# A Clever Approach to the Poison Bottle Dilemma

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## Chapter 1: The Poison Bottle Problem

The quest for unity can often be reflected through the lens of the binary system. As Buddha wisely noted, "Unity itself and the idea of Unity are already two." This brings us to a classic challenge in the realm of logic and testing: Imagine you have 1000 bottles, where 999 contain water and just one holds poison. With only 10 mice and a week to determine which bottle is toxic, how can you accomplish this task?

Before proceeding, I suggest you take a moment to think about the problem. Grab a pen and paper, and see if you can devise a solution!

### Understanding the Binary System

This problem is widely recognized, yet it harbors a clever solution! To start, since (2^{10} = 1024) which exceeds 1000, we can assign each bottle a unique 10-digit binary code as follows:

0000000001 0000000010 0000000011 0000000100 0000000101 … 1111100110 1111100111 1111101000

(Each code is derived by incrementing the previous one, with the final code representing the 1000th bottle. Each bottle has a distinct 10-bit binary identifier.)

Next, we label the 10 mice with numbers 1 through 10. Allow the first mouse to sample from the bottles corresponding to a '1' in the rightmost position of the binary code, the second mouse for the second position, and so on. For instance, if bottle number 0010011 is selected, the first, second, fifth, and eighth mice would drink from it.

After a week, the fate of the mice will reveal the binary code of the poisoned bottle. If the first mouse perishes, we conclude that the first digit of the toxic bottle's code is '1.' Conversely, if the second mouse survives, it indicates that the second digit is '0.' Thus, if mouse number K dies, the Kth digit from the right in the poisoned bottle's code is confirmed as '1'; if mouse number L lives, then that Kth digit is '0.'

In essence, we owe our gratitude to the brave mice for their contribution to this ingenious solution.

## Chapter 2: Visualizing the Algorithm

In this video titled "Binary Search Algorithm - Easy & Quick Review for The Night Before Exam," the presenter breaks down the algorithm efficiently, showcasing its relevance to solving such problems.

The second video, "This Algorithm is 1,606,240% FASTER," explores advanced algorithmic techniques and their practical applications, providing insights that enhance our understanding of problem-solving methods.