Problem 6.5. Masha told Sasha that she had thought of two different natural numbers greater than 11. Then she told Sasha their sum.

- Sasha, after thinking, said: "I don't know the numbers you thought of."
- To which Masha replied: "At least one of them is even."
- Then Sasha said: "Now I definitely know the numbers you thought of!"

What numbers did Masha think of? Enter the answers in any order.
Answer: 12 and 16.

Solution. The sum of two different natural numbers, both greater than 11, is a natural number not less than $12+13=25$.

Suppose Masha told Sasha that the sum of the numbers is 25. Then he would have immediately understood that these are the numbers 12 and 13. Contradiction.

Suppose Masha told Sasha that the sum of the numbers is 26. Then he would have immediately understood that these are the numbers 12 and 14 (two identical numbers 13 could not be). Contradiction.

Suppose Masha told Sasha that the sum of the numbers is 27. Knowing the parity of one of the numbers, Sasha would still not have understood the guessed numbers: either it was 12 and 15, or 13 and 14. Contradiction.

Suppose Masha told Sasha that the sum of the numbers is 28. Then he would have understood that it could have been either 12 and 16, or 13 and 15 (two identical numbers 14 could not be). Knowing the parity of one of the numbers, Sasha would have definitely understood that these are the numbers 12 and 16.

Suppose Masha told Sasha that the sum of the numbers is $S \geqslant 29$. Knowing the parity of one of the numbers, Sasha would still not have understood the guessed numbers. Indeed, at least two pairs fit all the above conditions: 12 and $S-12$ (where $S-12>12$), and 14 and $S-14$ (where $S-14>14$). Contradiction.