Why a formula says in one line what words take a paragraph to say

0:00Here is a month, laid out the way months always are. Seven columns, one for each day of the week, and the numbers running along the rows. That layout has a consequence worth noticing straight away. Every cell holds exactly seven more than the cell above it, because a week is how far it is to the same day again. Now lay a small window over the grid. Two cells across, two cells down.

0:30Put it here. It catches twelve and thirteen on the top row, and nineteen and twenty underneath. A square like that has two diagonals. One runs from twelve across to twenty. The other from thirteen across to nineteen. Add each pair, and watch what happens. Twelve and twenty. Thirty two. Thirteen and nineteen. Thirty two. The same total, out of two different pairs. That could be an accident, so slide the window somewhere else and try again.

1:05Four and five, over eleven and twelve. Four plus twelve is sixteen. Five plus eleven is sixteen. Again. Seventeen and eighteen, over twenty four and twenty five. Forty two, both ways. It keeps happening, and the temptation now is enormous. The temptation is to say: it works every time I try it, so it always works. That sentence has a hole in it, and the hole is what this whole video is about.

1:38So how many squares are there to check? Inside this month, nineteen. Nineteen complete two by two windows, and not one more. You could do all nineteen in an afternoon. And at the end of that afternoon you would know something true and almost useless. You would know it holds for nineteen squares. But the claim was never about nineteen squares. It was about squares like this one, wherever they happen to be.

2:07And to see how far that reaches, stop treating this as a month for a moment. Treat it as what it actually is. A grid, seven columns wide. The month runs out at thirty. The grid does not have to. Carry the numbering on. Thirty one, thirty two, and along to thirty seven. Then another row. Thirty eight to forty four. And another. And another. Nothing in the pattern ever needed the month. It needed seven columns, and the numbers running in order.

2:43So the grid goes on for as long as you like, and there is no last row. Which means there is no last square either. And now look at what checking can actually do. It can settle any square you name. It can never settle all of them, because there is no point at which you would be finished. So checking is not going to do this. Something else has to.

3:09Go back to one square, and this time refuse to say which one. Its top left corner holds a number. Which number? It does not matter. Call it a. That one decision is the whole trick, and it is worth being careful about what it does. a is not twelve. It is not any particular date. It is whatever the top left corner happens to hold, in whichever square you picked.

3:37Which means anything true about a is true about every one of those squares at once. One letter, standing in for endlessly many numbers. Now fill in the rest of it. Words first, symbols after. Three sentences, one for each remaining cell. The cell to the right of a is the next date along, so it holds one more. The cell below a is the same weekday a week later, so it holds seven more.

4:06And the cell diagonally across is one along and one down, so it holds eight more. Now write those down. The top row is a, and a plus one. The bottom row is a plus seven, and a plus eight. Four cells, and not one of them is a number. Nothing has been assumed about which square this is, and that is precisely the point. Now take the two diagonals, exactly as before.

4:36The first runs from a down to a plus eight. Add them. a, plus a plus eight. The two a terms are like terms, so they collect. Two a plus eight. The second diagonal runs from a plus one down to a plus seven. Add them. a plus one, plus a plus seven. Collect the letters, and you get two a. Collect the plain numbers, one and seven, and you get eight.

5:03Two a plus eight. The same expression. Not the same number, the same expression, and that is a much stronger thing to be. Stop and be precise about what just happened, because it is easy to undersell. Three lines of adding have settled a question that no amount of checking could reach. Not the nineteen squares in the month. Every square in a grid that has no last row. Including squares nobody has drawn, in rows nobody has written down.

5:37And it hands you the total as well. Eight more than twice the corner. Twelve in the corner gives thirty two, which is where this started. Ninety in the corner gives a hundred and eighty eight, and nobody has to draw anything. That is what naming the unknown bought. Not a shortcut for the arithmetic. A way of finishing something that had no end. The tactic is worth far more than the result, so try it on a different shape.

6:08Five cells in a cross. A centre, one either side of it, one above and one below. Here that is eight on top, then fourteen, fifteen and sixteen across, and twenty two below. Add all five. Seventy five. Which is five times fifteen, and fifteen is the one sitting in the middle. So do the same thing. Call the centre a, and say the rest in words first. Left is one less. Right is one more. Above is seven less. Below is seven more.

6:42Add them up. a minus seven, a minus one, a, a plus one, a plus seven. The ones cancel, the sevens cancel, and five a is left. Five times the centre, for every cross there is. One more thing, and it matters more than it looks. Anybody who memorises plus seven is going to be stranded, so be clear about where the seven came from. It came from the grid being seven columns wide. It is not a fact about calendars.

7:16Here are the whole numbers written four to a row instead. One, two, three, four. Then five to eight. Then nine to twelve. Same argument, different width. The cell below a now holds four more, not seven. So both diagonals of the little square come to two a plus five. The constant was never eight. It is always the width, plus one. And the entry in row r, column c is four, times r minus one, plus c, which puts a hundred and twenty four at row thirty one, column four.

7:55Here is the last thing, and it is really a claim about language. You could say all of this in ordinary words, and people often do. Take any two by two square anywhere in a grid seven columns wide, add the two ends of one diagonal, add the two ends of the other, and the totals will agree. That is a long sentence, and it is correct. Now the same thing in symbols. a plus, bracket, a plus eight, equals, bracket, a plus one, plus, bracket, a plus seven.

8:30Shorter. But shorter is not the interesting part. The interesting part is that the sentence can only assert it. The symbols can settle it. You cannot write out a check across endlessly many squares in words, because no sentence of any length performs one. Being brief and being general turn out to be the same property, and that is what algebra is actually for.