The 1000 Poisoned Wine Bottle Riddle

Find the poisoned wine bottle from 1000 similar wine bottles using 10 test strips that each change color when a drop of poisoned wine falls on it.

The Poisoned Wine Bottle Riddle

You have 1000 bottles of wine. One is poisoned. You also have 10 test strips that each can detect poison by changing its color with a drop of the poisoned wine on it. The poison takes an hour to show up on the strips. You need to find the poisoned bottle within an hour to save yourself from being poisoned. How do you do it?

Assume the 1 hour ticks just after you finish putting wine drops onto the strips.

Hint: First, form the primary condition for successful identification of the poisoned bottle before you think about the solution.

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Solution to the 1000 Poisoned Wine Bottle Riddle

Step 1: Form the primary condition and strategy for successful identification of the poisoned bottle

Primary condition: To test all the 1000 bottles using only 10 test strips:

• You need to identify one bottle among 1000 using a set of 10 strips in which only one combination of strips changed color receiving wine drops from it. No two bottles should produce the same strip result.
• It means: no two bottles should deliver drops to the same combination of strips.

Form the strategy that would meet the primary condition:

• Number the bottles from 1 to 1000.
• Place the strips one after the other and mark the positions by 1 to 10 from the right.
• Devise a method of putting drops from a bottle to the strips occupying a unique combination of positions. No two bottles should deliver its drops to the same combination of strip positions.

Each unique combination of 10 strip positions must then point to one and only one bottle among 1000.

This is the point of reasoning when the position of a strip gets primary importance.

• The only way to implement this strategy is to form a 10-digit unique number for a bottle that will deliver drops to the strips in positions having 1 in the 10-digit unique identifier of a bottle with rest of the digits marked 0 receiving no drops from this bottle.
• As you have put wine drops from a bottle marked by a unique combination of strip positions, no two strip combinations will change color. For example, the poisoned bottle 0000001010 will deliver its drops only to the 2nd and 4th strips. Only this combination of strips will change color and you will know 0000001010 is the poisoned bottle. Why use 1 and 0 in the numbering scheme? Because, using 1 and 0 is the easiest way to represent "yes" for receiving drops and "no" for not receiving drops from a bottle.
• This identifier will then form a 10-digit number with its digits 1 or 0 only.

Conclusion: Create a 10-digit numbering scheme with digits 1 or 0 so that each such 10-digit combination points to one and only one bottle among 1 to 1000.

A 10-digit number with 1's in certain positions of the number will deliver its drops to only those strip positions that are marked by 1 to 10.

For example: The bottle 1100101001 will deliver its drops to the strip positions 1, 4, 6, 9 and 10 from right. These 5 strips will get wine drops from this specific bottle.

Simplified problem:

How to devise a numbering scheme in which a 10-digit number with 1's and 0's as digits will be equivalent to a unique number that you and I will understand as the number of a specific wine bottle among 1000.

Step 2: Discover how to form the winning numbering scheme

From your experience in daily calculations, you already know how numbers are formed.

For example, you know a random number 1234 as a sum of an increasing series of numbers 0, 10, 100, 1000 starting from the right, each multiplied by the digit in a specific position:

1234 = 1x1000 + 2x100 + 3x10 + 4x1.

Why 10? Because any position of the number may have one of the 10 values from 0 to 9. Right? This is the positional base value of a number in this scheme. We multiply this base value of 10 with itself a number of times equal to the position minus one. This is the decimal number system you use daily.

Example: In the third position from the right, we multiply 10 with itself twice (three minus one), resulting in 10 x 10 = 100.

But what about the puzzle strips? How many values a strip position can take?

Revelation: The color of a test strip may remain either unchanged or will change when poisoned. A strip can then have only two states 0 for unchanged and 1 for changed.

This fits perfectly into our strategically arrived conclusion.

Extend the concept of forming the numbers such as 1234 to this new scheme with positional base value as 2 instead of 10:

• Positional value is 2 multiplied with itself the number of times of the position minus 1.
• Example: Positional value of 5th position is 2x2x2x2 = 16. Positional value 16 is based on positional base value 2. This forms the heart of a numbering scheme.
• Ultimate value of a number: Sum of the number in a position multiplied by positional value for each position.
• Example: 0000101101 = 0000 + 1x2x2x2x2x2 + 0 + 1x2x2x2 + 1x2x2 + 0 + 1 = 32 + 8 + 4 + 1 = 45. Positional value of the sixth position is 32, the number in sixth position 1 and positional base value for all positions 2.

Step 3: Actions to find the poisoned bottle

First step: Mark the bottles 1 to 1000 by numbers based on this new 10-digit numbering scheme:

1: 0000000001 : 000000000 + 1x1

2: 0000000010 : 00000000 + 1x2 + 0

3: 0000000011 : 00000000 + 1x2 + 1x1

.....

1000 : 1111101000 : 1x512 + 1x256 + 1x128 + 1x64 + 1x32 + 0 + 1x8 + 000.

Second step: Put wine drops from a 10-digit numbered bottle onto the strips at positions where the 10-digit number has 1.

Example: The 110th bottle is marked with the 10-digit number 0001101110. Put wine drops from only this bottle onto the strips at positions: 2, 3, 4, 6 and 7 from the right.

Third step: Waiting for 1 hour to know which strip combination has changed color.

Step 4: Find the poisoned bottle from the combination of strips in positions that changed color

If strips in positions 1, 3, 4, 7 and 9 changed color, corresponding 10-digit number having 1's in positions 1, 3, 4, 7 and 9 must have received drops from the poisoned bottle.

The number is:

0101001101 => 0 + 256 + 0 + 64 + 0 + 0 + 8 + 4 + 0 + 1 = 333.

Representing the values, 256, 64, 8, 4 and 1 in terms of binary number system,

• 333 = 1x2⁸ + 1x2⁶ + 1x2³ + 1x2² + 1x2⁰, where 2⁶ means 2 multiplied with itself 6 times, expressed in shorthand form mathematically,
• Or, 333 = 1x2^(9-1) + 1x2^(7-1) + 1x2^(4-1) + 1x2^(3-1) + 1x2^(1-1), where the first terms inside the brackets are positions of digit 1 in the 10-digit binary number, similar to decimal number system you use.

This is the way the numbers in the two-digit binary number system with 0 and 1 as digit values form.

Solution: 333rd bottle (in this example) has poisoned wine.

You have saved yourself from being poisoned.

Key idea

In the 10-digit binary number scheme, (2¹⁰ - 1) = 1023 non-zero decimal numbers 1 to 1023 can be represented. If the number of wine bottles were more than 1023, finding out a single poisoned bottle using 10 test-strips won't have been possible.

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Identifying the poisoned wine bottle using the binary number system and its equivalence to decimal numbers: Pictorial representation

This binary numbering scheme discovered by extending your idea about the numbers you use daily is the basis of the digital world today.

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Want to know more?

To solve a problem, people think in different ways. Here are a few examples:

1. A King, 1000 Bottles of Wine, 10 Prisoners and a Drop of Poison: Can you understand the math?
2. 1000 Wine Bottles: Do you appreciate the basis underlying the method?
3. Puzzle 19 | (Poison and Rat): The term "binary" you face at the very beginning. Are you aware of its mechanisms?

The overwhelming tendency to use math in solving problems ranging from puzzles to AI development, bypasses and curbs the power of human reasoning and inventive thinking. Be aware!

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