MHB Optimizing Prisoner Selection for Poisoned Wine Detection in Medieval Empires

  • Thread starter Thread starter alane1994
  • Start date Start date
  • Tags Tags
    Logic
alane1994
Messages
36
Reaction score
0
You are the ruler of a medieval empire and you are about to have a celebration tomorrow. The celebration is the most important party you have ever hosted. You've got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned.


The poison exhibits no symptoms until death. Death occurs within ten to twenty hours after consuming even the minutest amount of poison.


You have over a thousand slaves at your disposal and just under 24 hours to determine which single bottle is poisoned.


You have a handful of prisoners about to be executed, and it would mar your celebration to have anyone else killed.


What is the smallest number of prisoners you must have to drink from the bottles to be absolutely sure to find the poisoned bottle within 24 hours?

10 prisoners must sample the wine. Bonus if you worked out a way to ensure than no more than 8 prisoners die.
Number all bottles using binary digits. Assign each prisoner to one of the binary flags. Prisoners must take a sip from each bottle where their binary flag is set.
Here is how you would find one poisoned bottle out of eight total bottles of wine.

[TABLE="class: solutions"]
[TR]
[TD="align: right"][/TD]
[TD="align: center"]Bottle 1[/TD]
[TD="align: center"]Bottle 2[/TD]
[TD="align: center"]Bottle 3[/TD]
[TD="align: center"]Bottle 4[/TD]
[TD="align: center"]Bottle 5[/TD]
[TD="align: center"]Bottle 6[/TD]
[TD="align: center"]Bottle 7[/TD]
[TD="align: center"]Bottle 8[/TD]
[/TR]
[TR]
[TD="align: right"]Prisoner A[/TD]
[TD="align: center"][/TD]
[TD="align: center"]X[/TD]
[TD="align: center"][/TD]
[TD="align: center"]X[/TD]
[TD="align: center"][/TD]
[TD="align: center"]X[/TD]
[TD="align: center"][/TD]
[TD="align: center"]X[/TD]
[/TR]
[TR]
[TD="align: right"]Prisoner B[/TD]
[TD="align: center"][/TD]
[TD="align: center"][/TD]
[TD="align: center"]X[/TD]
[TD="align: center"]X[/TD]
[TD="align: center"][/TD]
[TD="align: center"][/TD]
[TD="align: center"]X[/TD]
[TD="align: center"]X[/TD]
[/TR]
[TR]
[TD="align: right"]Prisoner C[/TD]
[TD="align: center"][/TD]
[TD="align: center"][/TD]
[TD="align: center"][/TD]
[TD="align: center"][/TD]
[TD="align: center"]X[/TD]
[TD="align: center"]X[/TD]
[TD="align: center"]X[/TD]
[TD="align: center"]X[/TD]
[/TR]
[/TABLE]
In the above example, if all prisoners die, bottle 8 is bad. If none die, bottle 1 is bad. If A & B dies, bottle 4 is bad.
With ten people there are 1024 unique combinations so you could test up to 1024 bottles of wine.
Each of the ten prisoners will take a small sip from about 500 bottles. Each sip should take no longer than 30 seconds and should be a very small amount. Small sips not only leave more wine for guests. Small sips also avoid death by alcohol poisoning. As long as each prisoner is administered about a millilitre from each bottle, they will only consume the equivalent of about one bottle of wine each.
Each prisoner will have at least a fifty percent chance of living. There is only one binary combination where all prisoners must sip from the wine. If there are ten prisoners then there are ten more combinations where all but one prisoner must sip from the wine. By avoiding these two types of combinations you can ensure no more than 8 prisoners die.

 
Mathematics news on Phys.org
In fact no reply is needed - see Post #1 spoilers.
 

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
View full post »

Thread 'Fermat's Last Theorem'

Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
View full post »

Thread 'Useless continued fraction for 1'

I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.
View full post »

Similar threads

Logic puzzle : Smallest number of prisoners required?
Replies
14
Views
2K

Hot Threads

Recent Insights

Back
Top