4. On a circle, 1974 children are arranged and are playing a game of elimination as follows: the first child remains in the circle, the second child is eliminated, the third child remains, the fourth child is eliminated, and so on until only one child remains in the circle. Determine which child remains.
Solution. In the first round, children with even numbers are eliminated - only those with numbers of the form $2k+1$ remain, the smallest of which is 1 and the largest is 1973. In the second round, children with numbers remain such that the difference between adjacent numbers is equal to 4; the smallest of these numbers is 1, and the largest is 1973. We continue with this reasoning and obtain the table:

| Round | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :--- | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| Smallest Num. | 1 | 1 | 5 | 13 | 13 | 45 | 109 | 109 | 365 | 877 |
| Difference | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 |
| Largest Num. | 1973 | 1973 | 1973 | 1965 | 1965 | 1965 | 1901 | 1901 | 1901 | 1901 |

In the last, eleventh round, the child with the number 877 is eliminated, and the child with the number 1901 remains.

## Second Year