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Video: Puzzle 0002

Tim Burnham

In this video, we present a mind reading puzzle, which asks you to pick a number, do some simple arithmetic with it, then select the symbol corresponding to the result from a table. We then use some simple algebra to explain how we know the result.

06:13

Video Transcript

Welcome to puzzle time! Here’s puzzle number two. Let’s start our puzzle by looking at the magic mind reader. So what I want you to do is to choose any two-digit number, and then add together both those digits, and then subtract that total from your original number. So for example, if you picked twenty-three I want you to add together both those digits, so two plus three is equal to five, and then I want you to subtract that total from your original number, so five from twenty-three. So the number you’ve got to calculate in this case will be twenty-three take away five, okay? Do that with your digits now.

So whatever answer you got from that last calculation, I want you to look at this table and look for the number that you got; so for example if you got an answer of thirty-three, the figure that you would see is a little star; if you got an answer of seventy-five your figure that you got would be a little rose. So look up your number now and think very carefully about the symbol, the shape that you got. The mind reader says it was a flag. Okay, let’s have another go.

Choose any two-digit number, add together both of those digits, and subtract that total from your original number. So again if you started off with twenty-three, I hope you didn’t, then you would add together the two and the three to make five and take that number away from twenty-three to get your new number. So that’s okay. Go on, you can use a calculator; I don’t mind. So you’ve got your number. I now want you to look up that number on this chart, and I want to look at the symbol that’s beneath that number, and I want to think really carefully about that symbol. The mind reader says it was a bell. Okay, one last go then. Let’s try that once more.

Choose another two-digit number, add together both those digits, and subtract that total from your original number. Okay, take that result, look at that number on this table, and then look at the symbol immediately beneath it; think carefully what’s the symbol; project what’s that symbol. A mind reader says it was a book

So this puzzle is this question: how does the mind reader work? If it didn’t work, I suspect it’s your adding up that’s not quite right; but if it did work, there’s a very specific reason why it works. So what we’re gonna do is use a bit of algebra to analyse what was going on. So you might want to pause the video now and try and work this out for yourself. I’m gonna hold on for three seconds and then I’m gonna tell you how to solve this puzzle.

Okay, let 𝑎 be the first digit of my number; let 𝑏 be the second digit of my number So in the example we used in the question, t- if twenty-three was your number, 𝑎 would be two and 𝑏 would be three. Now if you think about it, the two is in the tens column and the three is in the ones column of this number. So 𝑎 equals two, that represents two lots of ten; 𝑏 equals three that represents three lots of one. So if I want to create an algebraic expression for this number in terms of 𝑎 and 𝑏, it’s gonna be ten lots of 𝑎, which represents the first digit, and one lot of 𝑏, which represents the second digit. So my number there ten 𝑎 plus one 𝑏 or just ten 𝑎 plus 𝑏.

Now the mind reader asked us to add the two digits together and take that away from the number we first had, so we got ten 𝑎 plus 𝑏 as our first number and we’re taking away the sum of 𝑎 plus 𝑏. We’ve added those two digits together, that’s 𝑎 plus 𝑏, and we’re taking that result away from ten 𝑎 plus 𝑏. So if we look inside the brackets there, we’re taking away the whole of 𝑎 plus 𝑏 that means we’re taking away 𝑎 and we’re taking away 𝑏.

And look, we’ve got 𝑏 and we’re taking away 𝑏. Well 𝑏 take away 𝑏 is nothing, so those two things cancel out. We’ve got ten 𝑎, and we’re taking away 𝑎, and ten 𝑎 take away 𝑎 is nine 𝑎. So it doesn’t matter what two digits I started off with my original number. When I go through the maths, if I do the maths correctly, I’m always gonna end up with an answer which is nine times whatever my first digit was. This process of adding the two digits together and taking it away from the original number means it doesn’t matter what the second digit was; that’s just gonna be ignored. The final answer is always just gonna be nine times the first digit.

So the first digit couldn’t be zero because it had to be a two-digit number, so the possible values for the first digit are one, two, three, four, five, six, seven, eight, nine; if I multiply each of those by nine, I’ve got my nine times table So in the mind reader table that you looked up, every multiple of nine had the same symbol.

So it doesn’t matter what digits you started off with. If you did the maths correctly, I knew what symbol you were going to have chosen. Second time round, I just switched all the symbols to be bells in the nine times table, and the third one was books. And with a little bit of algebra, we’ve created a trick that at first glance looks like absolute magic.