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Math Help - card 'trick' and group theory

  1. #1
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    card 'trick' and group theory

    Here's a card 'trick' I think you may find interesting, if you've not seen it before. Can anyone explain, using group theory, why this works?.

    I just thought it was fun.

    PH, you may like this in particular.

    "Ask friends to pick out seven cards from a deck and then to select one
    to look at without showing it to you. Take the six cards face down in your left hand
    and the selected card in your right hand, and announce you will place the selected
    card in with the other six, but they are not to know where. Put your hands behind
    your back and place the selected card on top, and bring the seven cards in front in
    your left hand. Ask your friends to give you a number between one and seven (not
    allowing one). Suppose they say three. You move the top card to the bottom, then
    the second card to the bottom, and then you turn over the third card, leaving it face
    up on top. Then repeat the process, moving the top two cards to the bottom and
    turning the third card face up on top. Continue until there is only one card face
    down, and this will be the selected card."
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  2. #2
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    Quote Originally Posted by galactus View Post
    Here's a card 'trick' I think you may find interesting, if you've not seen it before. Can anyone explain, using group theory, why this works?.

    I just thought it was fun.

    PH, you may like this in particular.

    "Ask friends to pick out seven cards from a deck and then to select one
    to look at without showing it to you. Take the six cards face down in your left hand
    and the selected card in your right hand, and announce you will place the selected
    card in with the other six, but they are not to know where. Put your hands behind
    your back and place the selected card on top, and bring the seven cards in front in
    your left hand. Ask your friends to give you a number between one and seven (not
    allowing one). Suppose they say three. You move the top card to the bottom, then
    the second card to the bottom, and then you turn over the third card, leaving it face
    up on top. Then repeat the process, moving the top two cards to the bottom and
    turning the third card face up on top. Continue until there is only one card face
    down, and this will be the selected card."
    Too much for me to read. I will return to it again. (In fact I am quite talented with card magic, one of my other things I like doing besides for math, but I never use card math tricks because they are way too noobish, when I show off).

    Basically, the way I see it, is that 7 is a prime number. And the movements of 3 cards, or whatever. Form a group of order 7. But by Lagrange's theorem this group is cyclic. Hence the operation of the elements 7 times will return back the identity element, the selected card.

    EDIT: It does not work for 1 (identity element).

    REDIT: Then you can use "magician's choice" and say "Ah, the first card is it!". Even better.

    RE-REDIT: It should work with any prime number. So try it with 5, 11, 13, it works!
    -------
    Some things involving cards are truly remarkable. There are a number of very interesting properties the cards must obey when they are riffle shuffled. Very mathematical, that I use as a tool. I never tried to prove them they seem complicated, and also seem group theory related.
    Last edited by ThePerfectHacker; December 9th 2006 at 02:21 PM.
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  3. #3
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    That's what I got from it. That every non-zero element of Z_{7} has order 7.

    Yes, PH, I knew you'd see it right off.

    I just thought it was interesting how cards can be used to demonstrate group theory.


    BTW, PH, do you know who Criss Angel(Mindfreak) is?. He has got to be the most outstanding magician/illusionist I've ever seen.
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  4. #4
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    Quote Originally Posted by galactus View Post
    BTW, PH, do you know who Criss Angel(Mindfreak) is?
    Of course I know who he is. Watched him.
    I do not like his style, he has a very unusual preforming style. Of course, I do not know how he does his illusion, because I am not an illusionist, and he does them well. But his "street magic" is not improptu, he prepares it in advance und makes it seems as if it was sudden.
    My favorite trick he ever done (think it was him), was disappearing in the middle of New York City. He was surrounded by an audience and crawled underneath a garbage can when it was removed he was gone and standing far far away.

    I do not really like stage magic so much because they rely on more of gimmicks then on their hands but my favorite is Lance Burton. He used to be my favorite when I was a little boy, he both has great slieght of hand and performance skills.

    Have you ever heard of Franz Harary? Another one I like. He is the one who did mega illusions. I seem some of them. Making the Sphinx vanish, making the NASA Explorer vanish and letting an audience member pass through the center, making an apartment building vanish, moving a mountain.
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    I have a brother-in-law who is a professional magician, and according to him, Angel actually *buys* many of his tricks from stores. According to my brother, he's more of an actor than magician.
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    Quote Originally Posted by The Pondermatic View Post
    he's more of an actor than magician.
    Illusionists are almost always actors.

    Real mathemations are those that can use sleight-of-hand.
    ---
    But it really is based on what the audience likes more, they like illusions more than anything. So they win out. Also get paid a lot.
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    that was a pretty cool trick, do you guys know know any more of these?
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    A silly card trick

    Here's one that I've won a few beers with.
    [Warning: do not pull this on a large tattooed biker named Bruno.]

    Toss nine cards face up on the table.
    Ask your victim to select one mentally.

    Gather up the cards and shuffle them.

    Casually, split the packets into groups of three cards.

    Pull off the top three cards and fan them toward him.
    . . Ask "Is your card one of these?"
    If he says "No", place them on the table and show him the next set of three cards.
    . . Ask the same question.

    When he says "Yes", place that set between the other two sets.


    Deal the cards face down, one at a time, into three piles: .[1] [2] [3]
    . . The cards are dealt onto piles 1-2-3-1-2-3-1-2-3 in that order.

    Pick up the first pile and fan them towards him.
    . . Ask "Is your card one of these?"
    Again, when he says "Yes", place that set between the other two sets.

    Secret: At this point, his card is the fifth (middle card) of the packet.


    Deal the cards face up, slowly, into a 3-by-three array:
    . . [1] [2] [3]
    . . [4] [5] [6]
    . . [7] [8]

    After dealing the eighth card, pause dramatically.
    Keep the ninth card face down, but hold it boldly.

    Announce: "I bet the next card I turn over is your card." **

    When he accepts, reach over and turn card [5] face down.

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    **

    Warning!

    After your bold announcement, your victim may laugh
    . . and point to his card and say "It's this one!"
    Then, if course, it's too late to say "I knew that."

    So after your announcement, be prepared.
    If he says anything other than "I'll take that bet" (or the like),
    . . interrupt him and get pushy: "Take the bet? .Yes or no!"
    Your rudeness should prompt him to take the bet.

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    Here is a card trick I love to do when I am working with a lot of people. Not mathematical but it works.

    Klicken Heir.
    ---
    Soroban probably knows this one.
    ---

    If you really like it you can buy a trick packet that does it. But I do the packet change through sleight-of-hand.
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  10. #10
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    Yup, I've already seen that one (PerfectHacker's trick). It took me a while to figure out the first time I saw it though.
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  11. #11
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    Another silly card trick

    The Three-Card Wonder

    This is a simple trick with a truly silly secret . . .
    . . but I've had a lot of satisfaction fooling my friends.

    Effect: The victim selects any three cards from a deck and remembers one of them.
    You place the three cards in your pocket and have him concentrate on his card.

    You remove a card from your pocket and return it to the deck.
    You remove another card from you pocket and return it to the deck.

    You ask him the name of his card.
    When he tells you, you remove the last card from your pocket
    . . and it is his chosen card.

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    Secret: Secretly place any two cards in your pocket in advance.
    Your right trouser pockets is probably the most convenient one.
    The faces of the cards should be toward your leg.

    Presentation: Have the victim shuffle the deck and pick any three cards.
    Tell him to remember one of the cards.
    Take the rest of the deck and place it on the table.

    When he has decided on a card, take the three cards from him
    . . and glance at them casually.
    Ask him if he wants to change his mind, if he wants three different cards.
    . . This is misdirection while you are memorizing the order of the three cards.

    Place the three cards in your pocket with the faces toward your leg,
    . . the same pocket with the two extra cards.

    Have him concentrate on his card.
    Reach into your pocket and remove one of the extra cards.
    Look at it (don't let him see it), put it on the deck,
    . . and say, "No, that's not it."
    Pretending to divine his thoughts, remove the second extra card.
    Look at it, put it on the deck, say "That's not it either."

    Reach in your pocket and position your fingers
    . . so can you grab any of the three cards easily.

    Ask him, "What was your card?"
    When he tells you, pull out that card and toss it on the table face up.


    At this point, the trick is over . . . You may retire to thunderous applause.
    . . Your victim may say, "Let's try it again!", hoping to trip you up.
    Note that you still have two extra cards in your pocket.
    . . You can immediately repeat the trick.

    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

    It is tricky to memorize three cards while "glancing casually".

    I look at the values. .If they are three different values,
    I simply arrange them in numerical order before placing them in my pocket.

    If two of them have the same value,
    . .I have some silly "rules" for the suits.

    If they have different colors (one red, one black),
    . . I place the red card further away from my leg.
    [Why? .because Red is hot.]

    If they have the same color (both red or both black),
    . . I think of the sharp points on the suits.
    With a \spadesuit and a \clubsuit, I place the spade further way.
    . . It has a sharp point; the club is rounded and safe.
    With a \heartsuit and a \diamondsuit, I place the diamond further away.
    . . It has 4 sharp points; the heart has only one.


    These rules work for me, but you can make up your own set of rules.

    One of my friends has his own set of images for the suits:
    . . \begin{array}{cccc}Heart: \\ Diamond: \\ Club: \\ Spade:\end{array}<br />
\begin{array}{cccc}Love\\Wealth\\War\\Death\end{ar  ray}

    He said that Spade = Death has to do with grave-digging.

    Of course, he prefers red cards to black cards.
    With two red cards, he prefers Love to Wealth (or so he says).
    With two black cards, both choices are grim, but he prefers War to Death.
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  12. #12
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    My favorite card trick....

    After the deck has been shuffled. You deal the cards face down the person says which are red and which are black. When this is complete you incrediblaly show that everything he said was true.
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  13. #13
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    Quote Originally Posted by ThePerfectHacker View Post
    Here is a card trick I love to do when I am working with a lot of people. Not mathematical but it works.

    Klicken Heir.
    ---
    Soroban probably knows this one.
    ---

    If you really like it you can buy a trick packet that does it. But I do the packet change through sleight-of-hand.
    I kneww it
    the cards was different all along from the second deck
    but u still got me
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  14. #14
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    Hello, ThePerfectHacker!

    My favorite card trick...

    After the deck has been shuffled, you deal the cards face down.
    The person says which are red and which are black.
    When this is complete, you incrediblaly show that everything he said was true.

    I think I know this one. .It's called "Out of This World" created by Paul Curry.
    . . I can do with a shuffled deck if I use a "stripper" deck.




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  15. #15
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    Quote Originally Posted by Soroban View Post
    Hello, ThePerfectHacker!


    I think I know this one. .It's called "Out of This World" created by Paul Curry.
    . . I can do with a shuffled deck if I use a "stripper" deck.
    Ha ha, you are a cheater.
    By that way, the deck is awesome. I just never use it.
    My philosophy, is the best tricks are those done with a standard deck.

    The way I do it is with false shuffling, Zarrow, pull-through, .... and so on. I can even let a spectator shuffle the cards! My method is after a spectator shuffles a do a different trick instead, and as I am doing other tricks I slowly rearrange the cards until of them are in the order I need.
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