River Crossing Puzzle Solution

This is a classic river crossing puzzle. The goal is to move a boatman, a dog, a cat, and a container of milk to the other side of a river. The boat can only carry the boatman and one other item per trip. The key constraints are that the cat cannot be left alone with the dog, nor alone with the milk, on either river bank.

Constraints Analysis

• Boatman must operate the boat.
• Boat capacity: Boatman + 1 item.
• Constraint 1: Cat (C) and Dog (D) cannot be left alone together.
• Constraint 2: Cat (C) and Milk (M) cannot be left alone together.

Step-by-Step Solution

We denote the starting bank as Bank 1 and the destination bank as Bank 2. The items are Boatman (B), Dog (D), Cat (C), and Milk (M).

1. Trip 1: B takes C across (Bank 1 → Bank 2).
   • Status: Bank 1 = {D, M} (Safe) | Bank 2 = {B, C}
2. Trip 2: B returns alone (Bank 2 → Bank 1).
   • Status: Bank 1 = {B, D, M} | Bank 2 = {C} (Safe)
3. Trip 3: B takes D across (Bank 1 → Bank 2).
   • Status: Bank 1 = {M} (Safe) | Bank 2 = {B, C, D} (Safe as B is present)
4. Trip 4: B returns with C (Bank 2 → Bank 1). This step is crucial to prevent C and D being left alone.
   • Status: Bank 1 = {B, C, M} (Safe as B is present) | Bank 2 = {D} (Safe)
5. Trip 5: B takes M across (Bank 1 → Bank 2).
   • Status: Bank 1 = {C} (Safe) | Bank 2 = {B, D, M} (Safe)
6. Trip 6: B returns alone (Bank 2 → Bank 1).
   • Status: Bank 1 = {B, C} | Bank 2 = {D, M} (Safe)
7. Trip 7: B takes C across (Bank 1 → Bank 2).
   • Status: Bank 1 = {} | Bank 2 = {B, D, M, C} (All items successfully transported)

Conclusion

Following these steps ensures all constraints are met at every stage. The minimum number of crossings required is 7.