### Video Transcript

In this video, we’re going to look at a mind reading number trick.

Okay Sue, I want you to think of a number from one to 15.

Okay Bob, I’ve thought of a number from one to 15.

Right then Sue, I’m gonna read your mind. Is your number on this card?

Yes it is Bob!

Excellent Sue, and is your number on this card?

Yes it is Bob!

Splendid Sue, and is your number on this card?

No it isn’t Bob!

Magnificent Sue. Lastly, is your number on this card?

Yes it is Bob.

Well done Sue! Your number was 11.

Gosh Bob, that’s amazing! You’re right! How ever did you do?

Well Sue, I’ll tell you shortly after I try my trick on the people watching this video. The problem is that I can’t hear them. So it’s going to be a bit tricky to do, but let’s give it a go.

Okay folks, I want you to think of a number from one to 15. Now if your number is on this card, if it’s one, three, five, seven, nine, 11, 13, or 15, I want you to write down the letter A. And if it’s on this card, if it’s two, three, six, seven, 10, 11, 14, or 15, then I want you to write down the letter B. Now if it’s on this card, if it’s four, five, six, seven, 12, 13, 14, or 15, I want you to write down the letter C. And if it’s eight, nine, 10, 11, 12, 13, 14, or 15, I want you to write down the letter D.

Okay, if I’d heard your answers, I’d be able to instantly tell you what number you’d thought of. But because I can’t, this is gonna be a bit painful. You should have one or two or three or four letters written down in front of you now. Now if you’d just written A, then the number you thought of was one. If you just wrote B, then the number you thought of was two. If you wrote A and B, then you were thinking of three. If you just wrote C, then you were thinking of four. But if you wrote A and C, then you were thinking of 5, while if you wrote B and C, you were thinking of six.

Now if you’d written A and B and C, you were thinking of seven. But if you just wrote D, you were thinking of eight. If you wrote A and D, you were thinking of nine; B and D, 10; A, B, and D, 11; C and D, 12; A, C, and D, was 13; B, C, and D was 14. And if you wrote all four letters, you were thinking of 15. Well it’s a bit less impressive done that way, but how does this work? Well, it’s all down to binary.

Now we normally do our mathematics in base 10 or the denary system with the digits zero, one, two, three, four, five, six, seven, eight, and nine. If we look at the place value of our digits in base 10, we have a ones column, telling us how many ones our number has, a tens column, telling us how many tens our number has, a hundreds column telling us how many hundreds our number has, a thousands column, telling us how many thousands we’ve got, and so on.

Now the column values are powers of 10. The ones column is 10 to the power of zero. The tens column is 10 to the power of one. The hundreds column is 10 to the power of two. The thousands column is 10 to the power of three, and so on. So a base 10 number like one, two, three, four means we’ve got four ones, four times 10 to the power of zero. We’ve got three tens or three times 10 to the power of one, two hundreds or two times 10 squared, and 1000. So that’s one times 10 to the three.

And to evaluate that, we’ve got one times a 1000 plus two times a 100 plus three times 10 plus four times one, or 1234 as we’d normally say that. But in the binary system, base two, we only use the digits zero and one and the columns have different values: ones, twos, fours, and eights. And that’s basically two to the power of zero, two to the power of one, two to the power of two, and two to the power of three, and so on.

Then let’s have a look at how we’d represent the base 10 numbers, one to 15. In binary, the base-10 number one is exactly the same in binary. We’ve got one one. But to represent the decimal number two, we don’t have a two digit in the binary system. So we have to say we’ve got one of the twos from the twos column but no ones to add on to that. So one zero is two in binary. To represent the decimal number three in binary, we need to count out.

We’ve got one two and one one. When we add them together, two and one make three. So in binary, three would be one one. Then the decimal number four, well we can put a one in the fours column cause it’s got one of those. But we don’t need any twos and we don’t need any ones to add to that to make it four. So the decimal number four is represented in binary as one zero zero.

Then five would be one zero one. Six would be one one zero, that’s a four and a two and no ones. Then seven is one one one. Eight is one zero zero zero. We’ve got one eight and no fours, twos, or ones. Nine is one zero zero one. 10 is one zero one zero. 11 is one zero one one. And 12 is one one zero zero. 13 is one eight, one four, no twos, and a one. 14 is one one one zero. And 15, we need an eight, a four, a two, and a one. So it’s one one one one.

Now let’s write the column values as the first number on each of the cards. So that’s one, two, four, and eight. Now we can write all of the numbers that have a one in the ones column on the first card. And we’ve already written the one on there. So next up is the three, then the five, then the seven, then the nine, 11, 13, and 15. Now we can write all the numbers that have a one in the twos column on the second card.

Well we’ve already written the two, but we need to write the three, the six, the seven, and then the 10, 11, 14, and 15. Now we’ll write all the numbers that had a one in the fours column on the third card. So that’s four, five, six, seven, 12, 13, 14, and 15, and then all the numbers that have a one in the eights column on the fourth card. So that’s eight, nine, 10, 11, 12, 13, 14, and 15.

Now if we remember how to work out the decimal value of a binary number, what we have to do, for example with 13, we’ve got one eight. We’ve got one four, no twos, and one one. So we just need to add those digits together. Eight plus four — we don’t need any twos, but we need a one — eight and four is 12 plus one is 13. And that means to work out what number your friend was thinking of, you just need to add up the top left number of each of the cards that they said their number was on.

So remember, in our first example, Sue said yes to cards A, B, and D. So I just had to quickly add up one, two, and eight. And that makes 11. Easy! Now here’s your challenge. Can you make a new set of five cards to tackle numbers in a puzzle from one to 31? You’ll need to make up a table of the binary versions of the numbers from one to 31 and then work out which numbers go on the five cards with one, two, four, eight, or 16 in their top left corner.

Pause the video now if you want to have a go and then check back to see if you were right. Okay, here are my five cards with one, two, four, eight, and 16 in the top left corner. What numbers did you have for card one? Well it’s all the odd numbers: one, three, five, seven, nine, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, and 31. And all the numbers that had a one in the twos column were two, three, six, seven, 10, 11, 14, 15, 18, 19, 22, 23, 26, 27, 30, and 31.

All the numbers with a one in the fours column were four, five, six, seven, 12, 13, 14, 15, 20, 21, 22, 23, 28, 29, 30, and 31. All the numbers with a one in the eights column were eight, nine, 10, 11, 12, 13, 14, 15, 24, 25, 26, 27, 28, 29, 30, and 31. And all the numbers with a one in the 16s column were 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, and 31.

And here’s a handy top tip. If you write the top left corner number — one, two, four, eight, 16 — on the back of the appropriate card, you don’t even have to look at the numbers on the front of the card to show your friend when you do this trick. So it’ll look even more impressive. Good luck and have fun!