The Rectangle Paradox. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
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The Rectangle Paradox. Focus Secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

When summing the areas of the parts, the permutation of triangles B and C in the upper part of Fig. 1 results in an apparent loss of one square unit.

Focus Paradox with Rectangle
Ris.1

This is due to the areas of the shaded parts: on the top of the figure there are 15 shaded squares, on the bottom - 16. Replacing the shaded pieces with two figures of a special kind covering them, we arrive at a new, striking form of paradox. Now we have a rectangle that can be cut into 5 parts, and then, swapping them, make a new rectangle, and, despite the fact that its linear dimensions remain the same, a hole with an area of one square unit appears inside (Fig. 2).

Focus Paradox with Rectangle
Ris.2

The possibility of converting one figure into another, of the same external dimensions, but with a hole inside the perimeter, is based on the following. If you take the point X exactly three units from the base and five units from the side of the rectangle, then the diagonal will not pass through it. However, the polyline connecting the point X with the opposite vertices of the rectangle will deviate so little from the diagonal that it will be almost imperceptible. After interchanging triangles B and C in the lower half of the drawing, the parts of the figure will slightly overlap along the diagonal.

On the other hand, if in the upper part of the figure we consider the line connecting the opposite vertices of the rectangle as an exactly drawn diagonal, then the XW line will be slightly longer than three units. And as a consequence of this, the second rectangle will be slightly higher than it seems. In the first case, the missing unit of area can be considered as distributed from corner to corner and forming an overlap along the diagonals. In the second case, the missing square is distributed over the width of the rectangle. As we already know from the previous one, all paradoxes of this kind can be attributed to one of these two construction options. In both cases, the inaccuracies of the figures are so slight that they are completely invisible.

The most elegant form of this paradox is the squares, which, after redistribution of parts and the formation of a hole, remain squares.

Such squares are known in countless variants and with holes of any number of square units. Some of the most interesting of them are shown in Fig. 3 and 4.

Focus Paradox with Rectangle
Ris.3

Focus Paradox with Rectangle
Ris.4

You can point to a simple formula that relates the size of the hole to the proportions of the large triangle. The three sizes that will be discussed, we will denote by A, B and C (Fig. 5).

Focus Paradox with Rectangle
Ris.5

The area of the hole in square units is equal to the difference between the product of A and C and the nearest multiple of size B. So, in the last example, the product of A and C is 25. The nearest multiple of size B to 25 is 24, so the hole is one square unit. This rule applies regardless of whether the real diagonal is drawn or the point X in fig. 5 is applied neatly at the intersection of the lines of the square grid. If the diagonal, as it should be, is drawn as a strictly straight line, or if the point X is taken exactly at one of the vertices of the square grid, then no paradox is obtained. In these cases, the formula gives a hole of zero square units, denoting, of course, that there is no hole at all.

Author: M.Gardner

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