## Interns or Slaves?

My classmate Misha gave me a math problem. Although I liked the math part, I hated the setup. Here is the problem using the new setup:

You are hoping to get very rich one day and you base your hopes on your new invention: a diet pill. The pill works beautifully and doesn’t have side effects. Patients simply take a pill when they get hungry. Within one hour they will fall asleep and will be unable to awake for exactly two hours, at which time they will awake by themselves feeling completely sated.

You have just arrived at your lab, when you realize that one of your interns has misplaced your bottle of wonder pills. After some investigation, you come to the conclusion that your bottle is on the shelf with 239 other bottles that contain a placebo. Unfortunately, those placebo bottles were custom-designed for your statistical tests to exactly match your medicine bottle.

You need to bring the medicine bottle to your boss in two hours. While you panic, all your five interns volunteer for experiments, hoping to be mentioned in your paper. Can you find your bottle in two hours?

The problem Misha gave me had 240 barrels of wine, one of which contained deadly poison and five slaves who could be spared.

I do not like killing people even when they are imaginary. But while I was slow in inventing my own setup, the original version of the puzzle started making the rounds on the Internet. So I decided to kill the problem by writing the solution.

The strategy is to give out some pills immediately, wait for one hour and see who falls asleep. The next step is to give some other pills at the beginning of the next hour to some of the interns who are awake.

Let’s count the information you can get out of this. Each intern will experience one of three different situations: falling asleep in the first hour, in the second hour, or not falling asleep at all. Thus, you can have a total of 3^{5} = 243 different outcomes.

If you had more bottles than 243, there would be no way to distinguish between them. The fact that you have 240 bottles might mean that 243 will work too, but apparently the designer of the puzzle didn’t want to hint into powers of three and picked the largest round number below 243. These considerations should increase your willingness to look into this problem base 3.

Let us label the bottles with different 5-character strings containing three characters 0, 1, and 2. Now we can use the label as instructions. The first character will be associated with the first intern and so on. Suppose the fourth character on the bottle’s label is 0, then the fourth intern doesn’t need to struggle with digesting a pill from this particular bottle. If the fourth character is 1, then the fourth intern gets the pill in the first hour. If the fourth character is 2, the fourth intern gets the pill in the second hour.

Note this minor detail: Suppose the fourth character on a bottle is 2, but the fourth intern is asleep by the second hour. That means, the bottle doesn’t contain the medicine, and we can put it aside.

At the end of two hours you know who fell asleep and when. This data will exactly match the label on the bottle with the medicine.

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## colorblind:

A cute, but not very challenging, problem. I certainly agree the morbidity of the problem as originally proposed would be off-putting.

Follow up problem: While meeting with the boss, one of your interns loses the real pills in amongst the orginal other 239. They are promptly fired. The boss is meeting with you again at the end of the day, 4 hours later. Find the pills.

On another note, the US diet pill industry is VERY poorly regulated, so I’m not sure they’d do the testing suggested above. Personally, I’d have said the other placebos were really just the old diet pills that gave a person gas/hives/some other relatively harmless malady (hence why they needed a replacement with something new like your new pill).

5 July 2010, 1:03 pm## Jonathan:

Of course, after the first hour you might be able to put aside many more bottles than just those assigned to a sleeping intern. Taking advantage of this would be a good idea if you don’t want to waste the placebos, or if there is any reason to prefer taking less than 100 of them, rather than up to 160.

7 July 2010, 1:50 am