Bad King and Wine Bottles puzzle
A bad king has 1000 bottles of very expensive wine. A neighbouring King plots to kill the bad king and sends a servant to poison the wine. Unfortunately the bad king’s guards catch the servant after he has only poisoned one bottle. Alas, the guards don’t know which bottle but know that the poison is so strong that even if diluted 1,000,000 times it would still kill the king. Furthermore, it takes one month to have an effect. The bad king decides he will get some of the prisoners in his vast dungeons to drink the wine. Being a clever bad king he knows he needs to murder no more than 10 prisoners – believing he can fob off such a low death rate – and will still be able to drink the rest of the wine at his anniversary party in 5 weeks time. Explain How ?
Number the bottles 1 to 1000, and write the number in binary format.
bottle 1 = 0000000001
bottle 250 = 0011111010
bottle 1000 = 1111101000
Now take your prisoner’s 1 through 10
Let prisoner 1 take a sip from every bottle that has a 1 in its least significant bit.
Let prisoner 10 take a sip from every bottle with a 1 in its most significant bit. etc.
Prisoner – 10 9 8 7 6 5 4 3 2 1
Bottle 924 – 1 1 1 0 0 1 1 1 0 0
In this, bottle #924 would be sipped by 10,9,8,5,4 and 3
That way if bottle #924 was the poisoned one, only those prisoners would die.
After four weeks,
line the prisoners up in their bit order and read each living prisoner as a 0 bit and each dead prisoner as a 1 bit.
The number that you get is the bottle of wine that was poisoned.
Lets there are three prisoners
1 &2 4
If poison is filled in bottle no:4 both 1 and 2 dies .
If only 1st prisoner dies the poison is in 1st bottle.
If only 2nd prisoner dies the poison is in 2nd bottle.
similarly interpret others as well
From this one can see that with 3 prisoners i can verify 7 bottles.
If you observe carefully this nothing but no of selecting 3 prisoners excluding null set.
So with 10 prisoners i can verify 1023 bottles.(1024(2^10)-1)
If the king gets 1000 prisoners to drink a very small amount(1ppm soln) from each bottle such that each prisoner drinks wine from only 1 bottle, he will get to know which bottle contains the poison. Because ultimately only 1 prisoner will die while the rest will not die as their wine didn’t contain the poison in the first place.
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