Why a "think of a number" trick always works, and how to invent one

0:00Think of a number. Do not tell me what it is. Double it. Add four. Halve what you now have. And take away the number you started with. You are looking at two. Try it again with a different number. Try it with seven, with thirteen, with a hundred. Seven gives two. Thirteen gives two. A hundred gives two. Three different numbers, three different journeys, and all three land on the same tile.

0:29So the question is not whether it works. The question is why it cannot fail - and whether you could design one of your own. Here is the tempting answer: it works because I tried it and it worked. But three successes are three successes. They are not a statement about every number, and there are infinitely many numbers you did not try. You could run a thousand of them and still be describing a thousand cases.

1:00What you want is a different KIND of statement - one sentence that covers every starting number at once. And there is a way to get one, which is the whole reason algebra is worth learning. Stop putting a particular number in. If you never commit to a value, you never have to check the values one at a time. Instead of seven, send a letter through the steps. Call the secret number n. Not a mystery to be solved - just a number whose value we are refusing to fix.

1:36Double it, and you have two n. Add four, and you have two n plus four. Halve it. Two n plus four, halved, is n plus two. And take away the number you started with. n plus two, less n, is two. No n anywhere. The answer is two, and the letter has gone. Watch what happens to the letter, and only to the letter. It starts multiplied by one. Doubling makes it two. Halving brings it back to one. Taking away the start brings it to nothing.

2:12That last step is where the trick is decided. Once n is multiplied by nothing, the answer CANNOT depend on what you started with. There is nowhere for your number to show up. And what is left over is the prediction. Now notice what the halving did to the four. People say the halving undoes the doubling, so the four goes away too. It does not go away. It becomes a two.

2:41If you treat dividing as an eraser you predict nought, and then you decide the trick is broken. So where did the two actually come from? From the four. The four is added before the halving, so half of it survives. Change the four and the prediction moves with it. Suppose you want the trick to land on three. Then you need to add six, because six halved is three.

3:08Want it to land on five? Add ten. The added number is always twice the answer you want, and nothing else in the chain has to change at all. One step controls the answer. That is not something you would ever notice by trying numbers. Now you can build your own, as long as you like. Think of a number. Treble it. Add twelve. Divide by three. Take away the number you started with.

3:39Follow the letter: three n, three n plus twelve, n plus four, and then four. The prediction is four. And the twelve became a four for the same reason the four became a two - it was divided along with everything else. There is only one condition, and it is worth saying plainly. The letter's multiplier has to reach nothing by the last step. Whatever number is left standing when it does is your prediction.

4:09Break that condition and there is no trick. Double, add four, take away the start, and you are left with n plus four. The answer still depends on the number, so nobody can predict it. Now the same tool, turned the other way round. This time the secret is a date - your birthday, say - and I am going to read it back to you off a single number.

4:36Take the number of your month. Multiply by five. Add six. Multiply by four. Add nine. Multiply by five. And finally add the day of the month. Tell me the total. Suppose the date is the twenty-sixth of January. One times five is five. Plus six is eleven. Times four is forty-four. Plus nine is fifty-three. Times five is two hundred and sixty-five. Plus twenty-six is two hundred and ninety-one.

5:10Two hundred and ninety-one. And from that, I can tell you the date. Send letters through it. Call the month m and the day d. Five m. Five m plus six. Twenty m plus twenty-four. Twenty m plus thirty-three. A hundred m plus a hundred and sixty-five. And then the day joins on: a hundred m plus a hundred and sixty-five plus d. Look at what did NOT happen. The month's multiplier never reached nothing. It reached a hundred.

5:47The first trick destroyed the information. This one keeps it, and keeps the two secrets apart. That is the same expression doing the opposite job, and you read both of them the same way - by watching the multipliers. Why a hundred? Because the three multiplications are five, four and five, and five times four times five is a hundred. And a hundred is exactly what you want, because it lifts the month clear of the last two places.

6:21The day is at most thirty-one, so it fits inside those two places with room to spare. The month sits above them. Nothing overlaps. Take two hundred and ninety-one and subtract a hundred and sixty-five. That leaves one hundred and twenty-six. The one in front is the month. The twenty-six behind it is the day. The twenty-sixth of January, written out in digits, and it was sitting there the whole time.

6:54Where does the hundred and sixty-five come from? It is not a magic number and you do not have to memorise it. It is what the additions leave behind after everything that comes AFTER them has acted. The six is added early, so it gets multiplied by the four and then by the five. Six times four times five is a hundred and twenty. The nine is added later, so it is only multiplied by the last five. Nine times five is forty-five.

7:27A hundred and twenty plus forty-five is a hundred and sixty-five. So the performer does one subtraction. That is the whole performance. Everything else was decided when the steps were written. Try three. Someone reports one thousand two hundred and sixty-nine. Subtract a hundred and sixty-five: eleven hundred and four. The fourth of November. Three hundred and ninety-four. Subtract: two hundred and twenty-nine. The twenty-ninth of February. Two hundred and ninety-six. Subtract: one hundred and thirty-one. The thirty-first of January.

8:12Now, the twenty-ninth of February is not a day that exists in every year, and the trick has no idea. It decodes a month and a day. It was never told about years. It does not check anything else either. Report a quantity of a hundred and fifty instead of a day and it climbs onto the month's shelf, and the answer comes back confidently wrong. The trick needs the second number to fit in two places. It does not verify that it does, and neither does the person performing it.

8:48That is worth knowing about any procedure that reads an answer out of a single number: it works inside conditions nobody stated aloud. So rebuild it your own way. Take the month. Multiply by ten. Add three. Multiply by ten again. Add the day. The month is still carried up by a hundred, so the two secrets still land on separate shelves. But the three is only multiplied by the second ten, so the additions accumulate to thirty, not a hundred and sixty-five.

9:27Same date, different total, different subtraction. Thirty is what you take off. And that is the point of the whole thing. Once you can read the expression, you are not the audience any more. The trick was never in the numbers. It was in knowing what the letter was going to be multiplied by.