Thales of Miletus

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Imagine a flat Astriaan valon 30 meters long, which rolls on three circles (see figure). The centers of the circles are located at a distance of 10 meters from each other. This shaft is moved by two Astrians. Once the position shown in figure reaches the rear wheel and throws it to his companion. He catches the wheel and places it in front of the shaft as shown in point figure. Valon rolls forward again (figure to the right) on three wheels. The next wheel is removed, it is thrown forward, etc. This procedure continues as many times as necessary. There is a small protrusion 55 meters ahead of the wheel indicated by the point. Calculate how many wheels will roll over it when the shaft moves. Find the answer first in your mind, and then with a pencil and paper. Try to generalize the solution to the case of $n$ identical wheels. (Interestingly, the size of the wheel radius does not matter.)

THALES OF MILETS

One of the first mathematicians known to history was Thales of Miletus. He lived in Greece in 624-547. B.C.

Thales had exceptionally versatile talents and interests. He studied politics, technology, philosophy, astronomy, mathematics, and trade. As a political activity, he became famous for the wise advice that he gave to his compatriots - the impact of the alliance that King Croesus offered them; as an engineer, he became famous for helping the wax of the same king Croesus cross the river without getting his feet wet. On his instructions, a canal was dug in which the river bed was temporarily diverted…

Thales the philosopher belongs to the thesis “Knowing Oneself.” Answering the question, as he projects and fairly, he said: “Refrain from doing what you consider to be done for others.” When Thales was asked what reward he would like to receive for his discovery in astronomy, the sage replied: “It is enough for me that if I tell others about my discovery, you will say that this is my discovery, and not yours.” Thales is a mathematician. He measured the height parameters from the shadow; established that a circle is divided in half by diameter, that the angles at the base of an isosceles triangle are equal, and that the vertical angles are equal. He also came up with the theorem that an inscribed angle resting on a semicircle is a straight line.

IS IT SUITABLE FOR 27!

The signs of divisibility by 3 and 9 are well known. If the sum of the digits of a number is divisible by 3 or 9, then the corresponding number is divisible by 3 or 9.

What do you think, if the sum of the digits of a number is divisible by 27, then is the number necessarily divisible by 27?