## Tuesday, September 25, 2012

### Myth of the Right Answer Redux

A classmate brought this riddle to a study group I'm in:
You have eight pills. One of them is poisonous. The poisonous pill weighs slightly more than the others, but otherwise they appear to be identical. You have access to a scale, but you may only use the scale twice (for some reason). How do you determine which pill is the poisonous one?
The solution produced by our group (four undergraduate math majors) is as follows:

Satisfied with this solution, the other members of my study group were ready to move on. This is the myth of the right answer.

We are programmed since elementary school to find "the" answer and move on to our next assignment. Each quiz, riddle, puzzle, and problem is simply an obstacle to overcome in the ongoing mission to meet our teachers', principals', parents', and professors' approval. Why should the authority figure determine when our problem is solved?

So it's great that we found the algorithm to solve this problem with two weighs. But why stop there? Some related questions:
• Given n pills, what is the minimum number of weighs required to finding the poison pill?
• Given n pills that can be weighed with w weighs, is there an alternate weighing scheme that can find the poison pill  in exactly w weighs?
• What if we don't know if the poison pill is heavier or lighter (only that it weighs a different amount)?
• What if every pill weighs a different amount, but the poison pill is still heavier than all the others?
• What if there are two poisoned pills
• Given n pills, m of which are poisoned, how many weighs are required to find the poisoned pills?
• What if the scale can only hold two or fewer pills at a  time?
• Given n pills, one of which is poisoned, and a scale that can only hold k or fewer pills, how many weighs does it take to find the poisoned pill?
• Given n pills, m of which are poisoned, and a scale that can only hold k or fewer pills, how many weighs does it take to find the poisoned pills?
• Given n pills, m of which are poisoned, and a scale that can only hold exactly k pills, how many weighs does it take to find the poisoned pills?
• Suppose we would settle for knowing which is the poisoned pill with probability p, what is the minimum number of weighs?
None of my classmates asked these questions, they were satisfied with just having the answer. To be clear: my classmates are not stupid. In fact, they're all quite bright. But they (we) have been programmed to find the answer, to report the answer, then to forget the question. Somewhere in all of this answer-fetishism we have forgotten how to ask questions. We have lost our curiosity.

A good question is more interesting than a satisfying answer. Why then do we let other people ask all the questions?