The square paradox. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
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The square paradox. Focus secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

By changing the position of parts A and C, as shown in the figure, it is possible to turn a rectangle of 30 square units into two smaller rectangles with a total area of 32 square units, thus obtaining a "gain" of two square units. Only the cells adjacent to the cut line play a role here. The rest are just for decoration.

Focus Paradox with area

In this paradox, there are two essentially different ways of cutting the figure into pieces. You can start with a large 3 x 10 unit rectangle (the top of the drawing), carefully drawing a diagonal in it, then two smaller rectangles (the bottom of the drawing will be 1/5 units shorter than their apparent size. But you can also start with a figure made up of two neatly drawn smaller rectangles of 2 x 6 and 4 x 5 units, then the segments connecting point X to point Y and point Y to point Z will not form a straight line. And only because the obtuse angle they form with the apex at point Y is very close to being unfolded, the XYZ polyline appears to be a straight line, so a figure made up of small rectangle pieces will not really be a rectangle, as the pieces will overlap slightly along the diagonal.

Such paradoxes can be presented in two ways. In one of them, the paradox is obtained due to a slight decrease or increase in the height (or width) of the figures, in the other - due to an increase or loss of area along the diagonal, caused either by the overlapping of the figures, as in the case just considered, or by the appearance of empty spaces, with which we'll meet soon.

By changing the size of the figures and the slope of the diagonal, this paradox can be given a very different design. You can achieve a loss or gain of 1 square unit, or 2, 3, 4, 5 units, etc. Of course, the further you go, the easier it will be to find where the missing squares go.

Author: M.Gardner

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