1. **Problem Statement:** Two married couples want to cross a river using a boat that can carry one or two people. Each husband is jealous and will not allow his wife to be with the other husband without him present, either on the shore or in the boat. We need to find a sequence of crossings that gets all four people safely to the opposite shore. 2. **Key Constraints:** - The boat can carry one or two people. - No wife can be left with the other husband without her own husband present. 3. **Modeling the Problem:** We represent each state as a tuple $(H_1, W_1, H_2, W_2, B)$ where each of $H_1, W_1, H_2, W_2$ is either 0 (left shore) or 1 (right shore), and $B$ indicates the boat's position (0 for left shore, 1 for right shore). 4. **Initial State:** $(0,0,0,0,0)$ all on the left shore. 5. **Goal State:** $(1,1,1,1,1)$ all on the right shore. 6. **Valid Moves:** Move one or two people from the shore where the boat is to the other shore, ensuring no jealousy violation. 7. **Jealousy Rule:** On either shore, if a wife is present without her husband, the other husband must not be present. 8. **Solution Path:** - Step 1: $W_1$ and $W_2$ cross to right shore: $(0,\cancel{0},0,\cancel{0},0) \to (0,1,0,1,1)$ - Step 2: $W_1$ returns: $(0,1,0,1,1) \to (0,0,0,1,0)$ - Step 3: $H_1$ and $H_2$ cross: $(0,0,0,1,0) \to (1,0,1,1,1)$ - Step 4: $W_2$ returns: $(1,0,1,1,1) \to (1,0,1,0,0)$ - Step 5: $W_1$ and $W_2$ cross again: $(1,0,1,0,0) \to (1,1,1,1,1)$ All four have crossed without violating jealousy constraints. **Final Answer:** The sequence of crossings described above successfully gets all four people across the river safely.