[Problem of the 5th Olympiad in Linguistics and Mathematics]

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indicated, as they say, an algorithmic insoluble problem. There are many different ways to determine that some books do not contain descriptions of algorithms for calculating decimals (if, for example, they contain meaningless text or they talk about life on Mars), but there is no universal recipe for determining in a finite time whether a book contains a description of such an algorithm.

This concludes our discussion of the possible meaning of the words in the title of the article. We are a little worried that after all that has been said, some readers begin to eradicate the “consideration” of infinite sets in their studies of mathematics, causing trouble for themselves and those around them. Therefore, we will make a few more comments, which, unfortunately, we will not be able to support here with arguments (and will limit ourselves to references to authorities). Most mathematicians are able to “consider” infinite sets, introducing small restrictions into their reasoning that, in their opinion, ensure the absence of contradictions. At the same time, of course, one comes to “examine” objects, verbal descriptions of which a person could not complete until the end of his days. Nevertheless, we found out that such “considerations” lie at the very basis of the edifice of mathematics that has been built over centuries, and its future fate depends on them. Although a number of mathematicians, with more or less consistency, are trying to present mathematics on the basis of an understandable framework that allows for a finite description, and along this path many very meaningful facts have been obtained that have made it possible to look at many things from a different perspective, however, the claims that the development of mathematics should take place within these frameworks have not received widespread support. “No one can drive us out of the series that Cantor created for us,” said one of the greatest mathematicians, David Hilbert.

We place one of the problems proposed at the 5th traditional Olympiad in linguistics and mathematics (at the first type in 1969).

Problem.
Letter 1.

Dear friend!
Some time ago I bought two very beautiful dogs: a shepherd and a dachshund, but quickly discovered that they were very poorly trained. The fact is that they really love to lie down. As a result, living with them in the same house is quite difficult. However, I do not despair, or I have established through practical testing that their behavior is subject to a certain law, incomprehensible but indisputable (although this may be explained by the fact that dogs used to perform in the circus), and that I can influence them by playing the piano or closing a real lamp.

During every minute, each dog either barks or remains silent: they do not detect any transitions. Their behavior in the next minute depends only on the events of the moment, and this dependence is as follows.

The shepherd dog in the next minute behaves in the same way as in the previous one: barks or is silent, unless in that previous minute the piano was played and the dachshund did not bark. In the latter case, the shepherd changes its behavior to the opposite: barking to silence and vice versa.

As for the dachshund, if the lamp was on in the previous minute, the dachshund will lie down or be silent depending on whether the shepherd dog barked or was silent, so the dachshund copies the shepherd dog with a minute delay. If, however, the lamp was not on, the dachshund would do the opposite of what the shepherd did.

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