4 7/12

In order for the average of three numbers to be 2, they must all add up to 6. That gives you this equation:
2/3 + 3/4 + n = 6

Solve for n = 55/12, or 4 7/12.

1. Fill the 3-gallon bucket.
2. Pour the 3 gallons of water into the 5-gallon bucket
3. Fill the 3-gallon bucket again.
4. Fill up the 5-gallon bucket with the 3-gallon bucket, leaving you with 1 gallon left in the 3-gallon bucket.
5. Empty out the 5-gallon bucket.
6. Pour the remaining 1 gallon of water from the 3-gallon bucket into the 5-gallon bucket.
7. Fill the 3-gallon bucket.
8. Pour the 3 gallons of water from the 3-gallon bucket into the 5-gallon bucket leaving you with 4 gallons of water in the 5-gallon bucket.

Alternate solution:

1. Fill up the 5 gallon bucket
2. Pour it into 3 gallon bucket, leaving 2 gallons
3. Empty out the 3 gallon bucket
4. Pour the 2 gallons in the 5 gallon bucket into the 3 gallon bucket
5. Fill up the 5 gallon bucket and pour it into the 3 gallon bucket until it’s full, leaving 4 gallons in the 5 gallon bucket.