Missing square puzzle
The Missing square puzzle is an optical illusion from the geometry. It looks as if the area of a triangle vary in size, depending on how to arrange the individual partial surfaces. The puzzle has probably 1953, the amateur magician Paul Curry in New York invented.
Description
Two identical, right-angled triangles are compared. The side lengths of 13 cm and 5 cm are known. ( The unit itself is unimportant. ) Both triangles consist of the same individual, colored patches here:
- A right triangle (in this case blue) with an area of
- Another triangle (in this case red) with an area of
- Further two surfaces (in this case yellow and green), which together form a rectangle with the size of accounts thereof to yellow and to green
The two total triangles look the same size, and they consist of the same colored surfaces. Nevertheless, the overall lower triangle a square of size remains. This looks strange, since the area of the whole is not expected to depend on how you fold it, the individual surfaces.
Problem
The surface area of the two triangles total can be calculated easily, because the other sides are known. These are the two side lines emanating from the right angle, respectively. The total area of a triangle should total amount accordingly.
However, to get different results when each together processes the individual subareas in the two total triangles. In the case of the upper triangle overall these are the four areas of color (red, blue, green and yellow). The sum is:
In the case of the lower triangle on the other hand, the total sum is different. Because of the four colored areas just adds one square centimeter of the missing square. The sum is then 33 cm ². At the top overall triangle so lacking in the sum of half a square centimeter, while the lower triangle is the sum total a half too much. This is the mathematical proof that something is wrong.
Solution
The viewer is optically deceived: The overall structure are not triangles, but actually quadrangles. The trick is that the red and blue triangle are only apparently similar in the geometric sense. Your angles are different in reality. Mathematically, this proved as follows:
- Blue triangle:
- Red triangle:
- To compare the angles of a triangle with short sides of length 13 and 5 (ie, corresponding to the total triangle):
The two total triangles thus have not three but four corners; it is a corner but barely visible. But is still at the transition from the red to the blue triangle. The upper edges of the red and blue triangle appear in the alleged total triangle as a long straight, as the hypotenuse of the alleged total triangle. In reality, the apparent long straight a kink, which is the fourth corner.
The apparent total upper triangle is a concave ( indented ) square, and the apparent lower overall triangle a convex ( bent- ) square. The surface areas of the two squares differ by 1 cm ². This corresponds to the missing square.
It is an optical illusion in that the top edge only appears as a straight line looks like. The eye suspects in the overall structure a triangle and is therefore inclined to overlook the kink. It assumes a uniform overall slope.
Can be prepared by this optical illusion is also a paper version. Here, the kink is obscured by a thick line border line. In addition, the cutting and joining is too imprecise to that you could see the difference.
Similar puzzles
A special geometric arrangement of Sam Loyd illustrates an expanded paradox. It seems as though even the same geometric parts occupy three different total areas in different arrangements. In reality, however, the parts do not touch completely and partially extend also over the checkered borders. By this blending, the different total areas can be achieved.