I recently came across a puzzle called the “Water Buckets” problem in this book, which totally stumped me.
You have a 12-gallon bucket, an 8-gallon bucket and a 5-gallon bucket. The 12-gallon bucket is full of water and the other two are empty. Without using any additional water how can you divide the twelve gallons of water equally so that two of the three buckets have exactly 6 gallons of water in them?
I and my nephew spent a good deal of time trying to solve it and ultimately gave up.
I remembered then that I have seen a programmatic solution to a similar puzzle being worked out in the “Functional Programming Principles in Scala” Coursera course by Martin Odersky.
This is the gist to the solution completely copied from the course:
package bucket
case class Pouring(capacity: Vector[Int], initialState: Vector[Int]){
type State = Vector[Int]
trait Move {
def change(state: State): State
}
case class Pour(from: Int, to: Int) extends Move {
def change(state: State) = {
val amount = state(from) min (capacity(to) - state(to))
state updated (from, state(from) - amount) updated (to, state(to) + amount)
}
}
class Path(history: List[Move], val endState: State) {
def extend(move: Move) = new Path(move :: history, move change endState)
override def toString = (history.reverse mkString " ") + "-->" + endState
}
val glasses = 0 until capacity.length
val moves = for {
from <- glasses
to <- glasses
if (from != to)
} yield Pour(from, to)
val initialPath = new Path(Nil, initialState )
def from(paths: Set[Path], explored: Set[State]): Stream[Set[Path]] = {
if (paths.isEmpty) Stream.empty
else {
val more = for {
path <- paths
next <- moves map path.extend
if !(explored contains next.endState)
} yield next
paths #:: from(more, explored ++ (more map (_.endState)))
}
}
val pathSets = from(Set(initialPath), Set(initialState))
def solution(target: State): Stream[Path] = {
for {
pathSet <- pathSets
path <- pathSet
if (path.endState == target)
} yield path
}
}package bucket
import org.scalatest.FunSuite
import org.junit.runner.RunWith
import org.scalatest.junit.JUnitRunner
@RunWith(classOf[JUnitRunner])
class PouringTest extends FunSuite {
test("Solution to the 12-gallon, 8-gallon, 5-gallon problem") {
val p = Pouring(Vector(12, 8, 5), Vector(12, 0, 0))
println(p.solution(Vector(6, 6, 0)))
}
}and running this program spits out the following 7 step solution! (index 0 is the 12-gallon bucket, 1 is the 8-gallon bucket and 2 is the 5-gallon bucket)
Pour(0,1) Pour(1,2) Pour(2,0) Pour(1,2) Pour(0,1) Pour(1,2) Pour(2,0)
If you are interested in learning more about the code behind this solution, the best way is to follow the week 7 of the Coursera course that I have linked above, Martin Odersky does a fantastic job of seemingly coming up with a solution on the fly!.
| Reference: | Solving "Water buckets" problem using Scala from our JCG partner Biju Kunjummen at the all and sundry blog. |
