A step to The smart Hassan poison tasters problem

Logic Level 5

You are the ruler of a great empire and you have decided to throw another celebration tomorrow. However, apparently you were forgetful of the near disaster that happened in the last celebration, and decide to hatch your own plan with poison. For during this celebration, one of your most hated rivals will come, thinking that you have finally given up your fight and you held this celebration in part to surrender to your rival's superiority. You decide to pour some drops of that same deadly poison (that have no symptoms except death 10 to 20 hours later) into his glass; however, you do not want to look guilty, so to push the blame away you will insert some poison in your own drink that will make you sick after 10 to 20 hours with some worrisome but definitely non-lethal symptoms.

You have already inserted the deadly poison in one glass and your non-lethal poison in another glass out of the $\boxed{1402}$ glasses, but because of your forgetfulness you forgot which glasses had the poisons! Certainly, you don't want another one of your beloved guests to drink a poisoned glass, or even worse, giving yourself the lethally poisoned glass.

Fortunately, you still have your supply of death-row prisoners who you are sure won't mind lending a helping hand by taste-testing the glasses. What is the least amount of prisoners needed to guarantee successful location of the lethally and non-lethally poisoned glasses?

Note: if the lethal poison acts before the non-lethal poison, the prisoner will die without any symptoms.

inspired from this problem but with more glasses.

Here is a harder version of this problem

The answer is 20.

1 solution

Mr X
Nov 24, 2016

Here is the solution for 1458 glasses:

It is clear that with 2 prisoners you can reduce the problem to the half number of glasses (for more details see the proof of this problem 1 ). By the same way, 3 prisoners reduce the problem to the third of the number of glasses. Now, just use 2 prisoners 1 time, and use 3 prisoners 6 times. So, for a total of $\boxed {20}$ prisoners you locate the lethal and non-lethal glasses.

Note: this is not the smallest number of prisoners needed. see this problem 2

Can you explain how "By the same way, 3 prisoners reduce the problem to the third of the number of glasses"?

Calvin Lin Staff - 4 years, 6 months ago

If we have 3 prisoners we can divide the glasses into 3 equal subsets. each prisoner will test one subset. So we can locate the lethal and non-lethal glasses each in a third of the initial number of glasses.

Note: equal subsets to 1 glass difference when the number of glasses is not a multiple of 3.

Mr X - 4 years, 6 months ago

I have another solution Let's take 11 prisoners and mark them as 1,2,4,8,16,...,1024 Now number the glasses from 1to 1402. Take any glass and give it to the prisoners whose sum is the desired number. Like the glass numbered 10
Should be given to prisoners 2,8. Now which which prisoner will die there sum will give the glass with lethal poison After 1 prisoner dies we one more to know in a similar way which is non lethal poison. So in total we need 12 prisoners.

Prayas Rautray - 4 years ago

You don't have enough time to make a second test after you finish the first one. All you have is 24 hours and the test takes up to 20 hours.

Any way, even if you can run a second test you will need the same number of prisoners who died. So your method needs 22 prisoners.

Mr X - 4 years ago

A step to The smart Hassan poison tasters problem

The answer is 20.

1 solution

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