2. There are 1000 crows sitting on a wire. At the end of each minute, every third (third, sixth, ninth, and so on) crow flies away.

a) Which crows, by their initial count, will remain on the wire in the end?

b) How many minutes will pass before the crows stop flying away?
Solution. a) It is clear that at the end there will be no more than two ravens left, and the first and second never fly away. Therefore, it will be precisely these two. Answer: 1 and 2.

b) It is easy to see that the number of ravens flying away in the next minute is one-third of their number, rounded down. Thus, it only remains to carefully count:

| minute | $\sqrt{1}$ |  | 2 | 3 | $\overline{4}$ | 5 | 6 | 7 | 8 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| ravens at the start | 1000 |  | 57 | 445 | 297 | 198 | 132 | 88 | 59 |
| will fly away at the end | 333 |  | 22 | 128 | 99 | 66 | 44 | 29 | 19 |
| minute | 9 | 10 | 11 | $\overline{12}$ | 13 | 14 | 15 | 16 | 17 |
| ravens at the start | 40 | 27 | 18 | 12 | 8 | 6 | 4 | 3 | 2 |
| will fly away at the end | 13 | 9 | 6 | 4 | 2 | 2 | 1 | 1 | 0 |

We see that the ravens stopped flying away on the 17th minute, thus we get the answer: 16 minutes.