Solution: Wine and mice

We now publish the solution to the Wine and mice divertimento.

Divertimento:

We collected 1000 bottles of wine to celebrate a party. The day before the event, we receive a warning that one of the bottles is poisoned. It is a slow-acting poison, which kills the person who ingests it within 24 hours.

We have 10 laboratory mice, which we assume can be given any amount of wine.

Can we identify the exact poisoned bottle?

Solution:

It is possible to label the bottles with ten-digit numbers made up of zeros and ones, because with ten digits, and two possibilities for each digit, there are \(2^{10}=1024\) numbers. Thus, we label the first bottle with the number \(0000000001\), the second bottle with the number \(0000000010\), the third with \(0000000011\), the fourth with \(0000000100\), and so on. The last bottle has the number \(1111101000\).

We also label each mouse with a number from 1 to 10. Mouse number 1 is given wine from all the bottles with a 1 in the first digit, mouse number 2 from all the bottles with a 1 in the second digit, mouse number 3 from all the bottles with a 1 in the third digit, and so on.

By observing which mice have died we form the following number: the \(i\)-th digit  of that number will be a 0 or a 1 depending on whether the mouse number \(i\) is still alive or has died. Obviously this number uniquely identifies the poisoned bottle.