Calendar magic: the position in the grid is the whole secret

0:00Here is a trick you can do on anyone with a calendar in front of them. Ask them to draw a square around four dates - two side by side, and the two directly underneath. Don't look. Ask only for the total. They say forty. You say: six, seven, thirteen and fourteen. You did not guess. You did not need to see the square. And the reason is not about calendars at all - it is about position.

0:30Four dates went in. One number came out. And that one number was enough to put all four back, which should bother you slightly, because adding things up normally throws information away. Look at how a month is laid out. Seven columns, one row for each week. That layout puts two fixed moves into it. Step one cell to the right and the date goes up by one. Step one cell down and it goes up by seven.

1:02Now be careful about the seven. It is not there because a week has seven days. It is there because the month is set out seven columns wide. Print the same dates ten to a row and stepping down would add ten. Hold on to that, because we are going to do exactly that at the end. Two moves, and neither of them ever changes. That is the whole machine.

1:29So take the square. Name only the top left date. Call it a. Not the answer - just a name. The one beside it is one step right, so it is a plus one. The one below is one step down, so it is a plus seven. And the fourth is one right and one down, so it is a plus eight. Look at what has happened. I named ONE cell and the grid handed me the other three. Four dates, and there is only one thing you don't know about them.

2:01That is the answer to what should have bothered you. Adding four numbers usually loses information, but there was only ever one piece of information in there to lose. Now add them up. a, plus a plus one, plus a plus seven, plus a plus eight. Collect the a's: there are four of them. Collect what is left: nothing, one, seven and eight, which come to sixteen. So the total is always four times the corner, plus sixteen.

2:33And that sixteen is worth staring at. It is not a fact about August. It is not a fact about any month. It is the total of the block's own offsets - how far each cell sits from the corner - and those depend on the shape, not on the dates. Every month of every year, on any seven-column calendar, that number is sixteen. Which means you can run it backwards. Someone hands you a total of thirty-six.

3:05Take the sixteen off: twenty. Share the twenty four ways: five. So the corner is the fifth, and the square is five, six, twelve, thirteen. Two steps. Subtract, then divide. That is the entire trick. And run it on the forty we started with: take off sixteen to get twenty-four, share four ways to get six - and back come six, seven, thirteen and fourteen, which is the square we began from.

3:37Now here is a question the trick invites but does not answer. What if your friend reports a total you cannot decode? Every total is four times something, plus sixteen - so every total is a multiple of four. No exceptions. The smallest one this month allows is the square on the first: one, two, eight, nine, which comes to twenty. The largest is a hundred and four. So if a friend reports thirty-eight, you know something is wrong before you start. Thirty-eight take away sixteen is twenty-two, and twenty-two does not share four ways.

4:18Either they added wrongly, or the shape they drew was not a square of four. There is a second way the trick fails, and this one is the one that catches people out. Take the ninth, which is a Saturday. Arithmetic says the square is nine, ten, sixteen, seventeen. But look at the grid. The tenth is not beside the ninth. It starts the next row. The square runs off the right-hand edge, and the step of one is broken.

4:51Every Saturday fails the same way. And so does the whole of the last full row - including the twenty-fourth, which surprises people. The twenty-fourth would need a thirty-first below it and a thirty-second beside that, and there is no thirty-second. Out of thirty-one dates, only nineteen can be the corner. The argument is exact, and so is where it stops. So far, one shape. But nothing we did was about squares.

5:24Take three in a row. The offsets are nothing, one and two, so the total is three times the corner plus three. Take three in a column. Nothing, seven, fourteen: three times the corner plus twenty-one. Take an L - the corner, the cell below, and the cell to the right of that. Nothing, seven, eight: three times the corner plus fifteen. Same method every time. Write every cell in terms of one of them, add, and read off the leftover. A different shape does not need a different idea. It needs a different number.

6:04Now do it to a three by three block, and something lovely falls out. Nine cells, so nine copies of the corner, and the offsets add to seventy-two. Nine times the corner, plus seventy-two. But seventy-two is nine eights - so that is nine lots of the corner plus eight. And the corner plus eight is the centre cell. One right and one down. So the total of a three by three block is simply nine times its middle date. Here is one: ten, eleven, twelve, seventeen, eighteen, nineteen, twenty-four, twenty-five, twenty-six. Its middle is eighteen, and it totals a hundred and sixty-two, which is nine eighteens.

6:53Which makes it very tempting to say the square of four is four times its middle. Don't. Four times the corner plus sixteen is four lots of the corner plus four. That much is true. But the corner plus four is not one of the four dates. They are the corner, plus one, plus seven, plus eight. There is no plus four among them. A square of four has no middle cell. Four things have nothing in the centre.

7:23Compare the column of three: three lots of the corner plus seven, and the corner plus seven IS its middle date. The shortcut is real when the shape has a middle, and only then. Now the promise from the beginning. Take the calendar away. Here is a grid nine columns wide, running two to fifty. Rows start at two, twelve, twenty-two, thirty-two, forty-two. One step right still adds one. One step down adds ten.

7:57So a square of four sits at nothing, one, ten and eleven - and the total is four times the corner, plus twenty-two. Two, three, twelve, thirteen. Thirty. Take off twenty-two, share four ways, and back comes the two. Not one line of the reasoning changed. Only the step down did. So what actually did the work here? Not the calendar. Not the seven. What did the work was that a grid has two constant moves in it, so every cell of a block is a fixed distance from any one cell of it.

8:35That is why four dates carry one piece of information between them, and why one total is enough to hand all four back. Leave the two steps blank and the method still stands. Whatever grid you are handed, measure them, write every cell in terms of one, add, and solve. Count the columns, and the grid tells you the second step itself. A hundred columns wide, and a square of four totals four times the corner plus two hundred and two.

9:07The trick was never the calendar. It was the position.