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A four card trick

January 24, 2012

A four card magic trick

Get four playing cards
Look at bottom card and remember it

Put the top card on the bottom 
Turn the top card over and replace on the top 
Cut as many as you like to the bottom
Turn the top two cards over together and replace on the top

Cut as many as you like to the bottom 
Turn the top two cards over together and replace on the top

Cut as many as you like to the bottom 
Turn the top card over and place it on the bottom 
Place the top card on the bottom 
Turn the top card over and replace on the top 
Turn the pack over

Spread them and you will find one card face down – your card!

Works every time and I’m not sure why – other than that it’s mathematical!

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2 Comments
  1. Gary Turner permalink

    Get four playing cards.1.
    Look at bottom card and remember it (this is ???your card???).2.
    Put the top card on the bottom of the pack.3.
    Turn the top card over and replace it on the top of the pack.4.
    Repeat as many of the following as you like, as many times as you like, in any order you like:a)
    Turn the entire pack over.b)
    Cut as many cards as you like to the bottom of the pack.c)
    Turn the bottom two cards over together and replace them on the bottom of the pack.d)
    Turn the middle two cards over together and replace them in the middle of the pack. 5.
    Turn the top card over and place it on the bottom of the pack.6.
    Place the top card on the bottom of the pack.7.
    Turn the top card over and replace it on the top of the pack.8.
    Spread the cards out and you will find one card that is facing the opposite way to the others ??? it will be your card!How it works:The set-upCall ???your card???, which is initially on the bottom in Step 1, ???Y???.Call the card that is initially on the top ???A???. (This is moved to the bottom in Step 2.)Call the card that you turned over in Step 3 ???b??? (the lower case denotes that the card is face up).Call the remaining card ???C???.In Step 1, the order of the cards is ABCY (the upper case B denotes that the card has not yet been turned over).Step 2 re-orders the cards to BCYA.After turning card T over in Step 3, the order of the cards will be: bCYA. Note that the ???wrong-way up??? card b is exactly 2 cards away from ???your card??? Y.The invariant transformations section (Step 4)Each of the operations in Step 4 is a group-theoretical ???invariant transformation??? in that it always leaves the ???wrong-way up??? card is exactly 2 cards away from ???your card???.a)
    Turning the pack over transforms bCYA into aycB (a, y and c are now face up while B is face-down); y is still exactly 2 cards away from the ???wrong-way up??? card B.b)
    When cutting as many as you like to the bottom, there are 3 possible transformations:i. Moving 1 card to the bottom transforms bCYA into CYAb.ii. Moving 2 cards to the bottom transforms bCYA into YAbCiii. Moving 3 cards to the bottom transforms bCYA into AbCYIn each case, the ???wrong-way up??? card b remains exactly 2 cards away from ???your card??? Y.c)
    Turning the top two cards over together and replacing them on the top transforms bCYA into cBYA. The new ???wrong-way up??? card c is still exactly 2 cards away from ???your card??? Y.d)
    Turning the bottom two cards over together and replacing them on the bottom transforms bCYA into bCay. The new ???wrong-way up??? card C is still exactly 2 cards away from ???your card??? y.e)
    Turning the middle two cards over together and replacing them in the middle transforms bCYA into byaC. The new ???wrong-way up??? card C is still exactly 2 cards away from ???your card??? y.Thus, however many of the above operations you perform, ???your card??? y will remain exactly 2 cards away from the ???wrong-way up??? card (although which card is the ???wrong-way up??? may change several times!)The d??nouement (climax)At the end of Step 4, the cards could be in any order, but ???your card??? Y (whether it is face-up or face-down) will still be exactly 2 cards away from the ???wrong-way up??? card. Since this is so, let???s call the ???wrong-way up??? card ???w??? (whether it is a, b or c) and the other two (???right-way up???) cards we will call ???R???. Then there are 4 possibilities for the order of the cards at the end of Step 4, which will be transformed by Steps 5-7 as follows:Step 4
    Step 5
    Step 6
    Step 7YRwR
    RwRy
    wRyR
    WRyRwRYR
    RYRW
    YRWR
    yRWRRYRw
    YRwr
    RwrY
    rwrYRwRY
    wRYr
    RYrw
    rYrwIn each case, the only card that is left a different way up for Step 8 is ???your card??? ??? either as a lower-case (???wrong-way up???) y amongst a set of capital (???right-way up???) R???s and W???s, or as a capital Y, the only ???right-way up??? card amongst a set of lower-case (???wrong-way up???) r???s and w???s. So, in each case, the one card that is facing the opposite way to the others is ???your card???.

  2. Gary Turner permalink

    Sorry! The options under Step 4 should have read:a)
    Turn the entire pack over.b)
    Cut as many cards as you like to the bottom of the pack.c)
    Turn the top two cards over together and replace them on the top of the pack.d)
    Turn the bottom two cards over together and replace them on the bottom of the pack.e)
    Turn the middle two cards over together and replace them in the middle of the pack.

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