Solving "Water buckets" problem using Scala

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
	}
	}

view raw waterbuckets.scala hosted with ❤ by GitHub

	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)))
	}
	}

view raw waterbucketstest.scala hosted with ❤ by GitHub

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)

?

1 2 3 4 5 6 7

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!.