Logical Problems
Logical problems are deductive puzzles that combine several rules into a single conclusion. The PMDC MDCAT 2026 syllabus expects you to handle truth-teller and liar puzzles, weighing puzzles, and multi-step problems where multiple constraints must be satisfied at once. Expect 1-2 of these in the Logical Reasoning section — they are usually the slowest items, so a systematic approach is essential.
Deductive Reasoning Puzzles
Deductive puzzles give you a small set of rules and ask you to find the unique solution. The recipe is the same for every variant: pick a starting assumption, propagate its consequences, and see whether you reach a contradiction or a stable solution.
The systematic approach
- Assume: Pick the variable with the fewest possible values and assume the first one.
- Propagate: Apply each rule one at a time and write down what each forces.
- Check consistency: If any rule is violated, the assumption is wrong — backtrack and try the next value.
- Eliminate: Continue until only one possibility survives all rules.
| Puzzle type | Setup | Best approach | Key trick |
|---|---|---|---|
| Truth-tellers / liars | One always lies, one always tells truth | Trial assumption + check for contradiction | Ask a "double-negative" question → both give the wrong answer → reverse it |
| Knights and knaves | Knights truthful, knaves lie | "What would I be if asked?" type meta-questions | "I am a knave" is a contradiction — nobody can say it |
| Hat puzzles | People see others' hats but not own | Reason from another person's silence | "What can each person not deduce?" is the information |
| Weighing (counterfeit coin) | One coin different in weight | Ternary search — split into 3 equal groups | 3 weighings handle 12 coins; 4 handle 27 (3n rule) |
| Ranking / ordering | Comparative statements (A > B, etc.) | Sketch a vertical line; place each item | Watch for transitivity: A>B and B>C → A>C |
| Matching grid | People × jobs × colours, etc. | Tick / cross grid; eliminate row by row | Each row and column must end with exactly one tick |
| If-then chaining | Series of conditionals | Chain transitively; use contrapositive | "If P then Q" = "If not Q then not P" |
| River-crossing | Move items across with constraints | BFS through allowed states | Think of forbidden combinations as "states to skip" |
You meet two people. One always tells the truth; the other always lies. You do not know which is which. The classic single-question solution: ask one of them, "If I asked the other person which path leads home, what would they say?" Then take the opposite path. Truth-teller and liar both produce a wrong answer, so you reverse it.
Each person sees others' hats but not their own. Reasoning chain: if A could see two of the same colour, A would deduce his own; A's silence is information for B. The trick is "what each person can deduce from another's silence."
Twelve coins, one is heavier or lighter, three weighings on a balance. Method: split into three groups of four. Weigh group 1 vs group 2; if balance, the odd coin is in group 3. Continue subdividing. The classic answer requires exactly three weighings and demonstrates ternary search.
Knights always tell the truth; knaves always lie. If a person says "I am a knave", we get a contradiction either way (a knight cannot say it, nor can a knave honestly call himself a knave). The statement is impossible — so the premise must be wrong somewhere.
Worked example — truth or lie
A says: "B is lying." B says: "C is lying." C says: "A and B are both lying." Who is telling the truth?
Try A truthful. Then B is lying (per A). Since B is lying, C is truthful (B's statement was false). But C says A and B are both lying — that contradicts A being truthful. So A is not truthful.
Try A lying. Then B is truthful (since A's claim was false). B truthful → C is lying. C is lying → "A and B are both lying" is false — meaning at least one is truthful. B is truthful — consistent. So only B tells the truth.
Multi-step Problem Solving
Multi-step problems give a list of rules — "A is taller than B; B is taller than C; D is shorter than C" — and ask for an ordering. Use a sketch or table; do not try to track everything in your head.
Place the items on a vertical line. Each rule moves one item above or below another. After all rules, read off the order. Example: A > B, B > C, D < C ⇒ A > B > C > D.
For "Five friends, five jobs, five colours" puzzles, draw a grid with rows = people, columns = attributes. Mark each rule as a tick (must) or cross (cannot). Eliminate row by row until each cell is determined.
Conditional rules can be chained. "If P then Q; if Q then R" gives "If P then R." Use the contrapositive when needed: "If not R then not P." This converts a list of conditionals into a clear pathway.
When stuck, eliminate the impossible. Even if you cannot determine the answer directly, ruling out three of the four options is just as good on the MDCAT.
Worked multi-step example
Rules: (1) Ali is older than Bilal. (2) Bilal is younger than Asad. (3) Asad is younger than Ali. (4) Daud is older than Ali.
Step 1. From (1): Ali > Bilal. From (3): Ali > Asad. From (4): Daud > Ali. Combining: Daud > Ali > Asad and Ali > Bilal.
Step 2. Rule (2) places Asad > Bilal. So the full order is Daud > Ali > Asad > Bilal. Daud is the oldest, Bilal the youngest.
- Assumption
- A tentative starting value for the most-constrained variable.
- Contradiction
- A pair of statements that cannot both be true; signals a wrong assumption.
- Backtracking
- Returning to the last choice point and trying the next option after a contradiction.
- Constraint
- A rule that limits the values a variable can take.
- Process of elimination
- Removing options that violate any rule until only valid ones remain.
Worked MCQs
Five MCQs that capture the high-yield testing patterns for logical problems. Read the explanation even when you get the answer right — it's where the deeper concept lives.
Q1. Five students — P, Q, R, S, T — ranked their test scores. P scored higher than Q. R scored higher than P. S scored lower than Q. T scored higher than R. Who scored the lowest?
Chain: T > R > P > Q > S. T scored highest; S scored lowest. Build the chain rule by rule and the answer reads off the bottom.
Q2. Three people — X, Y, Z — are either truth-tellers or liars. X says, "Y is a liar." Y says, "Z is a liar." Z says, "X and Y are both liars." Who tells the truth?
If X is truthful, Y lies, so Z is truthful, but Z says both X and Y lie — contradiction. So X lies, meaning Y is truthful, so Z lies, meaning Z's claim that both X and Y lie is false — consistent because Y is in fact truthful. Only Y.
Q3. If "All men are mortal" and "Socrates is a man" then which of the following must also be true?
By contrapositive of "All men are mortal", anything not mortal (immortal) is not a man. Since Socrates is a man, anything immortal is not Socrates. The other options reverse universals or add unsupported claims.
Q4. Among 9 identical-looking coins, one is heavier than the rest. Using a balance, what is the minimum number of weighings needed to guarantee finding it?
Split 9 coins into three groups of 3. Weighing 1: group A vs group B. If balanced, the heavy coin is in group C; otherwise it is in the heavier pan. Weighing 2: take that group of 3 and weigh any two coins; the heavier pan reveals it, or if balanced the third one is heavy. Two weighings suffice.
Q5. Four houses stand in a row. The red house is to the right of the blue house. The green house is immediately left of the white house. The blue house is at the leftmost end. The green house is to the right of the red house. What is the order from left to right?
Blue is at position 1. Red is to the right of Blue (so position 2, 3, or 4). Green is to the right of Red, and immediately left of White. The pair "Green, White" must occupy two consecutive positions. The only arrangement that fits is Blue-Red-Green-White.
Quick Recap
- For every logical puzzle, follow A-C-E: Assume, Check, Eliminate.
- In truth-teller/liar puzzles, test each candidate as truthful in turn and look for contradictions.
- Hat and weighing puzzles rely on what one's silence (or balance state) reveals.
- For ranking puzzles, draw a vertical line and place each item rule by rule.
- For matching puzzles, use a grid with ticks for "must" and crosses for "cannot".
- Use the contrapositive to chain conditional rules safely.
- When stuck, eliminate impossible options — partial certainty is enough on the MDCAT.