Cyclic number. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
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Cyclic number. Focus Secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

The spectator is given five cards of a red suit, having numerical values 2, 3, 4, 5 and 6. For himself, the magician takes six cards of black suit, placing them so that their numerical values correspond to the digits of the number 142857. Both the performer and the spectator shuffle their cards; in this case, the showman only pretends to shuffle, but in fact keeps the order unchanged.

This can be easily achieved by shifting cards twice, one at a time from one side of the deck to the other. The rapid execution of this operation gives the complete impression of a shuffle, although the whole effect is that the arrangement of the cards is reversed twice, thus leaving the original order unchanged.

The artist lays out the cards on the table in a row, face up, forming the number 142857. The spectator draws one of his cards and places it face up under the row laid out by the demonstrator. With the help of a pencil and paper, the spectator multiplies our number by the numerical value of the card drawn by him. While he is busy with this business, the demonstrator collects his cards, puts the next card on the first left, then the next one on it, etc., "takes off" them once and again puts them on the table in a pile (face down).

"Remove" the deck means: dividing the deck into two parts, swap them. If the cards of the deck are written sequentially on a circle (to form a "cycle"), then the "removal" operation, without changing the order of the cards in the cycle, changes only the origin.

After the spectator has performed the multiplication, the demonstrator takes his pile of cards and arranges them again from left to right, face up. The six-digit number that this produces is exactly the same as the result of the multiplication found by the viewer.

Focus secret:

The demonstrator collects cards of black suit without disturbing the order in which they were laid out.

Let's say the viewer multiplied our number by 6; then the product should end in two, since six times seven (this is the last digit of the multiplicand) will be forty-two. If removed so that the deuce is at the bottom, then after the cards are laid out in a row, it will be the last card and the number depicted by the cards will coincide with the answer received by the viewer.

The cyclic number 142857 is the reciprocal of the prime number 7 in the sense that it is obtained by dividing 1 by 7. Performing this division, we get an infinite periodic fraction with a period coinciding with our cyclic number. "Other, larger, cyclic numbers can also be obtained by dividing one by large prime numbers."

Author: M.Gardner

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