3. Ten consecutive natural numbers are written on the board. What is the maximum number of them that can have a digit sum equal to a perfect square?
Answer: 4.

Solution: Note that the sums of the digits of consecutive natural numbers within the same decade are consecutive natural numbers. Since there are 10 numbers, they span two decades. Also note that among ten consecutive natural numbers, there can be no more than 3 perfect squares, and three perfect squares can only occur if there are 9 numbers. Therefore, among the 10 consecutive natural numbers written on the board, there cannot be more than 4 numbers with a sum of digits equal to the square of a natural number. We can provide an example where there are exactly 4 such numbers: $\{1,2,3,4,5,6,7,8,9,10\}$.

Grading criteria. Full solution - 20 points. Only the estimate - 12 points. Answer + example - 9 points. Only the answer - 3 points.