With 10 prisoners we've effectively got 10 bits. One prisoner samples every second bottle. Prisoner 2 skips 2 then samples 2, then skips 2, and so on. The pattern continues:
- 1: even bottles only
- 2: groups of 2
- 3: groups of 4
- 4: groups of 8
- 5: groups of 16
- 6: groups of 32
- 7: groups of 64
- 8: groups of 128
- 9: groups of 256
- 10: groups of 512

Depending on who dies, we get a binary number from 0 to 1023 (although it should only get up to 999). If we've numbered the bottles accordingly, that's how we know which is poisoned).