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Many card magic tricks rely on mathematics. Here is one popular trick, demonstrated for us by magician Sean Bone.

The magician produces a deck of cards and invites two members of the audience to cut the deck. The first is instructed to take approximately one-third of the deck and look secretly at the bottom card. The second person is asked to take about half of the remaining cards and look at that bottom card.

The deck is then reassembled by the audience members replacing their piles of cards.

The magician then explains he has two helpers in the deck: the two jokers, who will inform him of where the chosen cards are in the pile. He fans the deck across the table face down. The two jokers are face up. The magician removes them and reassembles the deck. He holds the jokers to his ears and reports that they have told him the cards are to be found at two numbers, these being their positions from the top of the deck.

The magician counts cards off the top of the deck, and to the amazement and stupification of the audience, the cards are at the predicted locations.

The magician could write the numbers down before the trick. For that matter, he could write down one prediction, and redo the trick as many times as he likes, because the cards are always at the predicted locations!

How can this be? you ask, since the victims have free choice of where they cut the cards, surely the cards could be any of the 52 available?

Yes, this is true, but the magician does not need to know *which* cards they are, only *where* they are. And that is where some clever mathematics comes to his aid.

The fact is that given a certain set-up, the two cards will always be located at card number 18 and 43. The cards are chosen randomly, but how the deck is reassembled places them at those two locations, irrespective of their initial positions! How is that possible?

The deck has no particular order, except that the two jokers are placed upside down after card number 9, and card number 27. The deck therefore has this arrangement:

9 cards ♦ joker ♦ 18 cards ♦ joker ♦ 25 cards

Victim 1 cuts the deck at approximately one-third down. Unless his cut is grossly inaccurate, it is almost certain to fall between the two jokers. He lifts this part of the deck to his chest to steal a glance at the card on the bottom. Let us call this card A, and the pile he has in his hands is composed of two sections (9 cards and an unknown number of cards, we shall call x), separated by the first joker:

9 cards ♦ joker ♦ x cards, the last of which is card A

Victim 2 now cuts the remaining two-thirds of the deck into two. This cut is almost certainly going to fall in the last block of 25 cards. His card is card B, and the block he holds in his hand consists of:

(18-x) cards ♦ joker ♦ y cards, the last of which is card B

Notice that, although we do not know what 'x' is, we know that the two blocks of cards, the x cards below the joker in the first victim's block, and the part of the block above the second joker in the second victim's block, must add up to 18. Hence the assertion that the second victim's block starts with '18-x' cards then the joker.

The trickster now reassembles the deck, but invites the first victim to replace his block first, then the second victim's. This goes unnoticed by the audience, as it is in the same order as the cards were cut, but actually reverses the two blocks. The new deck is now composed as follows:

(18-x) cards ♦ joker ♦ y cards(B) ♦ 9 cards ♦ joker ♦ x cards(A) ♦ (25-y) cards

The deck is now spread across the table face down. The jokers appear face up. When the jokers are removed, the magician ensures the parts of the deck they delineate are kept separate. What was the middle section is now made the base, while the previous bottom section is placed on top of it, and the top section remains the top. The deck is composed as follows:

(18-x) cards ♦ x cards(A) ♦ (25-y) cards ♦ y cards(B) ♦ 9 cards

It should be clear that card A now lies at position (18-x)+x. The 'x's cancel and card A is found at card 18. Similarly, card B will always be the tenth card from the bottom, or card number 43.

Extension: it may be entertaining to experiment with different numbers of cards before and between the jokers, and even whether more shock and awe can be gained by changing the number of victims.

Question: Where will the selected cards appear if the jokers' positions are changed to following card 8 and card 28 respectively?

Answer: 20 and 44

The positions of the cards are determined by: B is located from the end of the deck at the number of cards before the first joker. Card A is located from the top of the deck at the number of cards between the jokers.

A variation of this trick might be to have the victims place the jokers at positions they choose, and all the magician need do is quickly subtract the larger from the smaller to obtain the position of the first card, and subtract the smaller chosen number from the end.

Content © Andrew Bone. All rights reserved. Created : July 9, 2014 Last updated :March 2, 2016

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1700 - 1782

Daniel Bernoulli was a major scientist and mathematician of the 18th century, a friend and associate of Leonard Euler, and a member of the illustrious Bernoulli dynasty of mathematicians.