The chessboard paradox. Effective trick and its solution

EFFECTIVE FOCUSES AND THEIR CLUES
Free library / Directory / Spectacular tricks and their clues

The chessboard paradox. Focus Secret

Spectacular tricks and their clues

Directory / Spectacular tricks and their clues

Focus Description:

Consider a class of paradoxes in which the "principle of covert redistribution" explains the mysterious disappearance or appearance of squares. One of the oldest and simplest examples of paradoxes of this kind is shown in the figure:

Focus The Chessboard Paradox

The chessboard is cut obliquely, as shown in the left half of the figure, and then part B is shifted down to the left, as shown in the right half of the figure. If the triangle protruding in the upper right corner is cut off with scissors and placed in an empty space that looks like a triangle in the lower left corner of the picture, then a rectangle of 7x9 square units will be obtained. The original area was 64 square units, but now it is 63. Where did the one missing square unit go?

The answer is that our diagonal line runs slightly below the lower left corner of the square located in the upper right corner of the board. Because of this, the cut off triangle has a height that is not 1, but 11/7, and thus the height is not 9, but 91/7 units. The 1/7 unit height increase is almost imperceptible, but when taken into account, it results in a required rectangle area of 64 square units.

The paradox becomes even more striking if, instead of a chessboard, we take just a square sheet of paper without cells, since in our case, careful examination reveals an inaccurate closure of cells along the cut line.

The connection of our paradox with paradox of vertical linesand becomes clearer if you follow the cells at the cut line. When moving upward along the cut line, it is found that parts of the cut cells above the line (they are darkened in the figure) gradually decrease, and gradually increase below the line. There were fifteen shaded squares on the chessboard, but on the rectangle obtained after rearranging the pieces, there were only fourteen of them. The apparent disappearance of one darkened cell is simply another form of the paradox discussed above. When we cut and then move the little triangle, we are effectively cutting piece A of the chessboard into two pieces, which are then swapped along the diagonal.

For the puzzle, only the cells adjacent to the cut line are important, while the rest do not matter, playing the role of decoration. However, their presence changes the nature of the paradox. Instead of the disappearance of one of several small cells (or a somewhat more complex figure, say, a playing card, a human face, etc., which could be drawn inside each cell), we are faced here with a change in the area of a large geometric figure.

Author: M.Gardner

We recommend interesting articles Section Spectacular tricks and their clues:

▪ four robbers

▪ The pencil goes through the handkerchief

▪ Appearance and disappearance of burning candles

See other articles Section Spectacular tricks and their clues.

Read and write useful comments on this article.

<< Back

Latest news of science and technology, new electronics:

Air trap for insects 01.05.2024

Agriculture is one of the key sectors of the economy, and pest control is an integral part of this process. A team of scientists from the Indian Council of Agricultural Research-Central Potato Research Institute (ICAR-CPRI), Shimla, has come up with an innovative solution to this problem - a wind-powered insect air trap. This device addresses the shortcomings of traditional pest control methods by providing real-time insect population data. The trap is powered entirely by wind energy, making it an environmentally friendly solution that requires no power. Its unique design allows monitoring of both harmful and beneficial insects, providing a complete overview of the population in any agricultural area. “By assessing target pests at the right time, we can take necessary measures to control both pests and diseases,” says Kapil ... >>

The threat of space debris to the Earth's magnetic field 01.05.2024

More and more often we hear about an increase in the amount of space debris surrounding our planet. However, it is not only active satellites and spacecraft that contribute to this problem, but also debris from old missions. The growing number of satellites launched by companies like SpaceX creates not only opportunities for the development of the Internet, but also serious threats to space security. Experts are now turning their attention to the potential implications for the Earth's magnetic field. Dr. Jonathan McDowell of the Harvard-Smithsonian Center for Astrophysics emphasizes that companies are rapidly deploying satellite constellations, and the number of satellites could grow to 100 in the next decade. The rapid development of these cosmic armadas of satellites can lead to contamination of the Earth's plasma environment with dangerous debris and a threat to the stability of the magnetosphere. Metal debris from used rockets can disrupt the ionosphere and magnetosphere. Both of these systems play a key role in protecting the atmosphere and maintaining ... >>

Solidification of bulk substances 30.04.2024

There are quite a few mysteries in the world of science, and one of them is the strange behavior of bulk materials. They may behave like a solid but suddenly turn into a flowing liquid. This phenomenon has attracted the attention of many researchers, and we may finally be getting closer to solving this mystery. Imagine sand in an hourglass. It usually flows freely, but in some cases its particles begin to get stuck, turning from a liquid to a solid. This transition has important implications for many areas, from drug production to construction. Researchers from the USA have attempted to describe this phenomenon and come closer to understanding it. In the study, the scientists conducted simulations in the laboratory using data from bags of polystyrene beads. They found that the vibrations within these sets had specific frequencies, meaning that only certain types of vibrations could travel through the material. Received ... >>

Random news from the Archive

Medical prostheses from Lamborghini 19.06.2017

The Italian company Lamborghini and the Houston Methodist Research Institute (USA) have entered into an agreement to jointly develop composite prostheses.

An Italian-American partnership will focus on the biocompatibility of advanced structural materials, which will eventually allow them to be used in implants and medical prostheses.

The Americans will share their developments in the field of nanotechnology and medical bioengineering in order to get the opportunity from the Italians to make implants and prostheses even lighter and stronger. For the production of prostheses, Lamborghini can use, for example, a forged composite, which is made of carbon fiber and epoxy, but according to a different technological process. First, the carbon fiber is divided into short lengths, which are distributed and filled with resin, and then crimped with special rollers.

Then the resulting sheet is cut along the desired contour and enters the mold, where it is baked for several minutes at a temperature of 135 ° C and a pressure of 80 bar. The result is a material with high mechanical strength and resistance to fatigue cracks. Lamborghini believes that the forged composite is quite useful not only for strong car bodies, but also for medical prostheses.

Other interesting news:

▪ Upgraded 32nm processor from Intel

▪ Lighting from a balloon

▪ Robot tightrope walker

▪ Chocolate in the morning helps women lose weight

▪ Laptop without power supply

News feed of science and technology, new electronics

Interesting materials of the Free Technical Library:

▪ section of the site Household electrical appliances. Selection of articles

▪ article Pig under the oak. Popular expression

▪ article Why does the eye of Medusa, held by the starry Perseus, wink? Detailed answer

▪ article Cymbopogon. Legends, cultivation, methods of application

▪ article Kvasovanie sheepskins. Simple recipes and tips

▪ article High-voltage rectifier diodes. Encyclopedia of radio electronics and electrical engineering

Leave your comment on this article:

All languages of this page

Home page | Library | Articles | Website map | Site Reviews