The King’s Poisoned Wine
The Strategy:
Number the Bottles: Label the bottles from 0 to 999. Convert each decimal number to its 10-digit binary equivalent (e.g., 0 = 0000000000, 3 = 0000000011, 999 = 1111100111).
Assign Prisoners: Assign each of the 10 prisoners a single binary digit position (Prisoner 1 for the first digit, Prisoner 2 for the second, and so on, up to Prisoner 10 for the tenth).
Create Mixtures: For each bottle, create a mixture: if a bottle’s binary number has a ‘1’ in a specific position, add a drop of that bottle’s wine to the cup for the corresponding prisoner.
Example: Bottle #3 (0000000011) would contribute a drop to Prisoner 9 and Prisoner 10’s cups. Bottle #1 (0000000001) only contributes to Prisoner 10’s cup.
Administer Simultaneously: Give each prisoner their unique mixture to drink at the same time.Observe & Decode: After 24 hours, the prisoners who die reveal the poisoned bottle. The pattern of dead (1) and alive (0) prisoners forms a binary number. Convert this binary result back to decimal to find the poisoned bottle.
Why It Works: There are \(2^{10}=1024\) possible unique outcomes (combinations of dead/alive prisoners) for 10 prisoners, which is enough to uniquely identify one of the 1000 bottles (since 1024 > 1000). Each bottle’s unique 10-digit binary code directly maps to a distinct set of prisoners who will drink from it, creating a unique “fingerprint” for that bottle.