Mathematical magic Tricks

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The Manual of Mathematical Magic McOwan and Parker The Manual of Mathematical Magic McOwan and Parker The Manual of Mathematical Magic McOwan and Parker PPPeeettteeerrr MMMcccOOOwwwaaannn wwwiiittthhh MMMaaatttttt PPPaaarrrkkkeeerrr The cont The cont The cont ent (imag ent (imag ent (imag es and t es and t es and t e e e xt) in t xt) in t xt) in t his book ar his book ar his book ar e cop e cop e cop yr yr yr ight © Queen Mar ight © Queen Mar ight © Queen Mar yy y , Univ , Univ , Univ er er er sity of London. Y sity of London. Y sity of London. Y ou may not ou may not ou may not rrepr repr epr oduce an oduce an oduce an y par y par y par t of t t of t t of t his cont his cont his cont ent wit ent wit ent wit hout wr hout wr hout wr itt itt itt en per en per en per mission. mission. mission. wwwwwwwww...mmmaaattthhheeemmmaaatttiiicccaaalllmmmaaagggiiiccc...cccooommm SSSeeerrriiiaaalll NNNooo... 111000888999The Manual of Mathematical Magic Mathematics and magic may seem a strange combination, but many of the most powerful magical effects performed today have a mathematical basis. Famous magicians such as Derren Brown and David Blaine use mathematics-based tricks in their shows, but mathematics is also the secret behind the technologies we use, the products we buy and the jobs we will have. Mathematics is the language we use to describe the world around us - it’s the basis of all the sciences. This book will show you how to perform some magical miracles to impress and entertain your friends. But it will also explain the mathematics behind the tricks and how that same mathematics is used in the real world. It also looks at the varied and exciting sorts of jobs that make use of the mathematics powering your magic. All the tricks in this book are self-working, which means you don’t need to know any clever sleight of hand, like dealing cards from the bottom of a deck. But be warned: knowing the mathematical secret isn’t the same as being able to perform the tricks well. To do that, you need to use your performance skills to create a sense of wonder. A good presentation is where your audience is pulled into the magic happening and astonishment happens. With an imaginative story, you can take a simple mathematical trick and turn it into a jaw-dropper. This book will give you some ideas for presentations, but be creative, come up with your own way. Thinking creatively about new ways to solve problems is the key to good magic but it’s also one of the key skills of a good mathematician and one of the useful employment skills you get from being good at mathematics. As we will see, many famous mathematicians were also magicians.It’s not a coincidence - they enjoyed their maths, and also enjoyed using it to entertain. And now you can too. As you start to work your way though this book, you will find magic that uses a whole range of mathematical ideas that you may already have come across, from simple addition and subtraction, to prime numbers, geometry, algebra and statistics. We hope that this book shows that all of maths can be exciting, magical and useful. There is even an advanced section where some maths that’s probably new to you is introduced, with stunning magic results. So have some serious fun, learn, practice, create and above all entertain. Some of these tricks are in the acts of professional magicians. So when you do perform them, please remember the Magicians Code: practise, practise, practise. And never reveal the workings of magic tricks to your audience The Symbols Throughout this book you will see four different symbols. Each one is used to explain a different aspect of a trick. The magic symbol is how the trick will actually look to your audience. This will explain how the trick should flow and will give you an overview of how it should come across to observers. This won’t tell you the sneaky things that you need to be doing This is where the sneakiness is explained. The presentation symbol is where the actual trick itself is explained. This will go step-by-step through what you need to do to make sure the trick works and how you can increase the effect on your audience. Don’t forget that the best magic tricks use a very simple device but then this is built up through a brilliant presentation. Once you grasp how the trick works, be sure to try expanding on your own presentation ideas All of the tricks in the book are based on mathematical principles. Normally this is explained in the presentation section, but when the maths is particularly important, or you need to learn some extra maths to make the trick work, you will see this, the maths symbol. Some of the best tricks that are based on maths are all around us every day, from our computers to the food we eat. Whenever the maths that is used in a trick has some particularly interesting applications to the real world or to different professions, you will see the applications symbol and a brief explanation. You can always do some more research into any of these applications and you will definitely find many more than we had room to mention. So on with the magic. Are you watching closely?Contents The Magic of Basic Mathematics 1 Addition – It All Adds Up To A False Cut 2 Addition and Subtraction - The Dented Card Trick 4 Addition and Subtraction – The Force of Nature 6 Multiplication – Showing Double Digit Dexterity 8 Multiplication and Addition – Doing Fibonacci’s Lightning Calculation 10 Division - The Fast Five Trick 12 Factorising - The Calculator Beating Trick 16 Addition, Subtraction and Psychology - The Teleporting Card 20 Even and Odd Numbers – The Piano Trick 24 Basic Mathematics- The Applications 26 Binary Numbers- The Super Memory Experiment 28 Binary Numbers - The Applications 32 Ternary Numbers - The Card at any Number Trick 34 Algebra - The Brain Control Experiment 38 Algebra – The Number of Matches Prediction 42 Algebra and Addition – The Amazing Coincidence 44 Algebra -The Applications 47 Prime Numbers – Twice the Impossible Location 48 Prime Numbers- The Applications 50 Geometry – The Find a Card by Psychic Aura Trick 52 Geometry - The Applications 54 Statistics- Quick Think of a Number, The Street Magic Stunt 56 Statistics - The Applications 58 Advanced Magic and Mathematics 60 Clock Arithmetic – The Fairest Test (ever) of Psychic Skills 62 Clock Arithmetic – Professional Deck Stacking 66 Modular arithmetic- The Applications 76 Final Words 78 Bonus Effects – The Maths Hustle (or How to Win with Maths) 80 Bonus Hustle Effect 1- The Glass Challenge 81 Bonus Hustle Effect 2 - The Napkin Game 83 Glossary of useful terms 84“Inventing a magic trick and inventing a theorem are very similar activities.” Mathematician and magician Professor Persi DiaconiAddition It All Adds Up To A False Cut For many of these tricks, you will need to have certain cards in certain places in the deck for them to work. Magicians call this positioning of cards in a deck stacking the deck. But, as your audience might suspect you’ve been up to something sneaky before the trick starts, that’s where false shuffles and cuts come into play. It looks like you are fairly mixing the deck while in fact you do no such thing and the cards are in the same order at the end as they were at the beginning. Here’s one of the simplest false cuts. It looks like a real gamblers cut but doesn’t change the order of cards in the deck at all. Put your pack on the table. Let’s call this pile A. Now cut off about the top third from this pile and place these cards to the right. Let’s call this pile B. Now cut half of what’s left in pile A and place this further to the right of pile B. We’ll call this pile C. All that remains is for you to pause, then reassemble the pack. Place pile B onto pile C, then take this combined pile and place it on the cards in pile A. To the audience this looks like a fair series of cuts, but if you try it with a pack you will discover that the pack is in exactly the same order at the end as it was in the beginning. 2 1 3 4 pile A pile B pile C 2Why? Well, all this false cut does is show that when you first cut the cards into piles, you put the top third in pile B, the middle third in pile C and leave the bottom third in pile A. So, going from left to right, your pack is cut into three piles A (bottom), B (top) and C (middle). When you gather them back together, you put B on top, then C in the middle and then A at the bottom – exactly where they came from It’s obvious that the order of the cards stays the same, it is just simple arithmetic, but done in an offhand, casual way while telling the audience you really are mixing the cards they will believe you. It is enough to confuse the onlookers, and a useful way to start your tricks too. It looks like a real gambler’s cut but doesn’t change the order of cards in the deck at all. 3Addition and Subtraction The Dented Card Trick You demonstrate that your ‘super sensitive fingertips’ can find the dent in a chosen card which has been tapped by a spectator, even when it happens with your back turned. First your spectator shuffles the deck. No funny business - it can even be their own battered deck. Then you ask them to choose any number between 1 and 10. You turn your back and tell them to count off the top cards, one by one, until they reach their secret chosen number and then tap and remember the card at that number. They must leave it where it is. With your back still turned they then replace the cards and give the deck to you. You dramatically turn to face them, explaining that you will shuffle the cards then try and read them with your sensitive fingertips. You will then move their card to a different position in the deck. Meanwhile, you have casually put your hands behind your back. As you are talking, your hands are doing some secret counting. Rapidly count off exactly 19 cards and as you do, reverse their order, then replace them on the top of the deck. Announce that you have been successful You found their ‘dented’ card and have moved it to the twentieth position in the deck. Bringing the pack into view ask your spectator what number they thought of. Suppose they tell you it was 7. You start to count cards out loud from the top of the deck beginning, “8, 9, 10 …” and so on ‘feeling’ each one as you go. When the count reaches 20, pause. What card did they ‘dent’? They name the card. You turn over the twentieth card and it’s theirs 4So how does it work? Your spectator chooses a number between 1 and 10, let’s call it X. Their card starts out in position X. Behind your back, you reverse count off the top 19 cards so the card that was at X from the top is now at position 19-X+1. (To see why we add 1, imagine they chose X=1. After the reverse count off, the chosen card is the top card. It doesn’t move to position 19-1=18, it moves to position 19. We have to add an extra 1 as we begin counting at 1 rather than 0. So their card is now at position 20-X.) In the example where your spectator chose X=7, after your hidden reverse count, the card is now at position 20-7=13. Your final countdown actually counts off 20-X cards, starting at 7 (the spectator’s X). Mathematically, this is written (20-X)+X and of course this is 20. So their card will always end up at position 20 no matter what number they originally chose Addition and Subtraction The Force of Nature While your spectator thinks they are completely in control, you are able to force them to select a card of your choosing. You can then reveal the value of this card in any clever way suitable. For example, you could have texted or emailed it to them earlier. Get them to read the message now. Seemingly allowing the spectator a free choice but actually ensuring the card you know is selected, is a basic magician’s tool and the basis of many a good trick. It’s called a force. There are loads of ways to do it but here is a simple mathematical way. For the first force, secretly put the card you want selected on the top of the deck. Ask your spectator to tell you a number between 1 and 10. Explain to them that their choice was free and that you want them to count down one card at a time to their chosen number. Before they do this and to help things along, you demonstrate what they need to do and count down that number of cards one at a time onto the table, so reversing their order. You then scoop up the cards from the table, pop them back on top of the deck and pass the complete deck to the spectator. Get them to do the counting and their freely chosen card is, exactly the one you knew it to be. This works because of simple addition and subtraction. Say they choose 4. Your example deal of four cards onto the table puts your force card (previously on top of the deck) at the bottom of the tabled pile. Putting this tabled pile back on the deck and letting the spectator deal again reverses that, so the 4th card down now becomes the top card on the pile on the table. The two deals cancel each other exactly. Done casually, the spectator won’t remember that you helpfully counted the cards out the first time so don’t remind them. Leaving a period of time between your first demonstration deal and their later actual deal is called time misdirection. Speak to them during this stage. Ask questions: why did you choose 4? Has that some special significance? And so on, to help them to ‘forget’ your casual useful demonstration. You can even throw in a false cut after you did the first deal to remind them they are counting out using a ‘shuffled’ deck and consequently to make your powers seem even more amazing. Clearly, to make this work, you need to know the number they selected. So in the presentations, make sure you say something like, “Give me any number…” rather than, “Think of a number…”, unless, of course, you can mind read You could also start to count down from 10 to 1 and ask them to stop you at any number, or ask them to write a number on the board. Get creative. But get that number 7Multiplication Showing Double Digit Dexterity Most people can do a speedy multiply by 10. You just add a zero to the end of the number – 23 multiplied by 10 is 230, simple. Now you can prove your superior mental superpowers by speedy multiplication of any two-digit number by 11. You explain to the audience that this is clearly far more difficult. They have the calculators on their mobile phones ready to check, but you do your calculations correctly before they even start to click the keys. Imagining the 11 times trick To give us this superpower, we make use of two things. One is maths and the other is the human brain’s power of imagination. To do a lightning calculation multiplying any two-digit number by 11, you need to use some visual imagery and use your imagination. Let’s take the number 2 for example. Now imagine a space between the two digits, so in your minds eye you imagine  2. Add the two numbers together and imagine putting the sum of them in the gap in the middle, so you see  (+2) 2. And that is it, you have the answer: 11x2 = (7) 2= 72. The double trouble trick But what if the numbers in the gap add up to a double digit? For example, suppose you want to multiply 98 by 11. So, you imagine 9 (9+8) 8. But that bit in gap in the middle gives you 9+8=17, so where do you put these digits? 9 (17) 8? Easy, just leave the second number (here the 7) in the gap as before and imagine moving the 1 up a place, so you have (9+1) 7 9 = 10 7 9 = 1079. Correct again. 8The maths behind this is fairly easy if you explore it. Suppose you have the number AB (that’s A tens and B ones) and you want to multiply by 11. First you multiply by 10. That’s easy, 10xAB = AB0 (A hundreds, B tens and 0 ones). Then you add another AB so you’ve got 11 lots of AB altogether, giving you A hundreds, (B+A) tens and (0+B) ones. This is exactly what all that sliding numbers around in your imagination has been doing without knowing it. Of course, if the middle A+B is more than 10 (ie it’s a double digit number), you just slide the first digit up to the hundreds column, and it’s sorted. Lightning calculations by mind power and a maths trick. Mathematicians make use of their imaginations all the time. Our brains are really good at imagining things and creating pictures in our heads. Often that’s the way we solve tricky problems or come up with clever visual imagination tricks like this one. 9Multiplication and Addition Doing Fibonacci’s Lightning Calculation On a piece of paper, write the numbers 1 to 10 in a column. You are now all set to amaze with the speed at which you can add ten numbers. Ask your friend to choose any two two-digit numbers and write the numbers down in the first two spaces of your column, one under the other. Your friend then makes a third number by adding these first two numbers together and writes it below the first two, in effect starting a chain of numbers. They make a fourth number by adding the second and third, a fifth by adding the third and fourth, and so on, until your column of ten numbers is full. To show how brilliant you are, you can turn away once your friend has understood the idea, say after the seventh number in the list. Now you can’t even see the numbers being written. Meanwhile, with your back turned, you are actually multiplying that seventh number by 11 to get the final answer. Let’s imagine your friend chose 1 and 21 to start with. The list would look like this: 1. 16 2. 21 3. 37 4. 58 . 95 . 153 7. 248 8. 401 9. 649 10. 1050 You now turn round and write the sum of all ten numbers straight away Lightning quick, you say it is 2728. Let them do it slowly on a calculator to show your brilliant mind skills are 100 per cent correct. The final answer just involves multiplying the seventh number by 11. Why? 10Well this chain of numbers where the next term is made by adding the previous two terms is called a Fibonacci sequence. Fibonacci sequences have special mathematical properties that most folk don’t know about… So let’s look at the trick. We start with the two numbers A and B. The next number is A+B, the next number is B added to A+B which is A+2B and so on. Going through the number chain we find: 1. A 2. B 3. A+B 4. B+(A+B) = A+2B . (A+B)+( A+2B) = 2A+3B . (A+2B)+( 2A+3B) = 3A+5B 7. (2A+3B)+(3A+5B) = 5A+8B 8. (3A+5B)+(5A+8B) = 8A+13B 9. (5A+8B)+(8A=13B) = 13A+21B 10. (8A+13B)+(13A+21B) = 21A +34B Adding up all 10 numbers in the chain gives us a grand total of 55A+88B – check it yourself. But look at the seventh number in your column… this line is 5A+8B. It is exactly the total of the chain but divided by 11 So working backwards, you can get the final total by multiplying the seventh term by 11. And the maths proves this lightning calculation will work for any two starting values A and B. It is up to you to present this trick in such a way that it looks like you are just very, very clever. Which of course you are, as you now know how to use a Fibonacci sequence for magic. 11Division The Fast Five Trick Look all around, fives are everywhere. Five fingers, five toes. Now think about division. Divide by 10? Easy you say – just move the decimal point one place to the left. But by ? Far more of a challenge Now you can impress you friends with your ability to divide any number by  at super speed - and do the calculation correct to three decimal places Not a lot of people know this, but dividing a large number by  can actually be done by a very simple two-step method. Step 1: multiply the number by 2. Step 2: move the decimal point. That’s all there is to it. Now you can impress you friends with your ability to divide any number by  at super speed. Of course, your claim to do the calculation correct to ‘three decimal places’ is just presentation to make it sound more impressive. It’s really easy. All you do is get the answer, say 23.7, and add two zeros after the 7 to give 23.700. So that’s the answer right to three decimal places as you promised. Our maths proof below will actually show us that we will only ever get an answer with one decimal place. The rest is just magical flim-flam 12It is always a good idea to start easy with a simple example to test our method works so let’s look at 10 divided by . First, our method says we double the number, so doubling 10 we get 20. Then we shift the decimal point, so 20 now becomes 2.0 or 2. The answer’s correct. Good start. Now for a harder example, say 190 divided by . This looks tough without a calculator, but following our method, step 1, double 190 to get 380, and step 2, move the decimal point, so 380 becomes 38.0 or just plain old 38. Harder still: how about 47 divided by . Again, step 1, double the number, so we have 47 x 2 = 9134. Then step 2, shift the decimal point to get 913.4. It’s correct - check with a calculator if you don’t believe it. And lets get silly: 1234789 divided by . Step 1, double it, so 2 x 1234789 = 24913 78. S  tep 2, shift the decimal point and the answer is 24913 7  .8. (You may really want to check this on a calculator) Will it always work? Could there be a number somewhere that you haven’t tested and that the trick doesn’t work for? That would be embarrassing so let’s look at the maths. That way, we can be sure as we can prove it will always work and save red faces all round. Dividing by five is like doing half of a divide by 10. Dividing by 10 is easy, you just shift the numbers down a ‘slot’, so 100 divided by 10 is 10.0, 23 divided by 10 is 2.3 and so on. Well, division has a rather neat trick or mathematical property of its own. As you know, fractions are just another way of writing divisions. So 100 100 / divided by 10 can be written as and 23 divided by 10 can be 10 23 written as . / 10 13Division The Fast Five Trick And if you do the same thing to the top of a fraction as you do to the bottom, the answer to your division (or the ratio of the top to the bottom) remains the same. So multiplying the top and bottom by 2 each time, we have 8 4 1 = = = 2. The r / / esult is always the same. 2 / 4 8 A / Taking any number A and dividing by  means that is e  xactly the same (2xA) as . Here we have just multiplied top and bottom by 2. / 10 And there we have it - that simple formula proves the maths. To divide ANY number A by , first double it, step 1 and then shift the decimal place, that is divide by 10, step 2. We can see it will always work. What we have done here is what mathematicians and computer scientists do all the time - no, not divide by  They to come up with a series of steps (which we call an algorithm) and then try to show mathematically that it will always do what is supposed to. A Challenge Try a bit of maths yourself. Here is a two step algorithm to multiply a number by : Step 1: divide by 2. Step 2: move the decimal point to the right. Can you work out if this will method will always work? 141Factorising The Calculator Beating Trick You start by asking for a random four-digit number from anyone in the audience. You write it up on a board. Now, you explain, in a moment the next volunteer will give you a second four digit number and you will multiply them in your head faster than anyone else can, even if they are using a calculator. But wait If they already know the next number, they can start the calculation early To make it fair, as soon as they say the second number, you will also pick a new four-digit number and multiply the first number by it. To make it even harder, you will perform both multiplications and then add the answers together, to get the total faster than anyone with a calculator. And sure enough, you do Make a big deal of the fact that you will be multiplying two random four-digit numbers faster than a calculator. Then play down everything else that you add to the calculation, make it look like an after-thought. Afterwards, people will only remember that you multiplied together random numbers given by the audience. Begin by writing the first number from the audience on the board. Indicate below where the next number will go, write up a multiplication sign next to it and show where you will write the answer. Then, to the right, write the first number again, with a new multiplication sign and indicate where you will put your random number and then that answer. Where the two answers go should line up so you can put an addition sign between them and an equals sign below. 1