Determining the Poisoned Bottle with Mice: An Optimized Approach

Imagine the challenge of identifying the single poisoned bottle among a set of n bottles. Utilizing mice for such a task is both an intriguing and efficient method. In this article, we will explore the binary testing method using mice, and contrast it with a time-sensitive approach that involves directly testing individual bottles one at a time. This will provide a deeper understanding of the puzzle and the limitations of different methods.

Binary Testing Method with Mice

In a situation where you have 20 bottles and one is poisoned, the optimal way to identify the poisoned bottle is by employing a binary testing method with mice. Here's a detailed explanation of how this method works:

Binary Representation

Each bottle can be represented using a unique binary number. Given there are 20 bottles, we need enough bits to represent numbers from 0 to 19, as 25 (32) is the smallest power of 2 greater than 20. This means we need five bits to uniquely identify each bottle.

Assigning Mice to Bits

With five bits, we can assign each of the five mice to one of the five bits. For each bottle, we examine its binary representation and give a drop of the wine from that bottle to the corresponding mice based on the bit value (1 or 0).

Testing and Interpreting Results

After the testing period, we observe which mice have survived and which have died. The resulting combination of dead and alive mice will give us a binary number corresponding to the poisoned bottle.

The minimum number of mice required to determine the poisoned bottle among 20 is 5. This method is efficient and ensures accurate identification without the need for mixing or other complex actions.

A Time-Sensitive Approach

Consider a scenario where you need to identify the poisoned bottle within a very short time frame, specifically within an hour. Here's a proposed method that involves directly testing individual bottles one at a time:

Direct Testing at a Rapid Pace

If you start the experiment precisely at noon and give one sample from each bottle to the rat one at a time, each bottle will be tested once per second. If a bottle is poisoned, the rat will die exactly k seconds after 1:00 PM, where k is the index of the bottle. This method can complete the experiment in one hour.

To complete the experiment within x hours, you can adjust the testing rate to n/(6), meaning testing at a rate of n bottles per hour instead of one per second.

Controversy and Spirit of the Puzzle

While the above method appears to satisfy the conditions of the puzzle, it raises a question about the true solution. The puzzle explicitly mentions the requirement for mixing small amounts of wine before consumption. If mixing small amounts of wine in different proportions does not alter the poison's effect, it raises the following question:

Why should mixing after consumption change the poison's effect? Has a critical rule been broken in the rapid testing method?

The focus here is on the puzzle's spirit rather than the exact mechanics. Mixing beforehand is a condition that ensures the poison remains distinct and measurable. Violating this can be seen as breaking the puzzle's spirit, even if the mechanical conditions are met.

Conclusion

The binary testing method using mice is an optimal approach for identifying a poisoned bottle in a set of 20 bottles. On the other hand, a time-sensitive approach that involves direct testing at different rates can also work but may not align with the spirit of the original puzzle. Understanding the spirit of the puzzle is as important as solving it using a method that fits all conditions.