The wine and the rats

There are 10 rats. Each rat can be either live or dead. Therefore, there are actually (2^10) - 1 = 1023 different dead/alive combinations of rats available which you can use to pinpoint which wine was poisoned.

• Number the rats 1, 2, 4, 8, 16, 32, 64, 128, 256, and 512. Notice that they are numbered after significant bits.
• Number the bottles from 1 to 1000 using binary.
• For each bottle: 
• Inspect the binary number assigned to it
• For each bit of the binary number, starting from the least significant bit:
• If it is 1, then have the rat with the corresponding significant bit taste it
• For example, if the LSB is 1, then have bat #1 taste it; if the second LSB is 1, have bat #2, taste it; if the third LSB is 1, have bat #4 taste it.

Claim: this algorithm can discern precisely which bottle was poisoned.

Proof: since it is known that exactly one bottle is poisoned, it does not matter if the binary encoding of two bottles share a bit or more.