Suppose that you have two water jugs, one which holds three gallons and one which holds five gallons. In addition, you have a water faucet at your disposal. Your job
is to find a method that produces exactly one gallon of water, another method that
produces exactly two gallons of water, another that produces exactly three gallons of
water, etc. up to exactly eight gallons of water. You may fill jugs from the faucet or from
one another. You may not, however, use estimates. For example, you may not
produce one gallon by estimating one-third of the three gallon jug.

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Explanation:

1: fill the 3-gallon jug, empty it into the 5-gallon jug. Fill the 3-gallon jug again, and finish filling the 5-gallon jug from the 3-gallon jug. The 3-gallon jug will hold 1 gallon. (3×2-5=1)

2: fill the 5-gallon jug, fill the 3-gallon jug from it. The 5-gallon jug will hold 2 gallons. (5-3=2)

3: fill the 3-gallon jug. It will hold 3 gallons.

4: fill the 5-gallon jug, fill the 3-gallon jug from it. Dump the 3-gallon jug and empty the remaining 2 gallons from the 5-gallon jug into it. Fill the 5-gallon jug and finish filling the 3-gallon jug from it. The 5-gallon jug will hold 4 gallons. (5×2-3×2=4)

5: fill the 5-gallon jug. It will hold 5 gallons.

6: fill the 3-gallon jug, empty it into the 5-gallon jug. Fill the 3-gallon jug again. The jugs hold a total of 6 gallons. (3×2=6)

7: Do the procedure for 4 gallons. The jugs together hold 7 gallons. (5×2-3=7)

8: fill both jugs. Together, they hold 8 gallons. (5+3=8)

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step-by-step explanation: