Mathematical Explorations of Card Tricks

Transcription

1 Joh Carroll Uiversity Carroll Collected Seior Hoors Projects Theses, Essays, ad Seior Hoors Projects Sprig 2015 Mathematical Exploratios of Card Tricks Timothy R. Weeks Joh Carroll Uiversity, Follow this ad additioal works at: Part of the Mathematics Commos Recommeded Citatio Weeks, Timothy R., "Mathematical Exploratios of Card Tricks" (2015). Seior Hoors Projects This Hoors Paper/Project is brought to you for free ad ope access by the Theses, Essays, ad Seior Hoors Projects at Carroll Collected. It has bee accepted for iclusio i Seior Hoors Projects by a authorized admiistrator of Carroll Collected. For more iformatio, please cotact coell@jcu.edu.

2 Mathematical Exploratios of Card Tricks Timothy R. Weeks Joh Carroll Uiversity Seior Hoors Project Sprig 2015 Barbara D Ambrosia Joh Carroll Uiversity Departmet of Mathematics ad Computer Sciece Project Advisor

3 W e e k s 2 Itroductio I this project, we explore various mathematical topics as they apply to a assortmet of card tricks. We will focus o a examiatio of theorems applied to the maipulatio of cards i a attempt to prove why certai card tricks work. These theorems utilize abstract algebra, probability, umber theory, ad combiatorics. May tricks ca be explaied this way, istead of sigularly by sleight of had or other magical methods. We will rigorously prove the theorems ad priciples that explai these cocepts, focusig primarily o the card tricks ad examples preseted i Mathematical Card Magic by Colm Mulcahy (2013). While the study of pure, theoretical mathematics is very iterestig, it is also helpful to see how the subject ca be applied. Applicatios of math are prevalet ad ca be very practical, such as i egieerig ad ecoomics. The field of mathematical magic combies the academic aspect of mathematics with the etertaimet of card tricks ad magic. Thus, this topic shows that uderstadig mathematics ca result i somethig that eve o-mathematicias ca ejoy. For a future teacher, this project could fit i well with a classroom talk i order to help demostrate how math ca be fu for kids. The required sleight of had ad quick thikig required by the magic aspect ca also prepare the speaker for future work as a teacher. Luckily, the topic of math as it relates to card tricks has bee well researched. The substace of this project ivolves examiig a book by Colm Mulcahy, a well-kow expert i the area of mathematical card tricks. Accordig to Mulcahy, work i mathematical magic started i the early twetieth cetury (Mulcahy xiii). However, math was a recreatioal activity possibly as far back as aciet Mesopotamia civilizatios ad the early Egyptia empires (Merzback 7). I the middle of the twetieth cetury, Marti Garder released large amouts of data o mathematical magic, ad this is whe the field truly bega to flourish (Mulcahy xiii). Mulcahy s

4 W e e k s 3 book goes ito depth both o the magical ad mathematical aspects of the tricks. While he presets theorems ad priciples, he does ot prove the results, or at least ot to the extet that would be expected i a upper level math class. Those rigorous proofs are the primary objective of this project. Mathematical Card Magic has such a wide rage of examples that it has ot bee ecessary to pull tricks from ay other sources i this oe semester project. We start with some mathematical ad card related priciples which will elimiate redudacy ad possible cofusio about vocabulary whe we move ito the explaatio of specific tricks. I each later sectio, we will describe a trick ad explai it mathematically. Whe selectig which tricks to iclude i this paper, we chose those that have iterestig mathematical foudatios. As a secodary cosideratio we thought of practicality of performace, based o the metal mathematics ad advaced sleight of had ecessary for performace. We coclude the paper with possible further ivestigatios related to this project.

5 W e e k s 4 0. Geeral Priciples Before examiig card tricks i the cotext of their mathematical basis, it is importat to have a basic uderstadig of some fudametal priciples. These priciples are a mixture of facts related to math ad/or cards. For example, we use the covetio that a full deck has 52 cards with four suits (hearts, diamods, clubs, ad spades). First, the shufflig of cards is clearly of high importace i card tricks. Uless otherwise stated, we assume that ay maer of shufflig is sufficiet. Some tricks, such as that i V.b, explicitly call for riffle shufflig. Riffle shufflig (or rifflig ) a deck of cards meas dividig it ito two packets, bedig the cards with each thumb, ad releasig the cards so that the cards itermix i a sigle pile. This type of shufflig is ofte doe i tadem with bridgig, which simply re-beds the cards to maitai their shape. A explaatio of riffle shufflig is foud o page 3 of Mulcahy ad may videos olie demostrate this techique. At other times fake shufflig will be required (such as i trick III.a). There are may differet ways to preted to shuffle a deck, with varyig degrees of difficulty. Mulcahy explais differet ways to fake shuffle i his sectio o shufflig (1-13). Sometimes, either istead of or i tadem with fake shufflig, the magicia must kow the order of the cards. We use the memoic word CHaSeD to describe a deck orderig which seems radom, but is easy to remember. The capital letters i the word CHaSeD refer to the four suits ad the order of the letters idicates the order of the suits i the deck. Specifically, i a packet of CHaSeD cards, the suits are i this order: Clubs, Hearts, Spades, Diamods. Sometimes these suits allow us to desigate magitude also, such as clubs beig less tha hearts, which is less tha spades, which is less tha diamods. Usig this, the magicia ca remember

6 W e e k s 5 which cards were preset i the packet. For example, if a ace, two, three, ad four are used, it is easier to memorize their suits if they are i CHaSeD order tha i aother order (Mulcahy 13). The dealig of cards is of special importace. May tricks (especially the tricks i Sectio I) are completely based o the way the cards are dealt. Hece, it is useful to ote that whe cards are dealt from the top of a deck ito a ew stack of cards, the order of the cards is reversed. Coversely, whe cards are dealt from the bottom of the deck, the cards remai i the same order. The former of these realizatios is especially importat for the ext sectio of tricks, which is based o COATig. COAT stads for Cout Out Ad Trasfer ad refers to coutig out k cards from the top of a sized packet ad trasferrig the resultig stack of k cards to the bottom of the packet (Mulcahy 35). As refereced above, the k cards dealt from the top will be i reverse order at the bottom of the deck. Mulcahy uses the term overcoat to refer to this process whe k is required. 2 k. We will simply use the term COAT, ad idicate the istaces i which 2 It is importat to kow how to cout the umber of cards i a ordered sequece. Whe we subtract two whole umbers, we are really coutig the umber of oe-uit gaps betwee those umbers o a umber lie, as opposed to coutig the umbers themselves. We use this priciple whe determiig the distace betwee two cards i a deck. For example 8 5 3, so there are two cards i the packet betwee the card i positio five ad the card i positio eight, ad there are four cards startig with positio five through positio eight. Hece, i geeral, there are k 1 cards i positios k through. This will be especially useful for the sectio o COATs.

7 W e e k s 6 Aother coutig idea that is useful whe doig card tricks is modular arithmetic. Modular arithmetic is sometimes referred to as clock arithmetic, because it fuctios similarly to the fact that two o clock is four hours after te o clock: o a 12-hour clock, I arithmetic modulo, the sum of two umbers is equivalet to the remaider whe the sum is divided by. For example has a remaider of 2 whe divided by 12, thus 14 2 mod12. Fially, the Pigeohole Priciple surfaces several times i this paper. This priciple states that if items occupy k spaces ad k, the clearly at least oe space must be occupied by at least two items. A discussio of this priciple appears i most discrete mathematics books.

8 W e e k s 7 I. COATs We begi our discussio of card tricks by lookig at two tricks that rely o properties of the COAT procedure. I.a. Four Scoop Triple Revelatio. This trick is a combiatio of the tricks Three Scoop Miracle ad Triple Revelatio preseted by Mulcahy (25, 37). Descriptio of the Trick Start by havig three voluteers each pick oe card at radom from a deck that is approximately 13 cards. They should look at ad memorize their cards. Have them place their cards o the top of the deck. Ask a voluteer for his favorite ice cream flavor. If ecessary, ask the voluteer to adjust the ame of the flavor so that it is log eough, i.e. more tha half the deck size (such as chagig mit to mit chip or peppermit ). Tell the audiece that you are goig to make a sudae ad eed to scoop the ice cream. As a demostratio, COAT the cards (as described i the Geeral Priciples sectio) while spellig the ice cream flavor oe card per letter. The istruct each of the voluteers, i tur, to COAT the cards as described above. The had the deck to the last voluteer who placed his card o top. Have him reveal the top card ad otice it is his card. The had the deck to the secod voluteer ad do the same. Fially, do this with the remaiig voluteer. Mathematical Aalysis This trick is clearly a applicatio of COATig. We are iterested i the top three cards ad their movemets throughout the deck. Let the deck cosist of cards ad the umber of letters i the flavor be k, with k. 2

9 W e e k s 8 Suppose the voluteers choose cards x, y, ad z, respectively. The cards x, y, ad z begi o top of the deck i this order. After the first (demostratio) COAT, the three bottom cards are z, y, ad x, i this order. The Save at Least 50% Priciple below demostrates that after 3 COATs, the cards that are origially o the bottom of the deck move to the top of the deck, but i reverse order. Therefore, after the three COATs performed by the voluteers, cards x, y, ad z are agai o the top of the deck, i their origial order. Save at Least 50% Priciple: If k cards from are COATed three times, the provided that k, the origial bottom k cards become the top k cards, i reverse order. That is to say, three 2 COATs preserve at least half the packet the bottom half oly i reversed order, at the top. Proof: Let the deck have iitial order a1, a2,, a, where a 1 is the top card (dealt first). Let 2 k. The, after oe COAT of k cards, the orderig of the deck is a, ak 2 k 1,, a, k a, ak 1,, a 1. Note that ak 1, ak 2,, a cotais k cards ad k k. So these cards, possibly with some additioal cards, will be COATed i the ext iteratio. Specifically, this ext COAT moves k k 2k cards i additio to the cards a,, 1 a. Thus, the last COATed card will be a i, where k 2k 1 1 k if k k i. if k k Therefore, after the secod COAT the orderig of the deck is k a k, k 1 a,, a 1, a 1, k a 2,, a k k, a, a 1,, ak 1.

10 W e e k s 9 Notice that, as uordered sets, a k, ak 1, a1, a 1, a 2, ak a1, a2,, a k, a k 1,, a k k k ad thus the sequece a k,, ak i the twice-coated deck cotais k cards. Therefore, the third COAT of k cards results i this orderig of the deck: a, 1 a,, ak 1, a k,, a 1, a k 1, a 2,, a k. Fially, sice the sequece a1, a2,, a k cotais k cards, it follows that the sequece 1 1 k a, a,, a cotais k cards. This secod sequece is clearly at the top of the deck ad cotais the cards that were origially o the bottom of the deck, but i reversed order. I.b. Ace Combiatio. This trick is the trick Ace Combiatio preseted by Mulcahy but with a slight variatio (41-42). Descriptio of the Trick Have a voluteer choose a three digit umber, abc, that will be the combiatio of a safe. Idicate that the keypad oly cotais prime ad composite umbers, so 0 ad 1 are ot available. Use the first two digits, a ad b, to cout out a packet of 2a b cards. Have the voluteer COAT a b cards c times, where c is the third digit, before hadig the combied packet back to the magicia. The magicia puts this packet out of his ad the audiece s sight (behid his back or uder a table), maipulates the cards, ad the shuffles the packet. The magicia ow produces the shuffled packet to reveal all four aces overtured while the other cards are still face dow. 1 1 The magicia also has the optio of simply revealig the aces, i the case of a less proficiet magicia.

11 W e e k s 10 Mathematical Aalysis ad Trick Explaatio Before begiig the trick, the magicia assembles the full deck with two aces o top ad two aces o the bottom. Upo gettig the combiatio, abc, from the voluteer, the magicia deals out a cards ad the gives the remaider of the deck to the voluteer to cout b cards off the top. While the voluteer is doig this, the magicia moves the bottom card of the first packet of a cards to the top that packet. After the voluteer gives back the deck, the voluteer puts oe of the packets o top of the other. While this is occurrig, the magicia removes a cards from the bottom of the origial deck ad adjusts the packet so that a ace is o top ad bottom. He the sets the remaider of the origial deck aside. The magicia places this secod a-sized packet o the opposite side of the reassembled deck from the other a-sized packet. Thus, the aces are i positios 1, a, ab 1, ad 2a b i a deck of size 2a b. Now the voluteer will COAT a b cards, c times. Sice clearly 2a b ab, the Special 4-Cycle Priciple 2 described below, with 2a b ad k a b, shows that each of these 4 positios will cotai a ace, after ay umber of COATs. Next the magicia puts the cards where he ad the audiece caot see them ad turs the top ad bottom cards over, thus turig two aces the opposite directio of the rest of the deck. The magicia the COATs the deck with a b cards ad flips the top card; thus a third ace is the opposite directio. After oe more COAT, the magicia agai turs over the top card ad thus all aces are facig the opposite directio. Fially, the magicia shuffles the deck to disguise how the cards were flipped ad presets a deck i which all of the aces are facig the opposite way from the rest of the deck.

12 W e e k s 11 Special 4-Cycle Priciple: Cosider a deck of cards. If k, the uder a sequece of four 2 COATs of k cards, the top card (which starts i positio 1) orbits through positios, k, ad k 1, i tur, before returig to the top of the deck. Cosequetly, the cards origially i positios, k, ad k 1 also cycle through these positios (ad positio 1) before returig to their origial locatios. Proof: Let the deck have iitial orderig a1, a2,, a. We refer to the proof of the Save at Least 50% Priciple for the orderig of the deck after successive COATs. After oe COAT, the orderig of the deck is a, ak 2 k 1,, a, k a, ak 1,, a 1 ad a 1 is i positio. Agai from the Save at Least 50% Priciple s proof, after two COATs the orderig of the deck is a k, k 1 a,, a 1, a 1, k a 2,, a k k, a, a 1,, ak 1 ad a 1 is i positio k. Next, after three COATs the orderig of the deck is a, 1 a,, ak 1, a k,, a 1, a k 1, a 2,, a k. Sice a1, a2,, a k is k cards, a, a 1,, ak 1, ak,, a 1 must be k cards. Therefore, k a 1 must be i positio k 1 ad clearly oe more COAT puts a 1 i positio 1.

13 W e e k s 12 II. Ditch the Dud This trick is exactly Ditch the Dud as preseted by Mulcahy, ad utilizes the game of poker (72). Descriptio of the Trick Ask for a spectator who likes poker, as you shuffle the deck. Have te cards dealt out ito a face-dow pile, ad have that pile further mixed. Pick up the cards ad glace at their faces briefly, remarkig o how radom they are, ad yet how they may result i two iterestig poker hads. Aouce which of you will wi. Deal the cards ito a face-dow row, ad alterate with the poker fa i takig cards from oe ed of the row or the other, util you both have five cards. Compare ad see who has the wiig poker had. Your earlier predictio turs out to be correct. Mathematical Aalysis ad Trick Explaatio This trick relies o kowig that the te cards dealt from the top of the deck cotai three distict sets of three of a kid, alog with oe o-matchig Joah card. Therefore, the magicia must guaratee that the origial shufflig of the large deck keeps this set itact (ote this is a set, ot a ordered set, ad therefore order eed ot be preserved). Upo the removal of the te cards from the full deck, they may be legitimately mixed, agai because this is a oordered set. Based o the Joah Card priciple below, the magicia will kow that the perso whose had cotais the Joah card will lose. The magicia must therefore be able to guaratee which had has this card. Whe showig the audiece that the cards are radom ad will make iterestig poker hads, the magicia glaces to see where the Joah card is. Therefore, whe dealig the cards face dow i a lie, the magicia kows which of these cards is the Joah card.

14 W e e k s 13 The magicia, by choosig first, ca also determie which cards are i each had. As Mulcahy poits out with his Positio Parity property, if oly the two cards o the eds ca be selected, the by always choosig the card ext to the card chose by the voluteer, the magicia is guarateed to take all of the cards i eve positios, or all of the cards i odd positios, depedig o the positio of the iitial card take. If the magicia selects card oe first, ad the follows the strategy above, he gets all odd positioed cards. Similarly, if he selects card 10 first, the the magicia gets all of the eve positioed cards. Hece, if the Joah card is i a odd positio, the magicia ca make sure that the had with all of the odd cards is the had he predicted to lose. Joah Card Priciple: If te cards cosistig of three sets of three of a kid ad oe omatchig card (a card that forms o pairs with the other cards) are divided ito two poker hads, the whoever has the o-matchig card loses, without fail. The o-matchig card is called the Joah card. Proof: Let te cards cosistig of three sets of three of a kid ad oe Joah card be radomly split ito two poker hads of five cards each. By examiig the two hads, it is clear that oe must have the Joah card. This Joah had will also have four of the remaiig ie cards. Note that these are four cards chose from a set of three matchig triples, ad thus: a) The best Joah had cotais a three of a kid ad aother card i additio to the Joah card. This leaves the other had with three of a kid ad a pair, so the Joah had loses. b) The secod best Joah had cotais two pairs. This leaves the other had with a three of a kid, so the Joah had loses. c) The ext best Joah had cotais oe pair, plus two mismatched cards i additio to the Joah card. This leaves the other had with two pair, so the Joah had loses.

15 W e e k s 14 Sice the four o Joah cards i the Joah had are chose from a set of three matchig triples, the Pigeohole Priciple tells us that the Joah had has at least two matchig cards. Thus (c) is the worst had the Joah had ca have. Hece, all possible hads are accouted for ad the Joah had will always lose.

16 W e e k s 15 III. Set Sums The tricks i this sectio use special sets whose sums help the magicia recogize the idetity of specific cards. III.a. Little Fibs. This trick is exactly Little Fibs preseted by Mulcahy (89). Descriptio of the Trick Give the deck several shuffles, the deal six cards face dow to the table, settig the rest aside. Tur away, requestig that those six cards be thoroughly mixed up. Have ay two cards selected by two spectators, who the compute ad report the total of the two card values. From that iformatio aloe, you promptly ame [the umber ad suit of] each card. Mathematical Aalysis This trick, like the trick i the last sectio, relies o a packet of kow cards that appear to be radomly shuffled to the top. Therefore, this trick requires some fake shufflig. Oce the magicia shuffles, the desired set of cards should be o top. The values of these six cards should form a set of 2-summers as defied below. I order to help the magicia remember the suit of each card, he puts the cards i CHaSeD order alog with umerical order i the trick s preparatio. A set of 2-summers is a set S where for every a, b, x, y S such that a b ad x y, if a b x y, the either a x ad b y, or b x ad a y. So, for example, 1,2,3,5 is a set of 2-summers, but 1,2,3,4 is ot sice Give a set of 2-summers, we ca elarge it usig the followig lemma.

17 W e e k s 16 Lemma: If B b b b 1, 2,, z is a set of 2-summers such that by by 1 1 y z 1, the for ay bz 1 with bz 1 bz 1 bz b1 Proof: Let B b b b Let z 1, for every iteger y with B B b z1 is a set of 2-summers. 1, 2,, z be a set of 2-summers such that for every 1 y z 1, by by 1. b. I order for z 1 to be a set of 2-summers, the sums b1 bz 1 B B b, b2 bz 1,, bz bz 1 must be distict ad differet from the sums of ay other two distict elemets of B. Sice b1 bi for 1i z, it suffices that bz 1 b1 bz 1 bz. Thus, if bz 1 bz 1 bz b1, it follows that b1, b2,, bz 1 is a set of 2-summers. We show below that the set of Fiboacci Numbers is a set of 2-summers. So ay set of cards with values equal to distict Fiboacci umbers will work for this trick. Fiboacci umbers as a set of 2-summers: Let F be the set of Fiboacci umbers: F 1,2,3,5,8,13,. The ay subset of F is a set of 2-summers. Proof: The Fiboacci umbers are defied iductively by f1 1, f2 1, ad f 1 f f 1 for 2 ad, therefore, as a set of distict itegers, F f i 2. Note that, 1,2 f f is a set of 2-summers. 2 3 Now assume i f f f is a set of 2-summers. The, to show that,,, 2 3 f2, f3,, f 1 is a set of 2-summers, the lemma tells us that it s sufficiet to show that f 1 f 1 f f2. Sice the recurrece relatio for the Fiboacci umbers is fk 1 fk 1 fk, f 1 f2 f 1 f f2., ad,,,, Thus, f 1 f 1 f f2 f f f f is a set of 2-summers. So, by iductio, F is a set of 2-summers. It is clear that ay subset of a set of 2-summers is also a set of 2-summers. Thus, ay subset of F is a set of 2-summers.

18 W e e k s 17 Therefore, if the set of cards give to the spectators is a set of Fiboacci umbers, the the spectators ca choose ay two cards ad the magicia ca idetify these cards based o their sum. For example, if the spectator chooses two cards ad reports the sum of 10, the the magicia kows that oly 8 ad 2 ca make this sum. Hece, the magicia kows the cards used were a 2 ad a 8. Usig a CHaSeD orderig, the magicia ca eve easily memorize the suits of the cards ad report this as well. I the case of the first six CHaSeD Fiboacci umbers, the magicia would reveal that the voluteer s cards were 2 ad 8. III.b. Cosolidatig Your Cards. This trick is a variatio of the trick Cosolidatig Your Cards by Mulcahy (93-4). Descriptio of the Trick After shufflig, deal out six cards face dow from the top of the deck. Tell the voluteer that oce you tur away, she is to select three cards from these six, which will be used to determie her credit ratig. After she selects her cards, tell her to add the values of the cards together, with red cards as egative values ad black cards as positive. Oce she reports the sum, the magicia either reveals that two cards cacel ad gives the suit of the remaiig card or gives the value ad suit of all three cards. Mathematical Aalysis This trick agai relies o a packet of kow cards that appear to be radomly shuffled to the top. Therefore, after the magicia performs some fake shufflig, the desired set of cards should be o top. This set, i ay order, cosists of 9, 3, A, A, 3, ad 9 (ote that these cards are listed here i CHaSeD order for easy recollectio). This packet is importat because,

19 W e e k s followig the covetio of red as egative ad black as positive, it is 3, 3,3, 3,3, 3. This allows for the use of the followig theorem. Balaced Terary Priciple: Every iteger ca be writte as a sum of distict siged powers of 3, ad this represetatio is uique apart from cacelatios (i.e., each iteger has a uique balaced terary represetatio, where 0 is the empty represetatio). For example, ad Proof: Assume k. We iduct o k. First, ote that ad it should be clear that there is o other balaced terary represetatio for 1, so 1 has a uique balaced terary represetatio. Assume that for every t with t k, t has a uique balaced terary represetatio. Let 0 such that 3 k ad some uique itegers 1,2 If r 0 ad q 1,. The, by the divisio algorithm, k 3 q r for 1 3 k q ad 0r 3. k 3 is a uique terary represetatio. If r 0 ad q 2 the k 1 3 3, a uique balaced terary represetatio. Now assume r 0. Sice r k, r has a balaced terary represetatio by the iductio hypothesis. Also, sice r 3, the largest power of 3 that could appear i a balaced terary represetatio of r is 3. First we assume this balaced terary represetatio of r does ot have a 3 term. If q 1, the 3 for k. If q 2, the k plus this represetatio for r gives a balaced terary represetatio r ad k has a uique balaced terary represetatio. Next we assume the represetatio of r has a 3 term. Sice r 3, the 3 term must be positive. Also, let x r 3, so the balaced terary represetatio for x does ot have a 3

20 W e e k s 19 term. Hece, k q r q x q x. If q 1 3, the k x provides a balaced terary represetatio for k, ad if q 1 2, the a balaced terary represetatio for k. k x provides Hece, it is clear that every positive iteger has a uique balaced terary represetatio. If k 0, the all of the coefficiets of powers of 3 are zero ad we have the uique empty presetatio. Fially, if k is a egative iteger, otice that k k, so we use the balaced terary represetatio for k to produce the balaced terary represetatio for k. Give this theorem, uless two cards cacel out, the magicia fids the balaced terary represetatio of the sum reported by the voluteer i order to determie which three cards were used. The magicia kows two cards cacelled out if the sum provided is a power of 3, sice the uique balaced terary represetatio of a power of 3 is simply that umber. If this is the case ad two cards cacel out, the the magicia reveals that two cacelled ad reports the value ad suit of the remaiig card. For example, if the voluteer reports a sum of 11, the the magicia otices that So the cards are 3, A, ad 9. O the other had, if the voluteer reports a sum of 9, the the magicia otices that ad therefore two cards must have cacelled out. So he reports that two cards cacelled ad the other card is 9.

21 W e e k s 20 IV. Mootoe Subsequeces This sectio utilizes subsequeces of cards which are either costatly icreasig or costatly decreasig. Specifically, we use the followig result origially proved by Paul Erdős ad George Szekeres i 1935 (Gasarch 1): I ay arragemet of 2 k 1 1 (or more) differet umbers, there are always at least k, ot ecessarily beside each other, that are i umerical order. Hece, there is always either a risig ru or a fallig ru of legth k (or more). (Mulcahy 269) We provide a proof i the case of k 3 as part of our discussio of the ext trick, but we do ot provide the proof of the geeral result sice it is outside the scope of this project. IV.a. Five that Jive. This trick is adjusted from Erdős Numbers by Mulcahy (274). Descriptio of the Trick A accomplice waits where he caot see or hear the trick as the magicia selects a voluteer. After shufflig the deck, the voluteer deals out the top five cards, sets aside the rest of the deck, ad shuffles these five cards. The voluteer the lays the cards face up, otes the radomess of the cards, ad turs the cards face dow oce more. After the accomplice eters the room, the magicia reveals two cards ad the accomplice aouces the idetity of the remaiig three cards (both umber ad suit). Mathematical Aalysis This trick relies o a packet of kow cards that appear to be radomly shuffled to the top. Therefore, this trick requires some fake shufflig to esure that the ecessary packet of cards is at the top of the deck. This packet should be five cards which both the magicia ad

22 W e e k s 21 accomplice have memorized. The magicia ad accomplice will have previously agreed o a liear order relatio o the cards i the packet, so that for ay two cards, oe is defied to be greater tha the other. Sice this is the k 3 case of the Erdős-Szekeres result, there is either a icreasig or decreasig subsequece of legth three. While the cards are face up, the magicia locates the mootoe subsequece. After the voluteer flips all of the cards face dow, the magicia flips the cards that are ot i the mootoe subsequece face up with the accomplice preset. The magicia reveals cards from right to left to idicate a icreasig sequece, ad reveals cards from left to right to idicate a decreasig sequece. The followig theorem proves this k 3 case, which guaratees that the magicia eed oly reveal two cards i this maer for the accomplice to aouce the idetities of the three cards that remai face dow. Special Case of Erdős-Szekeres: For ay sequece of five distict umbers, there is always a mootoe subsequece of legth three. Proof: Let a, b, c, d, e be a sequece of distict umbers. Assume a b. If b c, the,, a b c is a icreasig subsequece. Similarly, if b d or b e, the a, b, d or a, b, e is a icreasig subsequece, respectively. Otherwise, b max c, d, e. If c d, the,, b, c, e is a decreasig subsequece. Otherwise, c mi d, e b c d is a decreasig subsequece. Similarly, if c e, the. If d e, the,, b d e is a decreasig subsequece. The oly remaiig optio is if d e, i which case c, d, e is a icreasig subsequece. Now assume a b d or b e b. If b c, the,,, the,, a b c is a decreasig subsequece. Similarly, if a b d or a, b, e is a decreasig subsequece, respectively. Otherwise

23 W e e k s 22 b mi c, d, e. If c d, the,, b c d is a icreasig subsequece. Similarly, if c b, c, e is a icreasig subsequece. Otherwise, c max d, e. If d e, the,, e, the c d e is a decreasig subsequece. The oly remaiig optio is if d e, i which case b, d, e is a icreasig subsequece. three. Thus, i every case the sequece a, b, c, d, e has a mootoe subsequece of legth IV.b. Te Soldiers. This trick is adjusted from Te Soldiers by Mulcahy (264). Descriptio of the Trick This trick has the same procedure as the previous trick, Five that Jive, with the exceptios that te cards are take from the top of the deck istead of five, ad four cards are left to be revealed by the accomplice istead of three. Mathematical Aalysis This trick applies the k 10 case of the Erdős-Szekeres result. The proof of this case, like the geeral result, is outside the scope of this project. IV.c. Clear Cut Diamods. This trick is adjusted from Slippery Eough preseted by Mulcahy (271). Descriptio of the Trick The magicia selects a voluteer from the audiece ad gives him the deck to remove the diamods ad place them face-up i a lie, i whatever order he chooses. The magicia surveys the row of thirtee cards (possibly askig the audiece to do the same), the has the voluteer flip each card face dow. At this poit, a accomplice who has either see or heard what has

24 W e e k s 23 happeed so far is brought ito the room. The magicia turs over some of the cards (usually eight) ad the accomplice ames (correctly) the cards which are missig. Mathematical Aalysis This trick works because of statistics ad simple commuicatio. Sice there are thirtee diamods i a deck of cards, there are 13! possible arragemets of the cards i this trick. Of these possible sequeces of cards, there is approximately a 98.4% chace that the sequece has a mootoe subsequece of legth five (Mulcahy 271). If this is the case, the magicia uses such a sequece. If ot, the the magicia uses the k 4 case of the Erdős-Szekeres result, ad reveals ie cards istead of eight. The magicia reveals cards i the same maer as i Five that Jive. Sice all of the diamods are o the table, the accomplice ca easily determie which cards are face dow, ad ca use the magicia s cues to determie their order.

25 W e e k s 24 V. Error Correctig Codes This sectio deals with the mathematical idea of error correctig codes. These are codes that have a built i mechaism that eables the detectio ad correctio of errors. V.a. A Horse of a Differet Color. This trick is exactly A Horse of a Differet Color by Mulcahy (288-9). Descriptio of the Trick A audiece member is ivited to select ay three cards from the deck ad lay them i a face-up row o the table. You supplemet this row with three more face-up cards of your ow choosig. Say, Thik of these cards as six horses i a stream. Before that siks i, add, No doubt you ve heard the expressio, Do t chage horses i the middle of a stream. Actually, that s exactly what I wat you to do. Please chage ay oe horse for a horse of a differet color! The audiece member replaces ay oe of the cards o the table with a ew card from the deck, subject to the provisio that the ew card must ot be the same color as the oe it replaces. Your accomplice ow eters the room, ad soo idetifies which card o the table was switched. Mathematical Aalysis This trick uses the cocept of error correctig codes. Specifically, it uses a liear biary code. We begi by describig a seemigly easier trick, where the voluteer chooses oly two cards. We will the explai how to add a third card from the voluteer without addig ay actual complexity to the trick. The code, C, is defied by C : a, b a, b, a, b, a b where

26 W e e k s 25 ab, 0,1 ad additio is doe modulo 2. The followig logic proves that if a received code word is kow to cotai exactly oe error, the recipiet ca recover the correct code word. Assume x is the icorrect digit ad assume that c a bmod2 1) If the received (corrupted) code is,,,, for the origial a ad b. x b a b c or a, b, x, b, c, the positios oe ad three do ot match ad so oe of those positios cotais the error. Sice a b c but x b c, the assistat ca further pipoit the locatio of x. 2) If the received code is,,,, a x a b c or a, b, a, x, c, the positios two ad four do ot match ad so oe of those positios cotais the error. Sice a b c but a x c, the assistat ca further pipoit the locatio of x. 3) If the received code is a, b, a, b, x the the sum of positios oe ad two, ad the sum of positios three ad four, do ot equal the umber i the fifth positio, ad hece there is a error i positio five. Hece, the locatio of the error is idetifiable give oly the corrupted code ad the kowledge that the code cotais exactly oe error. Sice the code uses a biary system, by kowig the positio of the error, it is simple to correct the error by switchig the corrupted digit with the other elemet of 0,1. Give the assumptio that black cards have value zero ad red cards have value oe, the magicia uses the first two cards chose by the voluteer to choose three additioal cards. The voluteer the chages oe card s color (i.e., its value), thus itroducig a error to the code word. By applyig the logic preseted above, the accomplice ca the determie which card was switched.

27 W e e k s 26 Recall that i the descriptio of this trick, the voluteer chooses three cards, ot two. This extra card is igored i the magicia s choice of cards ad he simply chages the code to :,,,,,,, where a, b, z 0,1 C a b z a b z a b a b. The accomplice performs the same error checkig as before, except that if o error is detected i the code, the the error must be i the third positio, which is the positio ot checked by the code. Hece, the accomplice detects ad corrects ay sigle error. V.b. Ad Now for Somethig Completely Differet. This trick is exactly Ad Now for Somethig Completely Differet from Mulcahy (302). Descriptio of the Trick Give out a deck of cards for shufflig. Take it back, ad fa it to reveal that that the cards are all face up. Commet, These are t mixed up very well. Look, they all face the same way! Split the deck ear the middle, ad flip over oe half, before rifflig the two parts together. Perhaps had the deck out agai for additioal shufflig. That s better, you coclude, as you fa the cards agai to show that they are well ad truly mixed ow. Ivite a audiece member to select ay two cards from the deck ad place them side by side o the table. You rapidly supplemet these with two cards of your ow choosig, to form a row of four cards. Four radom cards, some Red, some Black, some face dow! Ad ow for somethig completely differet. Please chage ay oe card. For istace, you could just tur oe of these cards over, or you could switch a [face up] Red card there for a [face up] Black oe from the deck, or vice versa.

28 W e e k s 27 The audiece member does as istructed. Your accomplice ow eters the room for the first time, ad soo idetifies which card o the table was switched. Eve better, if the switched card is ow face dow, she ca tell whether it was origially Black or Red. Furthermore, if the switched card is face up, she ca tell whether it was origially a differet color or face dow. Mathematical Aalysis This trick uses a similar idea as i the previous trick, usig a terary liear code rather tha a biary code. The three properties of cards comprisig the terary system will be red (correspodig to 1), black (1) ad face dow (0). The code takes a ordered pair ab,, where ab, 1,0,1, ad turs it ito a ordered quadruple a, b, a b, b a, where a b ad b a are both reduced modulo 3 with the covetio of recordig 2 as 1 (ote that 2 ad 1 are equivalet modulo 3). I a correctly coded message s, t, u, v, with,,, 1,0,1 s t u v, t u v 0, sice 3 0mod3. Similarly, sice a b a b 2a b amod3 b b a b a b s u v ; s v t sice, a b a b ; ad s t u trivially. If we kow that a received code word cotais exactly oe error, we ca use the facts above to determie the locatio of the error. Lookig at the received message s, t, u, v, with exactly oe error, the followig are true: The error exists i the first positio if ad oly if t u v 0. The error exists i the secod positio if ad oly if s u v. The error exists i the third positio if ad oly if s v t. The error exists i the fourth positio if ad oly if s t u. This property shows that if oe of the equatios is true, the three variables used caot cotai the error ad the fourth variable must the cotai the error. However, this trick really

29 W e e k s 28 requires at most oe error. If o error is preset, all the above equatios will be true. Therefore, this trick has two variatios, oe where exactly oe error must occur ad oe where at most oe error must occur. Oce the accomplice determies the locatio of the error, the accomplice ca use the origial codig method to determie what the code word should have bee, ad hece whether the card was origially face dow, red or black. For example, if the preseted code is s, t, u, v 1,1,1, 1, the the accomplice otices that 1 1 1mod3 s u v ad the error must be i the secod positio. Sice u 11 0 s t (the equatio for positio three), we kow that positio two should have cotaied a zero ad, hece the card i that positio was origially face dow.

30 W e e k s 29 VI. Fitch Cheey s Five-Card Twist This trick is adapted from the trick with the same ame preseted by Mulcahy (306). Descriptio of the Trick The magicia selects a voluteer to shuffle the deck ad choose ay five cards. The magicia examies the cards, hides oe of the five cards, sets the remaiig four cards i a faceup row, ad has the voluteer retrieve a accomplice from outside the room. The accomplice briefly examies the cards ad idetifies the missig card. Mathematical Aalysis Ulike may of the other tricks preseted, this trick ivolves o fake shufflig or kow cards; the accomplice kows othig about the five cards before eterig the room. Clearly the magicia is usig the four remaiig cards to idetify the missig card. Hece, we will examie the decisio makig process for the magicia s selected four cards. First, sice there are five cards preset ad oly four suits, the Pigeohole Priciple tells us that at least oe suit must be used twice. Thus, the magicia hides oe of the cards from a duplicated suit. I order to commuicate the suit to the accomplice, the voluteer calculates the sum of the remaiig cards (with jack, quee, kig ad ace equal to 11, 12, 13, ad 1, respectively), the reduces the sum modulo 4 with 0 4mod 4 referrig to the fourth positio. The magicia places the card that determies the suit of the hidde card i the positio determied by this sum. The other three cards will commuicate the value of the hidde card. Note that there are 13 cards of each suit. Thus, for ay two cards with values a ad b, ab 6, ad so the hidde card ad the card that idetifies the suit of the hidde card are withi 6 of each other. Assumig that a b, the magicia hides a if ba, a 6 ad hides b

31 W e e k s 30 otherwise. For example, i choosig betwee hidig 3 or K, the magicia hides 3, because 13 3,9. By idetifyig suits as low to high followig the CHaSeD orderig, every card is uiquely higher or lower tha ay other card (i.e., 3 is greater tha 10 ). Thus, by applyig the followig rule, the magicia ca tell the accomplice what umber to add to the value of the visible suit card based o the relative degrees ad order of the remaiig three cards. Usig L, M, H for low, middle, ad high, respectively, where LMH meas the remaiig three cards are i the order low, medium, high from left to right: LMH: Add 1. LHM: Add 2. MLH: Add 3. MHL: Add 4. HLM: Add 5. HML: Add 6. Thus, the magicia ca order the four remaiig cards i a way that idetifies the hidde card, both i terms of suit ad value. For example, assume the voluteer selects 7, 7, 8, J, ad Q. The magicia the otes that two clubs are preset (the 7 ad 8) ad that 8 7,13. Hece, the 8 is hidde. Give the values of the remaiig cards, the magicia determies mod 4 ad puts 7 i the first positio to commuicate the suit of the hidde card. Sice 8 is oe card after 7, the magicia eeds to commuicate that 1 must be added. Thus, the order of the remaiig three cards is low, medium, high. Sice hearts are less tha spades, which are less tha diamos,

32 W e e k s 31 whe the accomplice walks i he will see 7, J, Q, 7 ad, i that order. The accomplice cocludes that the hidde card is 8.

33 W e e k s 32 Coclusio ad Persoal Reflectios Over the course of this examiatio of the mathematical bases of various card tricks, it became clear to me that these tricks use a wide rage of mathematics. The cocepts for each trick varied widely, from the combiatorics used i COATs ad the discrete mathematics of the Pigeohole Priciple, to error detectig ad correctig codes. This demostrated to me that mathematics is a wide ragig field, eve i applicatios as seemigly simple as card tricks. My ivestigatio raises the questio of how other areas of mathematics may lead to ew tricks. Thus, my further research ito this field might focus o ivetig ew tricks based o mathematics, whether simple or advaced, that I did ot examie i this paper. Additioally, certai tricks i Mulcahy s book seemed iterestig mathematically, but I excluded them from the paper due to their cofusig ature ad a lack of cosistet performace success. For example, Lucky Number Oe ad Thirtee requires either more work or a alteratio to make it a more easily accomplished illusio (Mulcahy 144-5). Additioally, i the future I would like to examie other sources of tricks such as Magical Mathematics by Persi Diacois ad Ro Graham, as well as various works of Marti Garder. The cotet of this paper is well suited for demostratig how mathematics ca be fu. Oe possible way I could spread this message is by usig this paper as a basis for oe or more presetatios i a high school mathematics class. Sice may of the cocepts of the paper are accessible to high school studets, the presetatio of these tricks i such a eviromet is practical. I would preset the tricks, followed by their explaatio i mathematical terms. This activity would both promote mathematics as ejoyable ad itroduce high school studets to mathematics based o proofs, ot just calculatios.

34 W e e k s 33 Aother directio for my future ivestigatio is tricks usig other types of cards. All of the tricks i this paper use a stadard deck of playig cards, which have three characteristics: suit, color, ad umber. Sice the suit determies the color, oe ca argue that each card has just two characteristics. It may be rewardig to ivestigate possible tricks ivolvig more complicated decks, such as the cards used i the game Set. These cards each have four attributes rather tha three. Overall, card tricks are iterestig mathematically as well as beig iterestig to lay people. Their study led me to a better uderstadig of how math ca be applied to seemigly urelated fields i additio to the educatioal beefits of demostratig mathematics i such a field to stimulate iterest. I gaied a deeper uderstadig of proof as a result of this study, movig away from provig typical mathematical results to provig card tricks. Through such a examiatio, I more fully uderstad the idea of proof.

35 W e e k s 34 Bibliography Boa, Miklos. Itroductio To Eumerative Combiatorics. Bosto: McGraw-Hill Higher Educatio, Prit. Gasarch, William. "Five Proofs of the Subsequece Theorem." (.d.):. pag. Uiversity of Marylad. Web. 8 Feb < Gilbert, Jimmie, ad Lida Gilbert. Elemets of Moder Algebra. 7th ed. Pacific Grove, CA: Brooks/Cole, Prit. Kuth, Doald Ervi, Tracy Larrabee, ad Paul M. Roberts. Mathematical Writig. Washigto, D.C.: Mathematical Associatio of America, Prit. Mulcahy, Colm Kevi. Mathematical Card Magic: Fifty-two New Effects. Boca Rato: CPC, Prit. Merzback, Uta C., ad Carl B. Boyer. A History of Mathematics. Third ed. Hoboke: Wiley, Prit. Richmod, Bettia, ad Thomas Richmod. A Discrete Trasitio to Advaced Mathematics. Belmot, CA: Thomso/Brooks/Cole, Prit. Rose, H. E. Liear Algebra: A Pure Mathematical Approach. Basel Birkh user erlag, Prit.