10.6. On the board, three natural numbers were written: two ten-digit numbers $a$ and $b$, as well as their sum $a+b$. What is the maximum number of odd digits that could have been written on the board?

(I. Bogdanov, P. Kozhevnikov)
Answer: 30.

Solution: Note that the number $a+b$ has no more than 11 digits, so in total, no more than 31 digits are written on the board. At the same time, all three numbers $a, b, a+b$ cannot be odd simultaneously. Therefore, one of their last three digits is even, which means that there are no more than 30 odd digits written.

Let's provide an example showing that there could be exactly 30 odd digits:

$5555555555+5555555555=11111111110$.

Remark. There are many examples with 30 odd digits, for instance, $3999999999+7999999999=11999999998$.

Comment. Only the answer without any correct explanation - 0 points. Only the proof that there are no more than $30-3$ points. Only a correct example with 30 odd digits - 4 points.