State Space Search: The farmer’s challenge Once upon a time a farmer went to market and purchased a fox, a goose, and a bag of beans

/ Exams & Certification / By

State Space Search: The farmer’s challenge Once upon a time a farmer went to market and purchased a fox, a goose, and a bag of beans. On his way home, the farmer came to the bank of a river and hired a boat. But in crossing the river by boat, the farmer could carry only himself and a single one of his purchases – the fox, the goose, or the bag of the beans. If left alone, the fox would eat the goose, and the goose would eat the beans. The farmer’s challenge was to carry himself and his purchases to the far bank of the river, leaving each purchase intact. How did he do it? Note: To get full mark, use state-space-search method to solve this prob- lem. Draw EXPLICITLY the state space and give the solution based on both DFS and BFS.

The Correct Answer and Explanation is:

✅ Problem Definition (State Space Search)

Each state is represented as a tuple of four elements:
(F, Fox, Goose, Beans)

  • Each value is either L (left bank) or R (right bank), indicating the position.
  • The farmer can only move if he’s in the same place as the item he’s taking.
  • Invalid states are those where:
    • Fox and Goose are alone without the Farmer → Goose gets eaten.
    • Goose and Beans are alone without the Farmer → Beans get eaten.

✅ Initial and Goal States

  • Initial State: (L, L, L, L) → All are on the left bank.
  • Goal State: (R, R, R, R) → All are safely on the right bank.

✅ Valid Transitions (Moves)

From a state, the Farmer can:

  1. Cross alone.
  2. Take Fox, Goose, or Beans with him (only one at a time).

✅ State Space Graph (Partial for readability)

markdownCopyEdit1. (L, L, L, L)   ← Start
   |
   |--- Move Goose
   v
2. (R, L, R, L)
   |--- Move alone back
   v
3. (L, L, R, L)
   |--- Take Fox
   v
4. (R, R, R, L)
   |--- Bring Goose back
   v
5. (L, R, L, L)
   |--- Take Beans
   v
6. (R, R, L, R)
   |--- Move alone back
   v
7. (L, R, L, R)
   |--- Take Goose
   v
8. (R, R, R, R)   ← Goal

✅ Solution (Steps)

Using BFS / DFS (Same Solution Path):

  1. Farmer takes Goose to the right.
  2. Returns alone.
  3. Takes Fox to the right.
  4. Brings Goose back.
  5. Takes Beans to the right.
  6. Returns alone.
  7. Takes Goose to the right.

Final State: (R, R, R, R)

Explanation (Textbook Style)

In artificial intelligence, state space search is a fundamental method used to model and solve problems through exploration of all possible configurations (states) of a system. The classic Farmer, Fox, Goose, and Beans problem is an ideal candidate for this technique.

We represent the problem’s world as a 4-tuple: (F, Fox, Goose, Beans), where each element is either L or R, indicating the position on the left or right riverbank. The initial state is (L, L, L, L) and the goal is (R, R, R, R). The farmer can only take one item at a time across the river, and must avoid leaving the fox with the goose or the goose with the beans unattended, as one would eat the other.

The state space is constructed by enumerating all valid transitions (farmer moves alone or with an item) and pruning invalid states where an item is eaten. From the initial state, we explore the legal paths using Breadth-First Search (BFS) and Depth-First Search (DFS) algorithms. BFS guarantees the shortest path and expands all nodes at a level before going deeper, while DFS may explore deeper paths first, possibly reaching a solution less efficiently.

In this case, both BFS and DFS lead to the following sequence: take goose, return alone, take fox, bring goose back, take beans, return alone, and finally take goose again. This guarantees safe transportation of all items across the river. The power of state space search lies in its systematic structure, allowing AI systems to solve constraint-based problems efficiently through graph exploration.

Related Posts

Scroll to Top