9.6. Ten-digit natural numbers $a, b, c$ are such that $a+b=c$. What is the maximum number of their 30 digits that can be odd?

(I. Bogdanov)

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# Answer. 29.

Solution. Note that if $a+b=c$, then all three numbers $a, b$, $c$ cannot be odd simultaneously. Therefore, among them, there is at least one even number, and the last digit of this number will also be even. Thus, among the 30 digits, there is at least one even digit, and the number of odd digits is no more than 29.

Example $1999999999+1111111111=3111111110$, shows that among the 30 digits, there can be exactly 29 odd digits.

Remark. There are many examples with 29 odd digits, for example, $3999999999+3999999999=7999999998$.

Comment. Only the answer without any correct explanations - 0 points.

Only proof that the number of odd digits is no more than $29-3$ points.

Only a correct example with 29 odd digits - 3 points. 9.7. Vasya wrote natural numbers from 1 to 81 in the cells of a $9 \times 9$ table (each cell contains one number, all numbers are different). It turned out that any two numbers differing by 3 are in adjacent cells. Is it true that there will necessarily be two corner cells, the difference of the numbers in which is divisible by 6?

(O. Podlipsky)

Answer. True.

Solution. Consider the remainders of the numbers in the four corner cells when divided by 3. By the pigeonhole principle, at least two of these numbers, $x$ and $y$, will have the same remainder, meaning the difference $y-x$ is divisible by 3. Without loss of generality, assume $x<y$.

Color the cells of our table in a checkerboard pattern with black and white such that the corner cells are black. Consider the cells with numbers $x, x+3, x+6, \ldots, y-3, y$. Any two of these numbers are in cells with a common side, meaning they are in cells of different colors. Therefore, all numbers in this sequence that have the same parity as $x$ are in black cells, and all others are in white cells. Since the number $y$ is in a black cell, it has the same parity as $x$, meaning $y-x$ is even. Thus, $y-x$ is divisible by 6.

Comment. Only the answer - 0 points;

Considering specific examples of number placement is not evaluated.

(1) It is correctly proven that the difference of some two numbers $x$ and $y$ in the corner cells is divisible by 3 - 2 points.

(2) The checkerboard coloring is considered, and it is noted that numbers differing by 3 are in cells of different colors - 2 points.

(3) It is then noted that all corner cells are of the same color, and it is claimed that the numbers in cells $x$ and $y$ have the same parity - another 2 points.

The points for advancements (1)-(3) are cumulative. Note that the proof of the evenness of the difference $x-y$ can be conducted without using the checkerboard coloring - such a proof should be evaluated accordingly.

Some remarks regarding these points, as well as some errors that may appear in the solutions, are provided below.

(1') After obtaining (1), it is claimed without justification that the numbers $x$ and $y$ have the same parity - no additional points are awarded.

(2') The idea of checkerboard coloring is not evaluated separately.

(3') Even after obtaining (2) and (3), the claim that $x$ and $y$ have the same parity requires some justification. Specifically, it is necessary to mention that there is a path from $x$ to $y$ where the numbers in adjacent cells differ by 3. (This mention can be implicit, for example, in an inductive argument.) If such a mention is absent, the problem is scored no more than 6 points. In particular, criteria (1)-(3) together give a sum of 6.

Some participants may believe that the difference between any two numbers in adjacent cells is 3. This fact is incorrect; any reasoning based on this fact is scored 0 points.

Some participants may conclude from the statement that any numbers differing by 3 are in cells of different colors (in the checkerboard coloring) that any two adjacent numbers have different parities. This conclusion is incorrect! If such an error is present in the solution, the problem is scored no more than 5 points. (5 points are awarded only if this argument is applied only to numbers having the same remainder when divided by 3.)

![](https://cdn.mathpix.com/cropped/2024_05_06_38e6a53fd9055131fdcdg-07.jpg?height=450&width=458&top_left_y=318&top_left_x=825)

Fig. 1