The Riddle - Farmer Wolf Goat Cabbage
A farmer is on the west bank of a river with a wolf, a goat and a cabbage in his care. Without his presence the wolf would eat
the goat or the goat would eat the cabbage. The wolf is not interested in the cabbage. The farmer wishes to bring his three
charges across the river. However the boat available to him can only carry one of the wolf, goat and cabbage besides
himself. The puzzle is: is there a sequence of river crossings so that the farmer can transfer himself and the other three all
intact to the east bank?
The Solution
It is a task that given an initial state, a final state, and a set of rules, one will start from the initial state and try to reach to the final state by passing through some intermediate states. Each move will transform from one state to the other, and there might be multiple valid moves from a given state. Such a collection of interconnected states can be represented by
State Space Graph.
Let's use 'f', 'c', 'g', 'w' to denote the farmer, the cabbage, the goat, and the wolf, and use '|' to separate the river where left of the '|' denotes west bank and right of the '|' denotes east bank. Initially, they are all at the west bank of the river, which is represented as 'fcgw |' as shown below. We can solve the riddle by figuring out what the possible and valid moves are, using either Breadth-First Search or Depth-First Search, on a state space graph shown below
Depth First Search
I use DFS to find the first possible solution to the riddle, where it looks like this
or with a possibility of backtracking like this
Breadth First Search
I can build a complete state space graph using BFS, excluding the red states as shown below
Sample Source Code
Implemented in a mix of Groovy and Java, where Java is mainly used for the POJOs, the sample source code, including the unit test, demonstrates how to build a complete state space graph (which is also a directed graph) and how to find all possible solutions based on the state space graph.
DirectedGraph.java
import java.util.*;
public class DirectedGraph<T> implements Iterable<T> {
// key is a Node, value is a set of Nodes connected by outgoing edges from the key
private final Map<T, Set<T>> graph = new HashMap<T, Set<T>>();
public boolean addNode(T node) {
if (graph.containsKey(node)) {
return false;
}
graph.put(node, new HashSet<T>());
return true;
}
public void addNodes(Collection<T> nodes) {
for (T node : nodes) {
addNode(node);
}
}
public void addEdge(T src, T dest) {
validateSourceAndDestinationNodes(src, dest);
// Add the edge by adding the dest node into the outgoing edges
graph.get(src).add(dest);
}
public void removeEdge(T src, T dest) {
validateSourceAndDestinationNodes(src, dest);
graph.get(src).remove(dest);
}
public boolean edgeExists(T src, T dest) {
validateSourceAndDestinationNodes(src, dest);
return graph.get(src).contains(dest);
}
public Set<T> edgesFrom(T node) {
// Check that the node exists.
Set<T> edges = graph.get(node);
if (edges == null)
throw new NoSuchElementException("Source node does not exist.");
return Collections.unmodifiableSet(edges);
}
public Iterator<T> iterator() {
return graph.keySet().iterator();
}
public int size() {
return graph.size();
}
public boolean isEmpty() {
return graph.isEmpty();
}
private void validateSourceAndDestinationNodes(T src, T dest) {
// Confirm both endpoints exist
if (!graph.containsKey(src) || !graph.containsKey(dest))
throw new NoSuchElementException("Both nodes must be in the graph.");
}
}
BoatLocation.java
public enum BoatLocation {
WestBank, EastBank
}
RiverRole.java, implemented Comparable interface in order to be used in a SortedSet
public class RiverRole implements Comparable<RiverRole> {
public final String name;
public final boolean canSailTheBoat;
public RiverRole(String name, boolean canSailTheBoat) {
this.name = name;
this.canSailTheBoat = canSailTheBoat;
}
@Override
public int compareTo(RiverRole o) {
return this.name.compareTo(o.name);
}
@Override
public String toString() {
return name;
}
}
RiverState.java
import java.util.Collection;
import java.util.SortedSet;
import java.util.TreeSet;
public class RiverState {
private SortedSet<RiverRole> westBank;
private SortedSet<RiverRole> eastBank;
private BoatLocation boatLocation;
public RiverState(Collection<RiverRole> westBank, Collection<RiverRole> eastBank, BoatLocation boatLocation) {
this.westBank = new TreeSet<RiverRole>(westBank);
this.eastBank = new TreeSet<RiverRole>(eastBank);
this.boatLocation = boatLocation;
}
public SortedSet<RiverRole> getWestBank() {
return new TreeSet<RiverRole>(westBank);
}
public SortedSet<RiverRole> getEastBank() {
return new TreeSet<RiverRole>(eastBank);
}
public BoatLocation getBoatLocation() {
return boatLocation;
}
@Override
public String toString() {
return "State{westBank=" + westBank + ", eastBank=" + eastBank + ", boatLocation=" + boatLocation + "}";
}
public boolean equals(Object o) {
if (this == o) return true;
if (!(o instanceof RiverState)) return false;
RiverState riverState = (RiverState) o;
return eastBank.equals(riverState.eastBank) &&
westBank.equals(riverState.westBank) &&
boatLocation.equals(riverState.boatLocation);
}
public int hashCode() {
int result;
result = westBank.hashCode();
result = 31 * result + eastBank.hashCode();
result = 31 * result + boatLocation.hashCode();
return result;
}
}
FarmerWolfGoatRiddle.groovy
import graph.DirectedGraph
import org.codehaus.groovy.runtime.MethodClosure
import static fun.rivercrossing.BoatLocation.EastBank
import static fun.rivercrossing.BoatLocation.WestBank
public class FarmerWolfGoatRiddle {
/* Farmer Wolf Goat Puzzle
A farmer is on the east bank of a river with a wolf, goat and cabbage in his care. Without his presence the wolf would eat
the goat or the goat would eat the cabbage. The wolf is not interested in the cabbage. The farmer wishes to bring his three
charges across the river. However the boat available to him can only carry one of the wolf, goat and cabbage besides
himself. The puzzle is: is there a sequence of river crossings so that the farmer can transfer himself and the other three all
intact to the west bank?
*/
RiverRole farmer = new RiverRole("farmer", true);
RiverRole wolf = new RiverRole("wolf", false);
RiverRole goat = new RiverRole("goat", false);
RiverRole cabbage = new RiverRole("cabbage", false);
RiverState initialState = new RiverState([farmer, wolf, goat, cabbage], [], BoatLocation.WestBank)
RiverState finalState = new RiverState([], [farmer, wolf, goat, cabbage], EastBank)
Closure<Boolean> areRolesHarmonisedOnRiverBank = { Collection<RiverRole> roles ->
// describe clearly what the rules are
if (!roles.contains(farmer)) {
if (roles.contains(wolf) && roles.contains(goat)) return false
if (roles.contains(goat) && roles.contains(cabbage)) return false
}
return true
}
public Collection<Deque<RiverState>> findAllPossibleSolutions() {
DirectedGraph<RiverState> graph = buildCompleteGraph()
Collection<Deque<RiverState>> solutions = []
Deque<RiverState> currentSolution = new ArrayDeque<RiverState>()
explore(graph, initialState, currentSolution, solutions)
return solutions
}
// using Depth First Search (Stack Implementation) to identify a solution
private void explore(DirectedGraph<RiverState> graph, RiverState currentNode, Deque<RiverState> currentSolution, Collection<Deque<RiverState>> solutions) {
currentSolution.push(currentNode)
graph.edgesFrom(currentNode).each { RiverState node ->
if (node == finalState) {
// add the current solution, including the final node, to the solutions
currentSolution.push(node)
solutions.add(new ArrayDeque<RiverState>(currentSolution))
currentSolution.pop()
} else {
// recursively explore
explore(graph, node, currentSolution, solutions)
}
}
currentSolution.pop()
}
// using Breath First Search to build the complete graph
private DirectedGraph<RiverState> buildCompleteGraph() {
DirectedGraph<RiverState> graph = new DirectedGraph<RiverState>()
graph.addNode(initialState)
Set<RiverState> visitedNodes = new HashSet<RiverState>()
// use a queue to keep the states to be visited in sequence
Deque<RiverState> currentStates = new ArrayDeque<RiverState>()
currentStates.add(initialState)
while (!currentStates.isEmpty()) {
if (currentStates.peek() != finalState) {
def currentState = currentStates.removeFirst()
if (!visitedNodes.contains(currentState)) {
visitedNodes.add(currentState)
def possibleStates = findNextPossibleStates(currentState)
def validStates = possibleStates.findAll { RiverState possibleState ->
// matched state should not be one of the visited states, and it should pass the validations
return !visitedNodes.contains(possibleState) && passRuleValidations(possibleState)
}
// add successors and edges
graph.addNodes(validStates)
validStates.each { graph.addEdge(currentState, it) }
// add to current states so that we can visit them next
currentStates.addAll(validStates)
}
} else {
currentStates.removeFirst() // remove the final state from the currentStates queue
}
}
return graph
}
private boolean passRuleValidations(RiverState state) {
return areRolesHarmonisedOnRiverBank(state.eastBank) && areRolesHarmonisedOnRiverBank(state.westBank)
}
private List<RiverState> findNextPossibleStates(RiverState currentState) {
// from the last visited state, find out who can sail the boat, and where is the sailor
List<RiverState> possibleStates = []
transitFrom(currentState, possibleStates)
return possibleStates
}
private void transitFrom(RiverState currentState, List<RiverState> possibleStates) {
// check where the boat is
BoatLocation boatOrigin = currentState.boatLocation
BoatLocation boatDestination
// we use method closure to get a copy of west bank or east bank roles
MethodClosure origin, destination
// multiple assignments
(origin, destination, boatDestination) = boatOrigin == WestBank ?
[currentState.&getWestBank, currentState.&getEastBank, EastBank] :
[currentState.&getEastBank, currentState.&getWestBank, WestBank]
def possibleSailors = origin().findAll { RiverRole role -> role.canSailTheBoat }
possibleSailors?.each { RiverRole sailor ->
// pick 0 passengers onto the boat
Collection<RiverRole> onBoat = [sailor]
possibleStates.add(createNextPossibleState(origin, destination, onBoat, boatDestination))
Collection<RiverRole> possiblePassengers = origin() as Collection<RiverRole>
possiblePassengers.remove(sailor)
// pick 1 passenger onto the boat
possiblePassengers.each { RiverRole passenger ->
onBoat = [passenger, sailor]
possibleStates.add(createNextPossibleState(origin, destination, onBoat, boatDestination))
}
}
}
private RiverState createNextPossibleState(MethodClosure origin, MethodClosure destination, Collection<RiverRole> onBoat, BoatLocation boatDestination) {
Collection<RiverRole> newDestination = destination() as Collection<RiverRole>
Collection<RiverRole> newOrigin = origin() as Collection<RiverRole>
// boat transits from origin to destination
newOrigin.removeAll(onBoat)
newDestination.addAll(onBoat)
return boatDestination == WestBank ? new RiverState(newDestination, newOrigin, boatDestination) : new RiverState(newOrigin, newDestination, boatDestination)
}
}
FarmerWolfGoatRiddleTest.groovy
import org.junit.Test
import static fun.rivercrossing.BoatLocation.EastBank
import static fun.rivercrossing.BoatLocation.WestBank
class FarmerWolfGoatRiddleTest {
FarmerWolfGoatRiddle riddle = new FarmerWolfGoatRiddle()
@Test
public void findAllPossibleSolutions() {
def solutions = riddle.findAllPossibleSolutions()
// the ordering of solutions will be random
assert solutions.size() == 2
assert solutions[0].size() == 8
assert solutions[1].size() == 8
def assertAllSolutions = { Deque<RiverState> solution1, Deque<RiverState> solution2 ->
assertFirstSolution(solution1)
assertSecondSolution(solution2)
}
// assert with the correct expectations regardless of which is the first solution
if (solutions[0].contains(new RiverState([riddle.wolf], [riddle.farmer, riddle.goat, riddle.cabbage], EastBank))) {
assertAllSolutions(solutions[0], solutions[1])
} else {
assertAllSolutions(solutions[1], solutions[0])
}
}
private void assertFirstSolution(Deque<RiverState> solution) {
assert solution.poll() == new RiverState([riddle.farmer, riddle.wolf, riddle.goat, riddle.cabbage], [], WestBank)
assert solution.poll() == new RiverState([riddle.wolf, riddle.cabbage], [riddle.farmer, riddle.goat], EastBank)
assert solution.poll() == new RiverState([riddle.farmer, riddle.wolf, riddle.cabbage], [riddle.goat], WestBank)
assert solution.poll() == new RiverState([riddle.wolf], [riddle.farmer, riddle.goat, riddle.cabbage], EastBank)
assert solution.poll() == new RiverState([riddle.farmer, riddle.wolf, riddle.goat], [riddle.cabbage], WestBank)
assert solution.poll() == new RiverState([riddle.goat], [riddle.farmer, riddle.wolf, riddle.cabbage], EastBank)
assert solution.poll() == new RiverState([riddle.farmer, riddle.goat], [riddle.wolf, riddle.cabbage], WestBank)
assert solution.poll() == new RiverState([], [riddle.farmer, riddle.wolf, riddle.goat, riddle.cabbage], EastBank)
assert solution.poll() == null
}
private void assertSecondSolution(Deque<RiverState> solution) {
assert solution.poll() == new RiverState([riddle.farmer, riddle.wolf, riddle.goat, riddle.cabbage], [], WestBank)
assert solution.poll() == new RiverState([riddle.wolf, riddle.cabbage], [riddle.farmer, riddle.goat], EastBank)
assert solution.poll() == new RiverState([riddle.farmer, riddle.wolf, riddle.cabbage], [riddle.goat], WestBank)
assert solution.poll() == new RiverState([riddle.cabbage], [riddle.farmer, riddle.goat, riddle.wolf], EastBank)
assert solution.poll() == new RiverState([riddle.farmer, riddle.cabbage, riddle.goat], [riddle.wolf], WestBank)
assert solution.poll() == new RiverState([riddle.goat], [riddle.farmer, riddle.wolf, riddle.cabbage], EastBank)
assert solution.poll() == new RiverState([riddle.farmer, riddle.goat], [riddle.wolf, riddle.cabbage], WestBank)
assert solution.poll() == new RiverState([], [riddle.farmer, riddle.wolf, riddle.goat, riddle.cabbage], EastBank)
assert solution.poll() == null
}
}
