I know that no one will be able to answer this question, but if you think your smart, do you think you can?

Question

Tom and Eileen are on a nature walk.They come to a stream and find 2 buckets.One is marked 4 gallons, and the other is marked 9 gallons.The buckets have no other markings.Tom and Eileen start talking about how many different gallon amounts they could measure with 2 buckets. Tom says,"You could only measure 4 gallons, 9 gallons, and 13 gallons"Eileen agrees that the total of the two buckets is the most you could measure but argues you could find other gallon amounts.For example, she says, that if you fill the 9 gallon bucket and pour 4 gallons out into the 4 gallon bucket you will have 5 gallons left in the 9 gallon bucket.

1)How many different gallon amounts can you find using only the 2 buckets?Explain how you arrived at your answer.

2)What if the buckets held 2 diff gallon amounts?you could always achieve the same variety of results regardless of the size of the bucket. Give specific examples to support your answer.Can you develop a rule? Give examples to support your answer.

Answer 1

1) You can come up with 1 gal (Fill the 9 from the stream, fill the 4 from the 9, empty the 4, fill the 4 from the 9, empty the 4, and you're left with 1 in the 9), 2 gal (Fill the 4, pour into the 9, fill the 4 again, pour into the 9, fill the 4 again, pour 1 into the 9 (left with 3 in the 4), empty the 9, pour the 3 into the 9, fill the four, empty into the 9, fill the 4, empty into the 9, fill the 4, empty 2 into the 9, left with 2 in the 4),...

You can get 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13 gallons this way.

2) Not true with all combinations. Consider a 4-gallon jug and an 8-gallon jug. You couldn't "factor" any smaller than 4 gallons.