9514 1404 393
Answer:
Fill the 19-gal bucket. Use that to fill the 8-gal bucket twice, so that 3 gal remains in the 19-gal bucket. Move that to the 8-gal bucket. Fill the 19-gal bucket again and use that to top off the 8-gal bucket. The 19-gal bucket now contains 14 gallons.
Explanation:
The problem is equivalent to solving ...
19a +8b = 14
One solution is (a, b) = (2, -3). That means you can obtain the required amount by filling the 19-gallon bucket 2 times, and using that source to fill (and empty) the 8-gallon bucket 3 times.
The third attempt to fill the 8-gallon bucket from the 19-gallon bucket will result in only 3 gallons in the 8-gallon bucket. The last 5 gallons needed to finish filling the 8-gallon bucket the third time come from the second filling of the 19-gallon bucket. After that transfer of 5 gallons, the 19-gallon bucket contains the required 14 gallons.
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We don't know what your steps are supposed to look like, but we can write them as ...
(19-gallon bucket, 8 gallon bucket)
(0, 0) . . . start
(19, 0) . . . first fill
(11, 8) . . . first transfer
(11, 0) . . . first dump
(3, 8) . . . second transfer
(3, 0) . . . second dump
(0, 3) . . . third transfer
(19, 3) . . . second fill
(14, 8) . . . 4th transfer -- one bucket has 14 gallons in it
Optional:
(14, 0) . . . third dump -- one bucket has 14 gallons in it