Yes, King Ross can determine the poisoned barrel within 48 hours using the four prisoners. The key is to leverage binary representation to test multiple barrels efficiently.

Strategy:

1. Number the barrels from 1 to 500.

2. Use the binary numbering system to represent each barrel. For example:

   • Barrel 1 = 0000001
   • Barrel 2 = 0000010
   • ...
   • Barrel 500 = 111110100

3. Assign each bit of the binary representation to a specific prisoner:

   • Prisoner 1 tests the least significant bit.
   • Prisoner 2 tests the second bit.
   • Prisoner 3 tests the third bit.
   • Prisoner 4 tests the fourth bit.

4. Each prisoner drinks a mix of wine from barrels where their corresponding bit is 1. For instance:

   • If the 1st bit (least significant) is 1, Prisoner 1 drinks from that barrel.
   • If the 2nd bit is 1, Prisoner 2 drinks, and so on.

5. Wait 12 hours for the results. Depending on which prisoners die, the binary pattern of deaths (alive = 0, dead = 1) identifies the poisoned barrel.

   • Example:
     • If only Prisoners 1 and 3 die, the binary result is 101, which corresponds to barrel 5.

6. Since the prisoners have 48 hours, this process can be repeated up to four times, allowing multiple tests if necessary. However, in this case, one round is sufficient to determine the poisoned barrel because all barrels can be tested at once using this method.

Conclusion:

By using binary representation and the 12-hour death confirmation, it is possible to determine the poisoned barrel within 48 hours with just four prisoners.

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Questions to Consider:

1. How would the strategy change if the death confirmation time were longer than 12 hours?
2. Could this method work with fewer prisoners? Why or why not?
3. What if the king had more than 500 barrels? How many prisoners would he need?
4. How would the setup change if the king wanted to save the prisoners' lives?
5. Could this strategy be adapted for scenarios with different constraints, such as delayed results?

Tip:

In mathematical problems like these, breaking them down into smaller logical components (like binary representation) can simplify complex scenarios.