Mathematical paradox as a means of brain training

Life is an amazing thing! Sometimes she presents us with a surprise in the form of a paradox, over which you can break the brain in an attempt to figure out where the correct answer lies. The very word "paradox" means a situation that may well exist in real life, but which at the same time can not be logically explained. In addition, there is such a thing as "aporia". This term refers to a fictitious situation, but it is understandable with the help of logical arguments.

Paradoxes are different: economic, legal, philosophical, chemical, physical, psychophysiological, logical, related to choice, statistical and mathematical. Apparently, they are not so little. Let's talk about how sometimes puzzles paradoxes in mathematics. By the way, they are good exercises for brain training. He, like the body, also needs to be trained. For example, try to solve the following comic mathematical paradox.

When he lived for a very long time, there was a gentleman who had a pair of excellent but small boots in the attic. He decided to get rid of them and instructed his servant to sell them at the market for 25 rubles. He went to carry out the assignment. In the bazaar the servant saw an invalid who did not have a left leg, and the right one was wrapped in some rags. It was autumn in the yard, the servant felt sorry for this man, and when he asked to sell him the right boot at half price, he agreed, although he understood that the remaining one was unlikely to be needed. So, he received 12.5 rubles. The servant was about to return to his master when a second one-legged invalid came to his eyes, already without a right foot, who also needed shoes. He sold the left left boot for 12.5 rubles and the satisfied returned to the master. The owner, after listening to this story, began to reproach the servant for the fact that he did not make the unfortunate discount. He gave him 5 rubles and instructed to find those two customers and divide the money between them. The servant, too, was not sewn up and decided that something must be done for his work. Therefore, he took 3 rubles, and each disabled person gave one ruble. Now, if we count together the money that the servant took and those that the disabled eventually paid, then we get 3 + 12.5-1 + 12.5-1 = 26 rubles. And the boots at first cost 25 rubles. Hence the question: where did the extra ruble come from? The answer to this mathematical paradox will be given a little later, so as not to deprive you of the pleasure of finding out for yourself how it happened. In the meantime, we note that in addition to joking, there are also serious paradoxes, over which many generations of scientists are tormented and who sometimes provoke heated arguments.

Take, for example, the paradox of time, which is still known as the paradox of twins. In 1905, none other than Albert Einstein, formulated a theorem that referred to the relativistic slowing down of time. Its essence is that if two identical clocks showing the same time are located at one point, and then one of them is moved along a closed curve at a constant speed, until they again find themselves in their original place, as a result they Will show a different time compared to the clock that was immovable. Strangely enough, but numerous experiments that have been conducted with macroscopic clocks and elementary particles suggest that this is quite possible and the theory of relativity also works in this case.

Well, it's time to explain our mathematical paradox with boots. Extra ruble appeared because to the amount of money paid by disabled people (23 rubles.) Were added three rubles a servant, and this is not true. The servant's money should not be added, but taken away. At first, the boots cost 25 rubles, but after the discount began to cost 20. So everything converges. If you managed to solve this comic mathematical paradox yourself, I congratulate you! And if not? Then try to solve other paradoxes in mathematics. In any case, this will increase the acuteness of your thinking. And who will refuse this?