Pistike counts in an original way on her fingers. She starts with 1 on her thumb, then 2 and 3 on her index finger, 4, 5, and 6 on her middle finger, 7 on her ring finger, and 8 and 9 on her pinky. Then she continues backward, 10, 11, and 12 on her ring finger again, 13 on her middle finger, 14 and 15 on her index finger, 16, 17, and 18 on her thumb, 19 again backward on her index finger, and so on. On which finger will she count 2004?
Solution. Let's create a table. Follow the counting and see at which number it happens again that Pistike counts a single number on the thumb:

| thumb | index finger | middle finger | ring finger | little finger |
| :---: | :---: | :---: | :---: | :---: |
| 1 | 2,3 | $4,5,6$ | 7 | 8,9 |
| $16,17,18$ | 14,15 | 13 | $10,11,12$ |  |
|  | 19 | 20,21 | $22,23,24$ | 25 |
| 32,33 | 31 | $28,29,30$ | 26,27 |  |
|  | $34,35,36$ | 37 | 38,39 | $40,41,42$ |
| 49 | $46,47,48$ | 44,45 | 43 | $\longleftrightarrow$ |

From the writing, we can read that this number is 49, which means the numbers repeat in groups of 48. Since $2004=41 \cdot 48+36$, it is enough to see where the number 36 is located in the table, because then, due to periodicity, 2004 will also fall there. Therefore, Pistike will count 2004 on the index finger.