirty-one and thirteen make forty-four. Twenty-eight and eighty-two make one hundred and ten. Twelve and twenty-one make thirty-three. Every one is a multiple of eleven. And you should want to know why it is eleven and not nine. Add the two expressions instead of subtracting. The tens parts give ten of each digit; the ones parts give one of each. So every digit is counted ten times and once: eleven times the total of the two.

4:48Ten minus one gave nine. Ten plus one gives eleven. It is the same two weights, doing the only two things you can do with two numbers. And the quotient this time is the total of the digits: four, then ten, then three. Three digits now. Take two hundred and thirty-four, and cycle the digits round: each one moves along a place, and the last one comes round to the front.

5:18That gives four hundred and twenty-three, and then three hundred and forty-two. Add all three and you get nine hundred and ninety-nine. Here is why. Look at any one digit across the three lines: it sits in the hundreds once, in the tens once, and in the ones once. So every digit is counted a hundred times, and ten times, and once. A hundred and eleven times, each of them.

5:47The total is a hundred and eleven times the sum of the digits. Two and three and four is nine, and that gives nine hundred and ninety-nine. And a hundred and eleven is three times thirty-seven. So the total is always a multiple of thirty-seven, and always a multiple of three - both from one factorisation, not two separate facts. One more. Take any three-digit number and write it out twice in a row, to make a six-digit number.

6:21Four six two becomes four six two, four six two. Now divide it by seven. Then divide that by eleven. Then divide that by thirteen. Nothing is left over at any stage, and what you hold at the end is the number you started with. The reason is the same as every other one. Writing the block twice pushes the first copy up three places, so the number is the block times a thousand plus the block times one: the block times a thousand and one.

6:57And a thousand and one is seven, times eleven, times thirteen. Dividing by those three, in any order, undoes the multiplication and hands the block straight back. Look at the four of them together. Nine is ten minus one. Eleven is ten plus one. A hundred and eleven is a hundred and ten and one added up. A thousand and one is a thousand and one added up. Every one of those numbers is built out of place-value weights and nothing else. Not one is a fact about nines, or elevens, or seven and thirteen. They are facts about what moving a digit does - and moving a digit only ever changes what it is multiplied by.

7:49That is why algebra was worth the trouble. Examples can tell you that something keeps happening. Only a statement with letters in it can tell you what would have to change for it to stop. The same move settles puzzles that look nothing like this one. Name what you do not know, write down what the situation says, and read off the answer. Three gardens, each with a pool that doubles any flowers dipped in it, leaving the same number at each and finishing with none. Start with seven, leave eight: seven doubles to fourteen, leave eight, six left; six doubles to twelve, leave eight, four left; four doubles to eight, and eight is what you leave.

8:37Horses and hens behind a fence: fifty-five heads, a hundred and fifty legs, twenty horses and thirty-five hens. Or without letters at all - fifty-five hens would be a hundred and ten legs, forty short, and each horse adds two. Two herds, one twice the other, made equal by handing over three: six and twelve, becoming nine and nine. Every one is the same act. The unknown gets a name before it gets a value.

9:11One last story, because here the answer is not the point. A traveller meets a genie by a tree. Each time he walks round it, the coins in his pocket double - and each time, he pays the genie eight. After the third walk he holds exactly eight, which is exactly what he owes. He hands them over and goes home with nothing. Work backwards and he began with seven. Seven doubles to fourteen, pay eight, six left. Six doubles to twelve, pay eight, four left. Four doubles to eight, and eight was the fee.

9:49But here is the thing. Doubling then paying leaves twice what he had, minus the fee - which beats what he had only when the fee is smaller than his holding. He was holding seven. The fee was eight. He was not unlucky, and he did not lose on the third circuit. He was losing from the first one, and the genie, who could see his pocket, had priced it that way.