# Farmers and Pesky Birds

Alice and Bob are two farmers each wanting to plant a (countably infinite) row of seeds, side by side in a field. Both of them have pesky birds that hinder their efforts in funny ways.

As Alice walks along the row, sequentially dropping her seeds, her bird picks up every fifth seed that she drops. So after Alice “finishes” planting her row of seeds, are there any seeds left? Sure… infinitely many of them.

But Bob’s bird behaves differently. Bob walks side by side with Alice, planting seeds in his row. 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?

No! Each of Bob’s seeds gets picked up by Bob’s bird, eventually! But that is strange: Alice and Bob are working simultaneously and their birds pick up seeds at the same rate… but Alice’s row still has seeds left! How can this be?

Presentation Suggestions:
Draw a picture of what’s happening, as in Figure 1. Be prepared for a discussion about infinity. Students will object that since Alice and Bob can never actually “finish” that there is no paradox. But this is not the real issue, because we can just have Alice and Bob plant the first seed in 1 sec, the second in 1/2 sec, the third in a 1/4 sec, etc. After 2 seconds they will be done planting.

The Math Behind the Fact:
The nature of this paradox lies in the counter-intuitive nature of infinite sets. An infinite set can be (and is in fact characterized by the fact that it can be) put into one-to-one correspondence with a subset of itself. So, both pesky birds have picked up sets of the same cardinality; one is just a subset of the other.

OK, if you liked that one, here’s a question to ponder: Suppose Charlie is a third farmer, planting seeds at the same rate as Alice and Bob, with a pesky bird that after each fifth seed that Charlie drops, picks up a random seed of those that remain? How many seeds will Charlie have left? An answer is in the reference. [Hint: what is the probability that Charlie’s first seed will be picked up?]

Su, Francis E., et al. “Farmers and Pesky Birds.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.

References:
S. Ross, A First Course in Probability, Prentice Hall, 1998.

Fun Fact suggested by:
Arthur Benjamin

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