On a night march, the Wolf and Fox patrols each received one identical candle. They lit the candles together at the start and set off. Each member of the patrol carried the candle for the duration it took for its length to be halved. The Wolves reached the finish line at the moment when the sixth carrier was passing the candle to the seventh. After that, their candle burned for another three minutes. The Foxes reached the finish line in two hours and 57 minutes.

- Which patrol reached the finish line first?
- By how many minutes did the winning patrol finish earlier than the other patrol?

(The candle burns evenly: it always burns the same amount in the same time.)

(M. Dillingerová)
First, we will determine how long the Wolves ran. The sixth carrier passed the candle to the seventh at the moment when it still had 3 minutes left to burn. He, therefore, carried it for 3 minutes and received it from the fifth carrier at the moment when it still had $3+3=6$ minutes left to burn. The fifth carrier, therefore, carried it for 6 minutes and received it from the fourth carrier at the moment when it still had $6+6=12$ minutes left to burn. We can proceed this way up to the first carrier:

| carrier no. | carried the candle (min) |
| :---: | :---: |
| 7 | 0 |
| 6 | 3 |
| 5 | 6 |
| 4 | 12 |
| 3 | 24 |
| 2 | 48 |
| 1 | 96 |

The march of the Wolves lasted $96+48+24+12+6+3=189$ minutes. (Alternatively, it can be calculated as $96+96-3=189$.)

The march of the Foxes lasted $120+57=177$ minutes.

Therefore, the Foxes were the first to reach the finish, and they were ahead by $189-177=12$ minutes.

Scoring. 1 point for recognizing that each carrier carried the candle for as long as it had left to burn when they passed it to the next; 2 points for filling out the table or its equivalent; 1 point for determining the duration of the Wolves' march; 1 point for each answer to the questions in the assignment.

Do not award any points for a solution that only has an unjustified conclusion that the Foxes were the first to reach the finish.