I encountered a puzzle that has stirred up a lot of controversy among my friends. Here's the problem:
"Bob is a farmer wanting to plant a (countably infinite) number of seeds in a row, but he has a bird that hinders his efforts. After every fifth seed that Bob drops, his bird picks up the first seed that remains in his row. After Bob has "finished" planting his row of seeds, are there any seeds left?"
Bob's seeds look like this (O indicates a seed remaining, and x is a seed that's been eaten):
x x x O O O O O O O O O O O O ...
The "official" answer given (at http://www.math.hmc.edu/funfacts/ffiles/30001.4-6.shtml) says that no seeds remain. I think that's wrong. I say that after Bob plants each 5 seeds, exactly 4/5 of the seeds planted remain, and this number always increases.
I think that they're answering the wrong question, (but answering the wrong question correctly.) I think they're answering the question "does any given seed eventually get eaten?" Yes. But the question was, "do seeds remain?" which is a very different question, which has to deal with the fact that seeds are planted and eaten at different rates.
What do you think? Send it to the e-mail address below.