## 31. Visit to Paris

Aunt Amelia visited Paris as a guest of her three little nephews and walked a lot with them around the city. After this, each of them told the following.

First nephew. We went up the Eiffel Tower, but we did not visit Montparnasse, however, we visited the Arc de Triomphe.

Second nephew. We went up the Eiffel Tower and visited Montparnasse. But we did not visit either the Arc de Triomphe or the ball game hall.

Third nephew. We did not go up the Eiffel Tower, but we visited the Arc de Triomphe.

If each child lied once (and only once), then which sights of Paris did they actually visit with Aunt Amelia?
31. Let $E$ denote the statement: they visited the Eiffel Tower; $A$: they visited the Arc de Triomphe; $M$: they visited Montparnasse, and finally, $P$: they visited the Ballroom. The nephews of Aunt Amelie claimed that:

the first: $E$ and $A$, but not $M$;

the second: $E$ and not $A, M$ and not $P$;

the third: not $E$ and $A$.

If $E$ is false, then $A$ is true, since the first nephew lied only once. Therefore, the second nephew lied twice, which is excluded. Hence, $E$ is true. Therefore, the third nephew lied when he said: not $E$. Then $A$ is true. Consequently, the first nephew lied when he said: not $M$. Thus, $M$ is true. The second nephew, who lied when he said: not $A$, told the truth when he said: not $P$. It follows that Aunt Amelie took her nephews to the Eiffel Tower, Montparnasse, and the Arc de Triomphe, but did not take them to the Ballroom.

established by the famous French scientist and philosopher René Descartes (1596-1650), and is sometimes called the Descartes-Euler theorem (but more often simply Euler's theorem). - Ed. note.

2 For a proof of Euler's theorem, see, for example: Rademacher H., Toeplitz O. Numbers and Figures.- M.: Science, 1966, topic 13, section 2; Golovina L. I., Yaglom I. M. Induction in Geometry.- M.: Fizmatgiz, 1961, § 2, example 12; Pólya G. Mathematics and Plausible Reasoning. - M.: IL, 1957, chapter II, and especially - Lakatos (or Lakatosh) I. Proofs and Refutations.- M.: Science, 1967.- Ed. note.