Must, Could and Cannot

Three strengths of claim, and they are not interchangeable. Must means true in every arrangement, could means true in at least one, and cannot means true in none. Because must is the strongest claim, it is the easiest to knock down — a single exception destroys it. Could is the weakest, so it is the hardest to knock down, and that makes it correct more often than students expect. People reject could options far too readily, because they sound non-committal and vague. But a weak claim being true is not the same thing as a weak answer.

Suppose an option claims that two numbers with an even sum must both be even. Four and six fit it. But three and five also sum to eight, and both of those are odd — so the claim is false, and one counter-example was all it took. You do not have to explain why the claim fails or how often; you just need the one case. So when an option says must, spend your time hunting for the exception rather than trying to justify the rule. Try the awkward values first: zero, one, negative numbers, two numbers that are equal, and the largest value the question allows.

Confirming a could works the same way, just in the opposite direction: build one arrangement that satisfies every condition in the question, and if it exists and breaks no rule, the option is correct and you can stop. The danger is building a case that quietly violates one of the given conditions, so list the conditions first and check your example against each one in turn. Notice that both moves are really the same move — construct one concrete case. For a must you want a case that fails; for a could you want a case that works. Concrete beats abstract, and it is far faster when you have seventy-five seconds.

If it rained, the match was cancelled. That statement lets you travel in exactly two directions. Forwards: if it rained, then the match was cancelled. And backwards from the negative: if the match was not cancelled, then it cannot have rained. What it never lets you do is go from cancelled back to rained, because something else could have cancelled the match — a flooded car park, an injured referee. Reversing an if-then looks perfectly sensible and is the most common reasoning error on this paper. Only the negative direction is allowed.

When a question gives several facts that have to fit together — three people and three colours, say — draw a grid. One tick per row and one per column, so every tick you place creates several crosses at once. Write in the crosses you are told about first, because the ticks then tend to appear on their own: a row with only one blank square left is solved, and solving it forces crosses across its column, which often solves the next row. That is the lesson. Notice which of must, could and cannot an option uses, settle it with one concrete case, never reverse an if-then, and draw a grid when the facts interlock. Quiz and exercises now.