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you have a stairwell outside with a certain number of wide steps on it. On each step is a 5 Liter bucket full to the 1 Liter mark with water. When you tip over a bucket, all of its contents are emptied, with 1/2 going down the stairs and away and the other 1/2 being distributed equally among all the lower buckets. In the case of the lowest bucket on the lowest stair, when you tip its ENTIRE contents will go away. Right after tipping a bucket, you place it upright again, ready to receive. If a bucket surpasses 5 Liters ever, the excess will run off and 1/2 will dissipate and the other 1/2 of the excess will be spread over lower buckets.

There are ten stairs; ten buckets; ten total liters of water.


(a) What is the method of tipping buckets to empty all buckets in as few of steps as possible?


(b) What is the method of tipping buckets so as to prolong the eventual "total-emptiness" of the system as long as possible? How many steps can it go?


example of tipping the topmost bucket where '1' is the bottom stair, '2' the second bottom-est, etc:

10: 1

9: 1

8: 1

7: 1

6: 1

5: 1

4: 1

3: 1

2: 1

1: 1

tip #10, 0.5 liters go away and 0.5 get split over the remaining 9, adding 1/18 to each 1

10: 0

9: 19/18

8: 19/18

7: etc

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2 answers to this question

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You tip the top, then the one after it, then the one after it..etc

10 9 8 7 6 5 4 3 2 1

10 tips, and you can't go less because you'll have to empty each bucket at least once!

You tip the first, then the second, then the first, then the 3rd, then the first, then the second...etc

Like this:






The equation is 1+2+4+8...which is 2n-1, in the case of 10 that'll be 1023 steps...

Edited by Anza Power

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