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Logic Puzzles and Their Abstract Representation
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Each of six friends is either a Knight or a Knave. Knights always tell the truth, while Knaves always lie. All friends claim, 'Exactly three of us are Knights.'
Let Ki be the proposition 'The ith friend is a Knight'. The puzzle is formalized as (K1 & K2 & K3 & ~K4 & ~K5 & ~K6) v ... v (~K1 & ~K2 & ~K3 & K4 & K5 & K6) to cover all combinations of three truths.
Five houses in a row are each painted a different color. No two adjacent houses can be of the same color. The second house cannot be red.
Let R represent 'Red', and Hi indicate the ith house. Formalize as H2 != R & (Hi != H(i+1) for i from 1 to 4).
If it rains, the grass gets wet. It's not raining. There are three possible causes for wet grass: sprinkler, rain, or a water truck spill. Only one can be the cause.
Let R 'It rains', W 'The grass is wet', S 'Sprinkler is on', and T 'Water truck spill'. Formulate as (R v S v T) -> W & ~R & (W -> (S v T) & ~(S & T)).
There are four houses, each house can only have one colored roof. The options are red, blue, green, and yellow. The blue roof cannot be on the first house.
Let B1, B2, B3, and B4 denote the proposition 'The nth house has a blue roof'. The constraint is represented as ~B1.
In a group of four people, exactly one is a knight who always tells the truth. One says, 'I am the knight', another says, 'I am not the knight', and the others remain silent. There can only be one knight.
Let K1, K2, K3, and K4 represent the proposition 'The nth person is the knight'. The puzzle can be formalized as (K1 v K2 v K3 v K4) & (~K2 -> K1) & (K1 -> ~K3 & ~K4) & (K2 -> ~K1 & ~K3 & ~K4)
You encounter two doors guarded by two guards. One door leads to a treasure, and the other leads to a trap. One guard always tells the truth and the other always lies. You don't know which is which, but you can ask one question to either guard to find the treasure.
Let T represent 'Truth-teller', L represent 'Liar', D1 'Door 1 leads to treasure', and D2 represent 'Door 2 leads to treasure'. The abstract representation is seeking a question that yields the same response (either D1 or D2) for both (T & D1 & ~D2) | (L & ~D1 & D2) and (T & ~D1 & D2) | (L & D1 & ~D2).
A five-person panel must sit in a row of five chairs, one person per chair. The middle chair must be occupied by a panelist wearing a green shirt.
Let G1 through G5 represent the proposition 'The nth chair is occupied by a person with a green shirt'. The puzzle is represented by G3 & ~(G1 v G2 v G4 v G5).
If a card has a vowel on one side, then it has an even number on the other side.
Let p be the proposition 'The card has a vowel' and q be the proposition 'The card has an even number'. The logic puzzle is represented by the implication p -> q.
There are three switches outside a closed room. One of the switches controls a light bulb within the room, the other two do nothing. You can flip the switches however you like but can enter the room only once to see the light.
Let S1, S2, S3 represent the proposition 'Switch n turns on the light'. The puzzle can be formalized as (S1 v S2 v S3) & ~(S1 & S2) & ~(S2 & S3) & ~(S1 & S3).
There are five consecutive numbers. The sum of the first and the second is 11, and the sum of the fourth and fifth is 14.
Let the sequence be a_1, a_2, a_3, a_4, a_5. These numbers satisfy a_n = a_{n-1} + 1 for n = 2,3,4,5 and a_1 + a_2 = 11 and a_4 + a_5 = 14.
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