To address the query "check state validity for each step you take," I will provide a valid sequence of steps for the river crossing puzzle and verify the validity of each state based on the given constraints. The puzzle involves a family—father (F), mother (M), two sons (S1, S2), two daughters (D1, D2), a servant (V), and a dog (G)—who must cross a river using a boat that can carry up to two entities. Only F, M, and V can operate the boat. The constraints are:
- **Constraint 1**: If the mother is absent and the father is present on any side, the father will hit the daughters.
- **Constraint 2**: If the servant is absent on any side, the dog will bite family members (F, M, S1, S2, D1, D2) present on that side.
A state is valid if, on each side of the river:
- The father is not present with any daughters without the mother.
- The dog is not present with any family members without the servant.
### State Representation
Each state is represented as \( (F, M, V, G, S_A, D_A, B) \):
- \( F, M, V, G \): 0 if on side A (starting side), 1 if on side B (destination side).
- \( S_A \): number of sons on side A.
- \( D_A \): number of daughters on side A.
- \( B \): boat location (0 for side A, 1 for side B).
### Validity Conditions
For a state to be valid:
- **Side A**:
- Not \( (F = 0 \text{ and } M = 1 \text{ and } D_A > 0) \) (father with daughters without mother).
- Not \( (G = 0 \text{ and } V = 1 \text{ and } (F = 0 \text{ or } M = 0 \text{ or } S_A > 0 \text{ or } D_A > 0)) \) (dog with family members without servant).
- **Side B**:
- Not \( (F = 1 \text{ and } M = 0 \text{ and } (2 - D_A) > 0) \) (father with daughters without mother).
- Not \( (G = 1 \text{ and } V = 0 \text{ and } (F = 1 \text{ or } M = 1 \text{ or } (2 - S_A) > 0 \text{ or } (2 - D_A) > 0)) \) (dog with family members without servant).
The goal is to get everyone from side A to side B in a valid sequence of steps. Below is a sequence of 11 crossings, with each state's validity checked.
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### Valid Sequence with State Validity Checks
#### **Step 1: Initial State**
- **State**: \( (0, 0, 0, 0, 2, 2, 0) \)
- **Description**: Everyone (F, M, V, G, S1, S2, D1, D2) is on side A, boat on A.
- **Side A**: F=0, M=0, V=0, G=0, S_A=2, D_A=2
- Father with daughters without mother: \( F = 0 \text{ and } M = 1 \text{ and } D_A > 0 \)? No (M=0).
- Dog with family without servant: \( G = 0 \text{ and } V = 1 \text{ and family present} \)? No (V=0).
- **Side B**: Empty, no issues.
- **Valid**: Yes.
#### **Step 2: V and G to B**
- **State**: \( (0, 0, 1, 1, 2, 2, 1) \)
- **Description**: Servant and dog cross to B, boat on B.
- **Side A**: F=0, M=0, S_A=2, D_A=2
- \( F = 0 \text{ and } M = 1 \text{ and } D_A > 0 \)? No (M=0).
- \( G = 0 \text{ and } V = 1 \text{ and family present} \)? No (G=1).
- **Side B**: V=1, G=1
- \( F = 1 \text{ and } M = 0 \text{ and } (2 - D_A) > 0 \)? No (F=0, 2-2=0).
- \( G = 1 \text{ and } V = 0 \text{ and family present} \)? No (V=1).
- **Valid**: Yes.
#### **Step 3: V to A**
- **State**: \( (0, 0, 0, 1, 2, 2, 0) \)
- **Description**: Servant returns to A, boat on A.
- **Side A**: F=0, M=0, V=0, S_A=2, D_A=2
- \( F = 0 \text{ and } M = 1 \)? No.
- \( G = 0 \text{ and } V = 1 \)? No (G=1).
- **Side B**: G=1
- \( G = 1 \text{ and } V = 0 \text{ and family present} \)? Yes, but no family members on B (F=0, M=0, 2-S_A=0, 2-D_A=0), so no issue.
- **Valid**: Yes.
#### **Step 4: F and M to B**
- **State**: \( (1, 1, 0, 1, 2, 2, 1) \)
- **Description**: Father and mother cross to B, boat on B.
- **Side A**: V=0, S_A=2, D_A=2
- \( F = 0 \text{ and } M = 1 \)? No.
- \( G = 0 \text{ and } V = 1 \)? No (G=1).
- **Side B**: F=1, M=1, G=1
- \( F = 1 \text{ and } M = 0 \text{ and } (2 - D_A) > 0 \)? No (M=1).
- \( G = 1 \text{ and } V = 0 \text{ and family present} \)? Yes, and F=1, M=1, but V=0, so dog bites family members.
- **Correction**: This state is invalid due to Constraint 2 on side B. Let’s adjust.
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### Adjusted Sequence
Let’s try moving V and G together initially, then manage family members carefully.
#### **Step 1: Initial State**
- \( (0, 0, 0, 0, 2, 2, 0) \)
- **Valid**: Yes (as above).
#### **Step 2: V and G to B**
- \( (0, 0, 1, 1, 2, 2, 1) \)
- **Valid**: Yes (as above).
#### **Step 3: V to A**
- \( (0, 0, 0, 1, 2, 2, 0) \)
- **Valid**: Yes (as above).
#### **Step 4: M and V to B**
- **State**: \( (0, 1, 1, 1, 2, 2, 1) \)
- **Description**: Mother and servant cross to B, boat on B.
- **Side A**: F=0, S_A=2, D_A=2
- \( F = 0 \text{ and } M = 1 \text{ and } D_A > 0 \)? No (M=1, but on B).
- \( G = 0 \text{ and } V = 1 \)? No (G=1).
- **Side B**: M=1, V=1, G=1
- \( F = 1 \text{ and } M = 0 \)? No (F=0).
- \( G = 1 \text{ and } V = 0 \)? No (V=1).
- **Valid**: Yes.
#### **Step 5: M to A**
- **State**: \( (0, 0, 1, 1, 2, 2, 0) \)
- **Description**: Mother returns to A, boat on A.
- **Side A**: F=0, M=0, S_A=2, D_A=2
- \( F = 0 \text{ and } M = 1 \)? No.
- \( G = 0 \text{ and } V = 1 \)? No (G=1).
- **Side B**: V=1, G=1
- \( G = 1 \text{ and } V = 0 \)? No (V=1).
- **Valid**: Yes.
#### **Step 6: F and V to B**
- **State**: \( (1, 0, 1, 1, 2, 2, 1) \)
- **Description**: Father and servant cross to B, boat on B.
- **Side A**: M=0, S_A=2, D_A=2
- \( F = 0 \text{ and } M = 1 \)? No (M=0).
- \( G = 0 \text{ and } V = 1 \)? No (G=1).
- **Side B**: F=1, V=1, G=1
- \( F = 1 \text{ and } M = 0 \text{ and } (2 - D_A) > 0 \)? No (2-2=0, no daughters on B).
- \( G = 1 \text{ and } V = 0 \)? No (V=1).
- **Valid**: Yes.
#### **Step 7: V to A**
- **State**: \( (1, 0, 0, 1, 2, 2, 0) \)
- **Description**: Servant returns to A, boat on A.
- **Side A**: M=0, V=0, S_A=2, D_A=2
- \( F = 0 \text{ and } M = 1 \)? No.
- \( G = 0 \text{ and } V = 1 \)? No (G=1).
- **Side B**: F=1, G=1
- \( F = 1 \text{ and } M = 0 \text{ and } (2 - D_A) > 0 \)? No (no daughters on B).
- \( G = 1 \text{ and } V = 0 \text{ and family present} \)? Yes, and F=1, so invalid (dog bites father).
- **Correction**: Invalid state.
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