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A friend asked me to solve this Mathematical Card Trick called “The Final 3″, to show why it works.

The Final 3 – Amazing Math Card Trick by mismag822

And here’s my solution

The trick begins with 52 cards, and the spectator takes 3 of them, let’s say X, Y, and Z.
Out of the other 49 cards; 40 are dealt FACE-DOWN into piles of 10, 15, 15;
leaving a remnant of 9 cards

1. Pile 1 has 10 cards, Pile 2 has 15 cards, Pile 3 has 15 cards
2. Spectator puts card X on Pile 1, making 11 cards on Pile 1.
• This means card X is in position 1 counting from Pile 1′s top
3. Spectator cuts Pile 2 anywhere and places on Pile 1.
• Let’s say she cuts it at Q cards from the top of Pile 2.
• Thus, Pile 1 is now 11+Q cards and card X is number Q+1 from the top of Pile 1.
• Pile 2 now has 15-Q cards
4. Spectator places card Y on Pile 2, making 15-Q+1 cards on Pile 2.
• Thus card Y is at position 1 from the top of Pile 2
5. Spectator cuts Pile 3 anywhere and places on Pile 2.
• Let’s say she cuts it at R cards from the top of Pile 3.
• Thus, Pile 2 is now 15-Q+1+R cards and card Y is now at position R+1 cards from the top of Pile 2
• Pile 3 now has 15-R cards
6. Spectator places card Z on Pile 3, making 15-R+1 cards on Pile 3.
• Thus card Z is at position 1 from the top of Pile 3.
7. Spectator places the 9 remaining cards on pile 3.
• Thus card Z is at position 9+1 = 10 from top of pile 3
• And pile 3 now has (15-R+1)+9 = 25-R cards
8. Dealer places pile 3 on pile 2, let’s call this Pile3+2
• Naturally, pile3+2 will have (pile2)+(pile3) cards, i.e. (15-Q+1+R)+(25-R) cards = 41-Q cards
• This pile3+2 will contain cards Y (from pile 2) and Z (from pile 3).
• Card Z position is still the 10th card from the top since pile 3 is on top of pile 3+2
• Card Y position is now shifted down by (25-R) cards (the height of pile 3, since pile 3 is upon pile 2) i.e. (R+1)+(25-R) = 26
9. Dealer places Pile3+2 upon Pile 1, lets call this Pile3+2+1
• Once again, Pile3+2+1 will have (pile3+2)+(pile1) cards i.e. (41-Q)+(11+Q) = 52 cards in all
• This Pile3+2+1 will have all three cards X,Y, Z from initial 3 piles
• Card Z and Y positions from the top will not change, since the Pile3+2 is on top of pile 1
• This means Card Z is still at number 10 and card Y is still number 26
• Pile 1 has been shifted down by (41-Q) cards which was the height of Pile3+2
• Thus Card X is now in position (Q+1)+(41-Q) i.e. 42
10. Dealer cuts 4 cards from the top of pile, to the bottom; this shifts card positions by 4
• Card Z, Y and X are now in positions 10 minus 4, 26 minus 4 and 42 minus 4 i.e. 6,22,38
11. Now for the fun part. Dealer deals cards into 2 groups, face up & face down
• The dealer deals face up, first
• Since there are two piles, there will be an odd pile (face up) and an even pile (face down)
• The dealer is seeking cards 6,22 and 38 so he must keep the face down pile
• The face up cards are 1,3,5,7…51 and face down cards are 2,4,6,…52
• THIS process has also reversed the direction of the numbering of the cards (first-in becomes last-out, like putting some stuff in a container, when you fetch it out the order is reversed)
12. Dealer repeats face-up and face-down alternate deals
• The dealer deals face up, first. Once again, an odd pile and an even pile
• The odd pile is the set of numbers 2M where M is odd e.g. 2,6,10,14,…50
• The even pile is the set of numbers 2N where N is even e.g. 4,8,12,16,…52
• Because of the reversal in (11e) above, odd pile is now face-down pile.
• The even pile is discarded
• Note that there is a second reversal due to the LIFO (Last In First Out)
13. Dealer does the face-up and face-down alternate deals again
• The dealer deals face up, first. Again, an odd pile and an even pile
• Due to the LIFO in (xi)(f), odd pile is now face-up pile, to be discarded
• The odd pile has 2,10,18,26,34,42,50 and the even has 6,14,22,30,38,46
• Keep the face-down even pile and discard the face-up (odd) pile
• Another LIFO reversal
14. Finally the dealer alternates deals for the last time
• The dealer deals face up, first; into 2 piles. Evens will be face-up now
• Since he is dealing 6,14,22,30,38,46 into two piles; odds will be 6,22,38 and evens are 14,30,46
• The dealer discards the 14,30,46 (even pile, face-up)
• Dealer presents 6,22 and 38 as the chosen cards
• But we already knew the chosen cards. We’ve been tracking them all along.
15. Important point here was to tell the spectator “Say stop when you see your card”
• This distracts the spectactor in case she is sneaky and wants to be counting/tracking the cards
• The spectator’s cards will NEVER come up; never any chance to say “stop”.

That’s all Folks!