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?
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
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?]
How to Cite this Page:
Su, Francis E., et al. "Farmers and Pesky Birds."
Math Fun Facts.