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 The Vanishing Egg Puzzle

     The question of "how does that egg come and go?" has befuddled countless numbers of puzzle enthusiasts over the past hundred years since it first made its appearance in 1880. Since that time, the principle behind "The Magic Egg Puzzle" has been incorporated into numerous other puzzle designs. The design seen here was taken from the original puzzle generously provided by the Slocum Puzzle Foundation.     

     Martin Gardner, the former Scientific American "Mathematical Games" columnist, has given an excellent explanation and history of this type of vanish. He wrote, "All geometrical vanishes involve the cutting and rearrangement of different parts of a figure. After the rearrangement is completed, a portion of the original figure (either part of its area or one of a series of pictures drawn on the figure) has apparently vanished without a trace! When the pictures are returned to their original form, the missing area or picture mysteriously appears once more." Gardner called this, "The Principle of Concealed Distribution."

     To understand what is going on here, it is best to reduce this puzzle to its most simple and elementary form. (illustration to come) This version is called the line paradox. Here we have ten vertical lines of equal length. They are placed on the rectangle so that if you follow the dotted diagonal from left to right, you find a progressive decrease in the length of the line segments above the diagonal and a corresponding increase in the length of line segments below.

     Use the slider button (applet to come) to slide the upper portion downward and leftward. If you now recount the lines you will find that one of the lines vanished, leaving only nine lines. Which line vanished and where did it go?Slide the lower part back to its former position and the missing line returns. Now ask yourself, which is the line that returned? These are puzzling questions and have vexed scores of people who have seen them through the ages in various forms. In this case, no individual line vanishes.What happens is that eight of the ten lines are broken into two segments, then these eighteen segments are re-distributed to form nine lines. Each of the nine lines is longer by 1/9 than each 9 of the ten lines. When you slide the lower part back up again, a tenth line appears, and now the lines are shorter by 1/10 than they were before. The total of all these small increases exactly equals the length of one of the original lines. Because the increase in the length of each line is very small, it is not immediately noticeable.

    There are many ways in which this line paradox can be embellished to make the vanish and reappearance much more interesting. Over the years, numerous artists have incorporated vanishing and reappearing rabbits, people, monsters, and various objects. These images need to drawn in such a manner that the pictures will fit properly in both configurations.

     The principle is very old and probably originated as a early method for counterfeiting money. William Hooper in his book Rational Recreations, published in 1794, described the paradox as "Geometric Money," It is possible to cut 9 bills into eighteen parts (following the pattern of the lines) and then to rearrange them to make ten bills. To foil this method, the two numbers on all U.S. currency is placed on opposite ends, one high and one low. In this way, counterfeit bills using this method are easy to detect since their numbers will not match correctly. In fact, in 1968, a man in London was sentenced to eight years in prison for using this scheme on British five-pound notes.

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