2 [Chess boards and chess pieces]

During a chess game, at some point, Hedgehog had twice fewer pieces on the board than Bear, and they were five times fewer than the free squares on the board. How many of Bear's pieces had been captured by this point?

#
According to the condition, the number of cells on the board (64) is $1+2+5=8$ times more than the number of pieces the Hedgehog has. Therefore, the Hedgehog has 8 pieces left, and the Teddy Bear has 16.

## Answer

None.

Author: $\underline{\text { Folklore }}$

Find the smallest number divisible by 45, whose decimal representation consists only of ones and zeros.

## Solution

A number is divisible by 45 if it is divisible by each of two coprime numbers: 9 and 5. Since the desired number is divisible by 9, the sum of its digits must be divisible by 9. Therefore, the number of ones in the desired number must be a multiple of nine. A number divisible by five can end in 0 or 5, but the latter case is impossible by the condition. Thus, the desired number must contain 9 ones and end with a zero.

## Answer

1111111110.