Daily Math Puzzle: 2026-06-10

Vinegar, Ginger, Sesame Oil — Let's assign single letters to the ingredients for easier analysis: S = Soy Sauce V = Vinegar H = Honey G = Ginger E = Sesame Oil And let's formalize the rules: 1. S → V 2. H ↔ ~G (Honey if and only if NOT Ginger) 3. E ∨ (S ∧ H) (Sesame Oil OR (Soy Sauce AND Honey)) 4. ~(V ∧ H) (NOT (Vinegar AND Honey) - meaning V and H cannot both be used) We are looking for a combination of *exactly three* ingredients. Let's start with Rule 4: `~(V ∧ H)`. This means Vinegar and Honey cannot be used together. So, either V is ON and H is OFF, or V is OFF and H is ON, or both are OFF. Case 1: V is OFF and H is ON. - From Rule 2 (H ↔ ~G): If H is ON, then ~G is ON, so G must be OFF. - From Rule 1 (S → V): If V is OFF, then S must be OFF (because if S were ON, V would have to be ON, contradicting V is OFF). - So far: S=OFF, V=OFF, H=ON, G=OFF. - Now check Rule 3 (E ∨ (S ∧ H)): (S ∧ H) becomes (OFF ∧ ON) which is OFF. For Rule 3 to be true, E MUST be ON. - Current set of ingredients: {H, E} (Honey, Sesame Oil). This is only 2 ingredients. Since we need *exactly three*, this case is invalid. Case 2: V is OFF and H is OFF. - From Rule 2 (H ↔ ~G): If H is OFF, then ~G is OFF, so G must be ON. - From Rule 1 (S → V): If V is OFF, then S must be OFF. - So far: S=OFF, V=OFF, H=OFF, G=ON. - Now check Rule 3 (E ∨ (S ∧ H)): (S ∧ H) becomes (OFF ∧ OFF) which is OFF. For Rule 3 to be true, E MUST be ON. - Current set of ingredients: {G, E} (Ginger, Sesame Oil). This is only 2 ingredients. Since we need *exactly three*, this case is invalid. Case 3: V is ON and H is OFF. (This is the only remaining possibility for Rule 4 to hold) - From Rule 2 (H ↔ ~G): If H is OFF, then ~G is OFF, so G must be ON. - So far: V=ON, H=OFF, G=ON. - Now check Rule 1 (S → V): Since V is ON, S can be either ON or OFF (S → ON is true whether S is ON or OFF). Subcase 3a: S is ON. - Current state: S=ON, V=ON, H=OFF, G=ON. - Now check Rule 3 (E ∨ (S ∧ H)): (S ∧ H) becomes (ON ∧ OFF) which is OFF. For Rule 3 to be true, E MUST be ON. - Ingredients: {S, V, G, E} (Soy Sauce, Vinegar, Ginger, Sesame Oil). This is 4 ingredients. Since we need *exactly three*, this subcase is invalid. Subcase 3b: S is OFF. - Current state: S=OFF, V=ON, H=OFF, G=ON. - Now check Rule 3 (E ∨ (S ∧ H)): (S ∧ H) becomes (OFF ∧ OFF) which is OFF. For Rule 3 to be true, E MUST be ON. - Ingredients: {V, G, E} (Vinegar, Ginger, Sesame Oil). This is exactly 3 ingredients. This subcase is valid! Let's double-check the unique valid combination: {Vinegar, Ginger, Sesame Oil} - V=ON, G=ON, E=ON - S=OFF, H=OFF 1. S → V: OFF → ON (TRUE) 2. H ↔ ~G: OFF ↔ ~ON (OFF ↔ OFF, TRUE) 3. E ∨ (S ∧ H): ON ∨ (OFF ∧ OFF) (ON ∨ OFF, TRUE) 4. ~(V ∧ H): ~(ON ∧ OFF) (~OFF, TRUE) All rules are satisfied, and there are exactly three ingredients. Thus, 'Vinegar, Ginger, Sesame Oil' is the only possible combination.