Mathematical Reasoning and Problem Solving
Core content
Candidates should reason logically, construct and critique arguments, and choose efficient strategies. Know the meaning of converse, inverse, contrapositive, necessary condition, and sufficient condition. Use counterexamples to disprove universal claims, and recognise proof by contradiction, induction, parity arguments, invariants, and the pigeonhole principle. Problem-solving strategies include restating the problem, drawing a diagram, working backwards, introducing variables, considering cases, estimating, and checking reasonableness. Modelling tasks require interpreting assumptions and units.
Trap watch
The converse of a true statement need not be true. A pattern that holds for several cases may fail later, so test edge cases and look for a structural reason. Hidden constraints such as integer values, positive lengths, or distinct objects can change the answer. Distinguish at least from exactly, and inclusive from exclusive counting. Percentage problems often use different bases. A calculation can be arithmetically correct yet answer the wrong question, so reread the final prompt.
Worked example
At a meeting, each of 10 people shakes hands once with every other person. Count pairs: the first person has 9 choices, the next 8 new choices, and so on, but each handshake is counted twice. The total is 10 x 9 / 2 = 45 handshakes.
Rule of thumb: restate the question in your own words, identify the unknown and constraints, then verify the result against the original conditions.