Mathematics is found in all of life, and much of magic uses math as an underlying principle. When the underlying principles are understood, a routine can be modified to fit a particular event or theme. When a trick is based on mathematics, or even has a mathematical component, then knowing the math gives you that ability to change your routine. A number of books have been written on this subject (a few are named at the end of this paper.)
This was the trick that introduced me to FCM. I was a contract programmer and working with Steve Blendermann, at StorageTek in Boulder, Colo. I'd worked with him enough to know he is a brother in Christ, and a Baptist, so when I spotted a deck of cards in his desk drawer, I couldn't resist a friendly jib.
He just laughed, and said "Let me show you what they're for" and did Last Two Cards Match with a Christian message. Fantastic - and I told him so. He just smiled and pulled an application for FCM from another desk drawer. I signed up on the spot. Thank you, Steve. (Steve is the Webmaster of the FCM web site.)
In the original form, five pairs of cards are chosen, and arranged in sequence: 1,2,3,4,5,1,2,3,4,5. (Any pairs can be used; I chose these for clarity in the explanation.) This small deck is cut any number of times. Finally five cards are dealt into another pile, which reverses the order of that pile. The spectator tells the magician which pile to use, and the magician begins to spell "Last", moving one card from top to bottom of that pile with each letter. The spectator may interrupt at any time, telling the magician to switch piles, and this can be done any number of times until the word "Last" has been spelled. The magician removes the top card from each pile and sets that pair aside. Then the process is repeated for the next word: "Two", and again with each of the words "Cards" and "Match". Now there are four pairs that have been set aside, and the magician is left with the last two cards, which are revealed to be matching (hence the name "Last Two Cards Match"). But the kicker is that each of the set-aside pairs is also a match!
This is an application of Modulo Arithmetic, which is the study of Remainders. The first move, dealing off 5 cards into a separate pile, reverses those cards. One pile is 1,2,3,4,5, and the other is 5,4,3,2,1. Moving the top card to the bottom 4 times brings the top card of both piles to match, and it doesn't matter which pile is chosen. Moreover, some of the moves can be made with one pile, and the rest with the other pile. If you don't understand, try it with all cards face up. The first pair is chosen by moving 4 times, or 5+4 times, or 10+4 times, and so on. If the name of the trick is used, the first and second pairs are chosen by "last" and "two", exactly what is needed to bring the two piles to a matching pair. The third pair is chosen by "cards," which is 5 letters, and is done when each pile has 3 cards. Thus "car" restores the order, and "ds" brings the top cards to match. And similarly, the last pair matches by any word with an odd number of letters.
For a Gospel message, use six pairs and the words "Jesus, Lord, Son of God."
Note too that the cards need not be playing cards; any pairs work fine. Jack Alexander made sets with Bible verses; but you can make anything you like with index cards. A stage production could use married couples, but you'd better be good at crowd control!
The first FCM meeting I ever attended was in Denver: the Rocky Mountain Chapter. The guest speaker used a dartboard with a "checkerboard" of four different colors on the reverse side. He used it to force red, but I noticed that the geometry of the device permitted a force to any of the four colors. I don't know if he knew that; my degree is in mathematics and I tend to notice such details.
The force for each color is achieved by choosing a different axis around which to turn the card. Turn the board so that the chosen number is at the top. Flipping the card top to bottom will produce one color, side to side a different color, and flipping around the two diagonals produces the other two colors. This is most easily learned by selecting the number "1" and watching how to flip the card to produce each color.
As Chukels, I've enjoyed using this board - a lot of clowning is possible with the dartboard. But probably this board is best used as part of a routine. For example: I have a trick with three cylinders riveted side-by-side, and balloons can be inflated in each of the cans, but a needle seemingly thrust though all three pops the outer balloons but not the center balloon (a variation of Andre Kohl's nail through a balloon in a can). I start with all three cylinders already loaded with balloons, each of a different color. I then pose this question: When we read the gospels, just who is this Jesus? There are three possibilities: he is a liar, a lunatic, or … just who he said he is: the Son of God - the Lord. I have three tags: Liar, Lunatic, Lord - and I assign these labels to the three balloons by using the dartboard to force the color for each (obviously forcing the "Lord" to the center). Running the needle through all three cylinders pops "Liar" and "Lunatic", but not "Lord".
In mathematics, the study of surfaces is called topology. One example is a Moebius Strip, which has one surface and one edge! The simplest example is s strip of paper, twisted once and the ends glued together.
Make a handy version of a Moebious Strip from a coat zipper, with Velcro at both ends, and both sides of one end. Then the zipper can become a simple loop, a Moebious strip with one twist, or with two twists. I used to carry one in the inner pocket of a story bag about Jonah and at the end of Jonah's tale; I'd discover the zipper.
When hooked in a simple circle, I'd unzip and say, "Sin separates us from God." Then as I unhooked the ends, re-zipped the zipper, and attached the ends together with one twist, I'd continue, "But if we turn back to God, and ask for forgiveness," as I unzip, "we can be forgiven and restored to a relationship with the Father." (Showing linked circles.)
"But" (as I restore and link with two twists): "St. Paul tells us that if we accept Christ as our Lord, then 'neither death, nor life, nor angels, nor rulers, nor things present, nor things to come, nor powers, nor height, nor depth, nor anything else in all creation, will be able to separate us from the love of God in Christ Jesus our Lord'" (showing one large loop).
Can you lay a piece of rope on a table, and pick it up by opposite corners, and then tie it into a knot without letting go of one end?
Answer: fold your arms first, and then without unfolding the arms, grab opposite ends of the rope with each hand, and unfold your arms. The fold is thus transferred to the rope. With a little practice and using a silk or a handkerchief), this can be done in such a fashion that others can't duplicate the feat.
That's the title of a book by Dr. George Gamov, written more than 50 years ago. In it he describes a Hottentot savage, who can only count to three. If he has more than three sheep, his herd is "many." Dr. Gamov went on to show that modern mathematicians were in the same category:
Surprisingly, all other "infinities" can be show to be the same as one of these three. Thus, Dr. Gamov claimed, mathematicians only counted to three.
But how does that relate to our discussion? Most people can only stay focused on three or fewer points at one time. Thus if a trick entails more than three significant steps, most of the audience will have forgotten the first. So if a trick can be structured such that the key "move" is followed by three or more apparently equal moves, most of the audience will have forgotten that key move.
Max Maven frequently uses this principle - he often has routines that require the audience to mentally follow four or more moves, then "mysteriously" produces the resolution. If you pay attention, it's no surprise, but apparently most people don't pay attention.
One of his routines requires that the viewer use 4 blank cards, and on the two sides of the first, write "1" and "2", and on the second "3" and "4", and so on. Then the viewer selects any card and turns it over, then subtracts 1 from that number. Finally, all the numbers are added together and the sum is used to predict something. Maven emphasizes that he can't know which card the viewer chose, or which side that viewer turned to for the subtraction, so apparently it's a big mystery how Maven can correctly predict something based that sum. (The Commutative Property of arithmetic guarantees that the sum will always be the same, 36, regardless of what order the numbers appear, so he knows exactly what each viewer has calculated by subtracting 1: 35!) Apparently he depends on the "1,2,3, Infinity" effect. And he continues as a regular on national TV.
There are several "1,2,3, Infinity" tricks to be found on the internet. A common one is "Pick a card, any card:" from a set displayed on your screen; for example:
You are told to remember that card and press Enter, which displays the next screen:
Was I able to read your mind?
Most people remember only the last three things they've seen (unless they've been warned and make an extra effort to remember). In this case, you are shown six cards, some black and some red, with at least one of each suit. And the directions tell you to concentrate on just one card; which forces your mind to note the position of that card together with its suit and value - thus almost guaranteeing that you won't remember what the other five cards were. You will only see that your card is missing. But, if you compare the two sets of cards, you see that they all different! All of the original six are missing!
Magic Meister (http://www.magicmeister.com/) offers several examples of mathematical magic. They're all good examples of "1,2,3,Infinity".
A much better internet trick (in my humble opinion) is
built by Andy Naughton, who gave me permission to use it in this lecture.
(Ref: http://www.cyberglass.co.uk/moviesflsh/mindreader.php),
Choose any two-digit number, add together both digits and then subtract the total from your original number. When you have the final number, find it on the chart and find the adjacent symbol (to the right). Concentrate on the symbol - concentrate - I'll read your mind. Your mind is like an open book - no, you're thinking of an open book - that's it! The symbol that you are thinking of is the "open book"!
If you haven't figured it out: within each group of ten numbers (the teens, twenties, etc.), subtracting the sum of the two digits produces the same number: a multiple of 9. All the teens map into 9, all the twenties map into 18, all the thirties map into 27, etc. All these numbers are the same symbol; all the other symbols are scattered randomly to confuse the pattern.
This example is especially good on the internet, because the pattern can be re-arranged between trials. A different symbol each time is then used for each iteration.
Finally, mathematics can be used in magic as the distracter, instead of the basis. For example, pass a spiral-bound notebook to someone in the audience and instruct them to write a three-digit number on the first page, then close the book and pass it to someone else. The second person is to write another three-digit number below the first and close the book again. This is repeated until four numbers have been entered, where upon the magician takes back the notebook, closed so that he can not see them, and he gives it to another member of the audience who is instructed to add the numbers. The magician holds a sealed envelope, which he claims is a prediction of that sum. The audience member adds the numbers and announces the sum, whereupon the magician breaks the seal and produces a piece of paper on which is written that sum.
This trick requires a spiral-bound notebook that looks the same front and back. Prior to the performance, the magician writes four 3-digit numbers in a column. Then he turns the book over, opens it to the now first page, blank, and hands it to someone in the audience. That person writes another number and closes the book, which is handed back to the magician. The magician repeats this four times. Then as he passes it to a fifth audience member to add the four numbers, he casually turns the book over. The back page has been prepared with four numbers that add to the magician's prediction, which has been sealed throughout the performance.
Explaination: The result of step 3 (adding together the digits of a multiple of nine) will always yield "9", so subtracting 5 yields 4, which in turn forces the letter "D".
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