Math CentralQuandaries & Queries


Question from Silje, a student:

Hi! How can I solve the following question without the use of a calculator?

"A cylinder is to be filled with peas. It is done like this: At 12:00 o'clock you put 1 pea in, at 12:01 you put 2 peas in, at 12:02 you put 4 peas in, at 12:03 you put 8 peas in, and so on.
This continues until 14:00 o'clock (two hours later), when the last peas are put in and the cylinder is full. At what time is the cylinder half full?"


First, we have to note that the question is very unrealistic; the cylinder would contain $2^{120}$ peas (about $10^{36}$ peas, a thousand times the mass of the solar system!) Makes the chessboard story (one grain of wheat on the first square, etc...) look pretty tame.

Now, how to solve it. Consider some smaller numbers: if your cylinder is full after the third minute (1+2+4 peas) about when is it half-full?

If it's full after the fourth minute, about when is it half-full?

If it's full after the fifth minute, about when is it half-full?

You should be able to answer the question yourself now!

Good Hunting!

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