Question

If you have a twelve gallon bucket, a seven gallon bucket, and a six gallon bucket, how can you get exactly nine gallons into one bucket

Answer 1

Completely fill the 7 gallon bucket.

Pour the water from 7 gallon bucket into 12 gallon bucket.

Again completely fill the 7 gallon bucket.

Pour the water from 7 gallon bucket into 6 gallon bucket.

Now in 7 gallon bucket you have 7 - 6 = 1 gallon.

Pour that 1 gallon into 12 gallon bucket.

Now in 12 gallon bucket you have 7 + 1 = 8 gallons.

Pour the water from 6 gallon bucket into 7 gallon bucket.

Now in 7 gallon bucket you have 6 gallons.

Completely fill the 7 gallon bucket.

Again pour the water from 7 gallon bucket into 6 gallon bucket.

Again in 7 gallon bucket you have 7 - 6 = 1 gallon.

Pour that 1 gallon into 12 gallon bucket.

Now in 12 gallon bucket you have 7 + 1 + 1 = 9 gallons.

Answer 2

There are a lot of ways doing these. Mine might be a long way, there might be a shorter way. Anyway, let the contents of each gallon be represented in this format:
(12-gallon bucket) - (7-gallon bucket) - (6-gallon bucket)

Fill the 12-gallon bucket. Leave the others empty. So:
12 - 0 - 0

Put the 12 gallon to the 7-gallon bucket. Remaining gallons in 12-gallon bucket is therefore 5:
5 - 7 - 0

Put the 7 gallon in 6-gallon bucket. Remaining gallons in 7-gallon bucket is therefore 1:
5 - 1 - 6

Put the 6 gallons from 6-gallon bucket to 12-gallon bucket:
11 - 1 - 0

Put the 1 gallon to 6-gallon bucket:
11 - 0 - 1

Put the 11 gallons to 7-gallon bucket. Remaining gallons in 12-bucket is therefore 4:
4 - 7 - 1

Discard the contents of 6-gallon bucket:
4 - 7 - 0

Put the 4 gallons to 6-gallon bucket:
0 - 7 - 4

Fill the 12-gallon bucket:
12 - 0 - 4

Put the 12 gallons to 7-gallon bucket. Remaining gallons in 12-gallon bucket is therefore 5:
5 - 7 - 4

Finally, put the 4 gallons from the 6-gallon bucket to 12-gallon bucket:
9 - 7 - 0

hope this helps~ `u`