Maths tricks for fast Calculation

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Published Date:02-07-2017
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1 Maths Made Magic Jason Davison is a Master of Mathematics (MMath) student at Warwick University. We hope you will be able to use this book This is where his interests in magic and in the classroom to help you teach many of Maths began to combine. He is a semi- the basic concepts in mathematics in an professional magician, putting his skills to engaging and entertaining way. All the tricks use at weddings and business events. He are self-working and easy to do, and have will be teaching with TeachFirst and is been tested in UK classrooms. For UK enthusiastic about teaching Maths using teachers the tricks have been mapped to magic tricks to make it fun for kids. many of the topics for mathematics at Key Peter McOwan is a Professor of Computer Stage 4 however teachers in other countries Science at Queen Mary, University of will also find this material useful. The book London. He has a particular interest may be photocopied for non-commercial in artificial intelligence. He uses his life use, and there are a number of easy to long interest in magic to help teach copy worksheets also included. mathematics and science concepts A sound understanding of basic and was awarded the IET Mountbatten mathematics underpins science and medal in 2011 for promoting computer engineering. We hope this book helps science to diverse audiences. you teach these subjects and gives you and your pupils the opportunity to develop presentation and communication skills. It may also open the door for others creating new mathematical magic and start some people on the road to a fascinating new hobby too.3/4 A handbook of magical mathematical tricks for you to learn 1 Queen Mary, University of London, The Further reading either to learn more Computer Science for Fun project (cs4fn), maths or to take up magic as a hobby Matt Parker, Jonathan Black, Heather Martin Gardner, Mathematics, Magic and Mackay, Richard Hardie, Sidney Stringer Mystery, Dover Publications (or any of his Academy, Howard Williams, Gemma Brittle numerous maths, magic and puzzle books) and her students at Beaverwood School for Karl Fulves, any of his ‘Self Working’ Girls, Chislehurst UK, Glendina Reid and series from Dover Publications Paul Curzon. Paul Zenon, Street Magic, Carlton This book was developed as part of the Books Ltd. Strategic Partnership between Queen Mary, University of London and the To learn about some amazing advanced University of Warwick. This collaboration is maths, tricks and the history of an agreement between two leading mathematical magic see Magical universities that aims to enhance each Mathematics By Persi Diaconis and institution's strengths in teaching, learning Ron Graham and research through an expanding range of joint initiatives. The Partnership is also Free to download pdf books intended to play a leading role in defining Manual of Mathematical Magic the future of universities in the UK and beyond. The Magic of Computer Science Vol 1 and Vol2 Illusioneering, the Magic of STEM This book is dedicated to Kinga Garriott de Cayeux. A new world of magic awaits.2 2 M Ma atth hs s M Ma ad de e M Ma ag giic c Within the pages of this book you will find In this book each nugget of magic and inventive illusions and mind-melting magic illusion is divided into 3 parts: “The Tale”, to amaze you, your friends and family. where you will be given the background You will also explore some fundamental and information needed to understand mathematics powering the tricks. Learning each magical effect, “The Magic”, where and entertaining go hand in hand in magic. the effect will actually happen, and Understanding the secret techniques, “The Secret”, where the illusion will be combined with fantastic presentation and explained further, and the hidden power practice can make any trick unbelievable. of mathematics will be made clear to you. The greatest magic tricks in the world If you are performing the tricks and always have simple but cleverly hidden illusions found here to your audience then solutions. Many are mathematical, but take care to not break the magician’s code. the presentation is so brilliant that the Practice hard and make sure the secrets audience’s minds don’t want to look for stay secret. Finding out your own style for the solution. An audience may even performing the tricks and illusions will help stumble upon the correct solution but will to give you a truly magical aura. There is immediately ignore it if your presentation enough material in this book for you to put makes them want to believe in magic. on a show, take up magic as a hobby or Magicians world-wide also know that the even start a magical career quality of your presentation can also affect The book will also let you explore some of the memories of your magic. Creating your great presentation means that when your the great theories of mathematics behind the tricks, mathematics which you will be magic is remembered in the minds of your spectators, it will be embellished, and able to use in whatever career you follow. Perhaps it will inspire you to discover seem even greater. This is what magicians call the art of creating astonishment. more and deeper astonishment in the realm of mathematics as well. We hope this book will help you entertain, amaze and learn.1 A handbook of magical mathematical tricks for you to learn 2 /2 The Prediction Hex and the Mathematics of Sequences 20 Transform Hocus-Pocus Changes and the Mathematics 1 of Transformations 21 /2 The Wizard of Mischief’s Payment and the power of the Mathematics The Vanishing Potion using the of Powers and Indices 23 power of the Mathematics of Area 3 Haunted Woods Magic and the 1 Thought Witchery using the power Mathematics of Locus 23 / 2 1 of the Mathematics of Algebra 4 /2 Thought Reading Sorcery and Enhanced Senses Incantation the Mathematics of Factors 25 using the power of the Mathematics 1 of Symmetry 6 /2 A Magic User’s Investigation of the 1 Mathematics of Sine and Cosine 26 / 2 Unique Conjuring using the 1 Mathematics of Inequalities 8 /2 Impossibility Enchantment and the power of the Mathematics Mind Control Spell and the of Probability 28 Mathematics of Fractions 1 and Percentages 10 / 2 Your Element of Destiny and the power of the Mathematics of Your Animal of Destiny and Pythagoras’ Theorem and the Mathematics of Circles, Simultaneous Equations 30 and Rounding 12 The Impossible Drinking Glass and The Challenge of the Magic User the power of the Mathematics of 1 and the Mathematics of Ratios Circumference and Circle Area 32 /2 and Algebra 14 Notes for Teachers 34 The Movement Charm and the Mathematics of Vectors 16 Mind Expansion Jinx and the Mathematics of Mental Methods 183 Maths Made Magic You master a spell to make a square vanish Now let’s look at the area of a triangle. Imagine we have a triangle that is 5cm If we have a rectangle that is 5cm tall tall and 13cm wide. What is the area of the and 13cm wide what is the area of the triangle? Well, imagine we had two of these rectangle? In this grid we can see that if triangles. We could then put them together the space between each parallel line is to make a rectangle that is 5cm tall and 1cm (parallel lines are two lines that point 13cm wide. in exactly the same direction) then each 2 square is a “square centimetre” or 1cm . So when we ask what the area of the rectangle is, we are asking how many square centimetres the rectangle contains. We know from before that a rectangle like 2 this has an area of 65cm . This is then also the area of 2 triangles. So if the area of 2 2 identical triangles is 65cm then the area of 1 triangle must be half of the area of the Imagine you have a group of apples and 2 rectangle. So it is 65cm divided by 2. This you arrange them so there are 5 rows and 2 is 32.5cm . each row has 13 apples. In total you then have 5x13=65 apples. The same is true if we now think of the squares in the rectangle. We can think of the rectangle as having 5 rows of squares. Each row has 13 squares in it. So the total number of square centimetres is 5x13=65. So the 2 area of the rectangle is 65cm .1 A handbook of magical mathematical tricks for you to learn 3 /2 So the area of the triangle should be the same but now there is a piece missing. Now for the magic Look at this triangle. This surely doesn’t make sense. This We can see it is 5cm tall and 13cm wide. picture seems to show that the area of the 2 I have cut the triangle up into different shapes from the first triangle (32.5cm ) 2 shapes . plus an extra 1cm gap still makes a 2 triangle that is also 32.5cm . Impossible. Where is this extra square centimetre coming from? This is a trick. The shapes to begin with don’t actually make a triangle. The longest edges (the “hypotenuses”) of the smaller triangles are not perfectly parallel. So the Look at what happens now when I move longest edge of the main big triangle is not the shapes around. We can make another perfectly straight. There is a small bend triangle the same size that is still 5cm tall which means that the area of all the and 13cm wide. shapes together in the first picture is a little bit less than the area of the triangle we think we are looking at. When we put the shapes in the second position the longest side of the triangle is still not straight; it bends outwards very slightly. The difference in area of the large triangles made by changing the smaller shapes4 Maths Made Magic 2 around is equal to 1cm . You can see One reason why the small shapes don’t 2 this extra 1cm along the long edge if perfectly make a large triangle is that the we overlap the two triangles made: two smaller triangles aren’t “similar”. If two shapes are similar then they have matching angles (but the lengths of the sides might be different – one of the shapes must look like a zoomed in/out version of the other shape). The angles in the small triangles don’t perfectly match the relevant angles in the big triangle we would like them to fit into. At the start of the trick I pretended that the triangles fitted perfectly, but they don’t. Clearly the total (sum) area of all the smaller shapes stays the same no matter what pattern they are placed in. If we work out this area, the area of the pink shape is 2 2 7cm , the area of the blue shape is 8cm , and the area of the green triangle is the height multiplied by the length divided by 2 = 2x5/2 = 5. In the same way the area of the red triangle is 3x8/2 = 12. So the total 2 area is 7+8+5+12=32cm which is half a square centimetre less than the area of the triangle we thought the shapes fitted into.1 A handbook of magical mathematical tricks for you to learn 4 /2 Mysteriously, a number thought of by you is drawn from your mind Algebra is where we study and use Now for the magic Let’s walk through numbers that are a little mysterious. When and explore…. we use algebra we might actually not know TRICK VERSION 1 Get your calculators out exactly what the numbers are. They could or switch your phone to calculator mode, be 3, 100 or -2.455, but we do know and think of any number. something about how the numbers combine together and using algebra Write this number down on a piece we can work them out. This may sound of paper. a little confusing, so let’s do some magic. Now add 5, Multiply by 2, Square (i.e. times the number you now have by itself), Minus 100, Divide by your original number, Minus 40, Divide by your original number, Add 4. If my powers of mind reading are strong enough you should now have the number 8. How did this work? Well we can actually explain it with algebra. First let’s try a simpler one:5 Maths Made Magic TRICK VERSION 2 Write down a number, Now to explain this let me first tell you that Add 5, unfortunately I am not a true mind reader. Multiply by 3, Although you may think your original Minus 15, choice of number would change the Divide by your original number, number at the end, in fact it doesn’t make Add 7. any difference what your start number is. Again, if my powers of mind reading are For example in version 3, I have no idea strong enough you should now have the what number you picked at the start. So for number 10. me you picked the number “?”. It’s just a question mark for me, I have no idea what Or an even simpler one: the number is. TRICK VERSION 3 Now let’s look at the third version Write down a number, of the trick. Add 5, Minus your original number. First I ask you to write a number, (for me this gives “?”) Again, if my powers of mind reading are strong enough you should now have the Then I ask you to add 5, number 5. Are you starting to see how I’m (for me this gives “? + 5”) tricking you? Then I ask you to minus your original number (for me this gives “? + 5 - ?”)1 A handbook of magical mathematical tricks for you to learn 5 /2 In algebra, we don’t usually use the One useful technique in mastering algebra “?” sign. We use “x” or another letter is called rearranging an equation. To start to represent the number we don’t know. off we can see what happens when we just use numbers, it’s obvious you get the same We can also “simplify” brackets. This answer, 2, if you write 3 + 7-8 = 7 + 3 - 8 means that if we had “3 × (x + 5)” we = 3 - 8 + 7 = 2. What’s important to could write it as “3 × x + 3 × 5”= “3x + remember here is that 3 and 7 are positive 15” without brackets. Check this is true numbers, so there is always a “+” sign in for yourself: front of them (but when we write the 3 or the 7 at the front of the equation that Look at: “+” sign is invisible, if we wanted to do it “x × (y + z) = x × y + x × z” longhand we could write + 3 + 7 - 8 = 2, but that could be confusing, it could look Choose any three numbers. Swap “x” like something was maybe missing at the for the first number, “y” for the second front, so we lose that + at the front, and so number, “z” for the third number. Then long as we remember it’s really there check that the equation above is true. everyone is happy). In our example the number 8 is negative, there is always a Let’s look at example 2 and use “x”: “-“ sign in front. As long as we keep the You write down a number: signs in front of the right numbers we can (for me this gives “x”) change the order as much as we like and it will give the same result. So we can change Then I ask you to add 5: the order of “? + 5 - ?” to “? - ? + 5” = “+ (for me this gives “x + 5”) 5” = 5. Then I ask you to multiply by 3: (for me this gives “3 × (x + 5)” = “3x + 15” from the rule above)6 Maths Made Magic Then minus 15: The equals sign means both sides of the (for me this gives “3 × x” or “3x”) equation are the same number. Take “x - 5”and add 5. This gives “x - 5 + 5” which Divide by your original number: equals “x”. (for me this gives 3) x - 5 + 5 = x Add 7: But from the first part we know that (for me this gives 10) x - 5 = 12, so replace the x - 5 with 12: 12 + 5 = x So again, the final number doesn’t have 17 = x an “x” in it which means the answer will always be 10 and it doesn’t matter what We have basically added 5 to both sides of the start number that you choose is. the equations. Now we know the mystery number x is 17. This trick gives you an introduction into algebra and how it works. In algebra there is a number (“x”) and you don’t actually know what the number is. Let’s say someone told you that this number minus 5 gives 12. Well then the number we don’t know is “x” so: x - 5 = 121 A handbook of magical mathematical tricks for you to learn 6 /2 An enchantment allowing you to increase the power of your senses to find a card just by looking at the faint fingerprint mark left by your spectator. M M There are three types of symmetry you’ll need to know about: Line symmetry, Plane symmetry and Rotational symmetry. Things are “line” symmetric or “plane” symmetric M M if you can place a mirror on a line or plane and the shape looks the same with or without the mirror. It is perfectly reflected In this diagram we try to find symmetry in the mirror. A “Plane” is a completely by placing a mirror on the diagonal line flat surface like a piece of card. between two corners. The red shape shows what the part of the rectangle above the For an example of line symmetry look at mirror looks like when it is reflected. You this diagram of a rectangle. Imagine we can see it does not match the rectangle. place a mirror at the line “m”. We can If the rectangle was square there would see how the rectangle has two lines of be diagonal symmetry as well. symmetry. The dotted lines in the lower diagrams are what the shape on the “m” side of the mirror would look like if it was reflected in the mirror. Notice that the reflected image looks the same as the M original rectangle. 7 Maths Made Magic Plane symmetry is similar to line symmetry Rotational symmetry is when we can fix a but we look at 3D shapes. For example, a point on a shape, rotate the shape around plane of symmetry of the cube is shown that point an angle more than 0 and less by the yellow plane . If there was a mirror than 360 such that the shape looks the where the plane is the cube would look the same after it’s been rotated. same if we looked through the mirror as it Look at this shape. If we rotate the shape would without the mirror. around the centre 120 degrees and 240 degrees the shape will look exactly the same, so it has rotational symmetry.1 A handbook of magical mathematical tricks for you to learn 7 /2 half of the card. Now put all these cards that are not “rotationally symmetric” on the Now for the magic Look through your top of the other cards. You are now ready pack of cards. You will notice that for for the trick. example the Jacks, Queens and Kings Spread the cards on a table face up to have rotational symmetry around the show your spectator that it is a normal middle of the card. You could turn the card pack. Then spread the cards face down on 180 degrees (upside-down) and it would the table and point at the bottom card on look the same. There are some cards this the deck. As you move your finger from the is not true for. Look at the 7s, they look bottom to the top card tell your spectator to different if you turn them upside-down, say stop whenever they want, but make they have no rotational symmetry. Now sure by the time you’ve said this, your remove all the cards that have no rotational finger is already in the top TWENTY TWO symmetry. cards (the ones not rotationally symmetric). (Try and do this yourself, but in case you When they say “stop” bring your finger need a hand, these will be: the Ace, down on a card. Get them to take that 3,5,6,7,8,9 of Hearts, Clubs, Spades and card. Collect the cards up again and turn 7 Diamonds). Now arrange these cards the deck around so that when they put so most of the suit symbols (e.g. the heart their card back in it will be the other way symbols) are facing the right way. For round (in terms of the symmetry). You example, if the 7 Clubs has 2 clubs facing can then give the pack a normal shuffle. up and 5 pointing down turn it so 5 are Your spectator can even normally shuffle pointing up and two down instead. Do them too. this for all the cards, the 7 of Diamonds is Now tell them you will look for their different because the diamonds look the invisible fingerprint on their card with your same either way but put it so the pattern of amazing heightened vision skills. Look at the diamonds is the same as on the Hearts and Clubs, with 5 of the symbols in the top8 Maths Made Magic the spectator’s fingers and “study” their fingerprints (this is just acting). Now That’s most of what you should know about deal the cards out one by one on the table, symmetry. But as well as this make sure “looking for fingerprints”. Secretly you will you understand the “order” of rotational know when you see their card as it is the symmetry. The order is the number of only non-rotationally symmetric card facing different positions you can put a shape the wrong way Magic. into by rotating it to give a shape that is the same. For example, a triangle has order 3 because you could turn it so that any of the three corners are at the top and the triangle would still look the same.1 A handbook of magical mathematical tricks for you to learn 8 /2 Somehow, I know the potion you will create Here is a spell using inequalities. The Now for the magic First you must create symbols you’ll need to know for this are your potion. To do this, shuffle your pack (and ≤ ≥). You place these symbols of cards. Each order the cards can be in where you might find an equals sign “=” represents a different potion. Once you’re (so between two mathematical phrases). happy with your potion you’ll need to find But instead of the phrases on both sides of the substance that potion creates with this the symbol being equal, one is larger than process: the other (but with ≤ ≥ both sides can also Put a Joker on the top of the pack and be equal). The arrow points to the smallest then turn the pack and hold it face up so side of the equation. You can think of it as you can see the faces. a greedy crocodile symbol that will always try to eat the bigger phrase. So for example Work out “14 - the value of the first card” we can say 74. We should also look at (Ace = 1, Jack = 11, Queen = 12, King = how these symbols work in Algebra. If we 13). If this new number is BIGGER than were asked to show a region where xy the value of the second card then put then we should draw the graph of x=y and these two cards to the back behind the then shade the region above the line as for Joker. If the new number is smaller or all these (x,y) co-ordinates xy. equal to the second card then get rid of the two cards on the table somewhere. Do this repeatedly until you get to the Joker then put the Joker to the bottom again.9 Maths Made Magic Now work out “11 - the value of the Do this repeatedly until you get to the Joker first card” (Ace = 1, Jack = 11, Queen = again. Hopefully you’ll now be holding a 12, King = 13). If this new number is few cards (if you don’t have any left shuffle SMALLER than the value of the second all the cards and try again). Now get rid of card then put these two cards to the back either the first two cards or the last two behind the Joker. If the new number is cards. Do this again until you only have two bigger or equal to the second card then cards left in your hand and add the value get rid of the two cards on the table of these two cards together. Now to find somewhere. what your potion was worth find this new number in the list. The sum of these two cards must be between 2 (if they are two Aces) and 26 (if they are two Kings): Gold 2 Iron 18 Moonlight 7 Stone 13 Silver 19 Silk 10 Grass 15 Starlight 5 Gold 2 Iron 18 Moonlight 7 Stone 13 Silver 19 Silk 10 Grass 15 Starlight 5 Bronze 14 Copper 4 Cotton 16 Ice 21 Diamond 6 Rubies 9 Ivory 26 Oak 8 Steel 23 Emeralds 17 Polish 3 Wool 20 Stone 12 Leather 22 Smoke 25 Pine 11 Tin 24 Rust 26 If the predictive powers of this spell book are strong then you will create Stone. Magic?1 A handbook of magical mathematical tricks for you to learn 9 /2 So all the pairs you To understand this trick we must work with now had inequalities. There are 52 cards in a pack. represented As this is an even number we can group co-ordinates the pack into pairs and move cards in each in this blue pair. Each pair in your shuffled pack section of represents a co-ordinate on a graph. the first The first card in the pair represents the graph. “x” co-ordinate, the second card is the “y” Second, I co-ordinate. So the pair of cards (Ace, Two) asked you to only keep the pairs such that represent the point that is 1 unit across “11-value of first card is smaller than the and 2 above the 0 point of the graph value of the second card”. Again, using co- (the 0 point is called the origin). ordinates, this means we only keep the co- ordinates such that y11-x. So again this First, I asked you to keep all pairs such would give the red graph that looks like that: “14 – value of first card is bigger this: than the value of the second card” and remove the others. Representing this with So all the the co-ordinates means we only keep the pairs you co-ordinates such that y14-x. Let’s now had also represent this on the blue graph; we draw represented the line of y=14-x and shade in the co- co-ordinates ordinates on the side of the line such that in the red the “y co-ordinate” is less than “14 – the x section of co-ordinate” to get this: the second graph. At this10 Maths Made Magic point the cards you held were co-ordinates find the corresponding value of y. This in both the red and the blue sections. As gives you the co-ordinate at x that is on they were in both sections, they must have the green line. Now add x and y and you been in the band where the section meets, should get 13. So when you eliminate pairs and that looks like this: and end up with just one pair, that pair will add to either 12 or 13. Now look at the list of items again, you’ll notice that both 12 and 13 are worth stone. So once again, the maths helped us to create an impossible feat. A lot of the co-ordinates in this band are not whole numbers. The only co-ordinates that are whole numbers and that are inside this band will be points that are on either the green or red lines. These are the lines y = 12 - x and y = 13 - x. For any x, y co- ordinate on these lines, x + y will then be 12 (for the first line) or 13 (for the second). You can check for yourself, pick any value of x. Use the line equation “y = 12 - x” to

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