Every card in JMC toolkit

1111

HintOnly one of the options is odd.

WhyChecking parity — odd or even — is a ten-second test. Odd plus odd is even and odd plus even is odd, so a sum of two odd numbers and two even numbers must be even, and 1111 cannot be the answer whatever the numbers were. Remember that 0 counts as even.

Work backwards from the answer choices

HintOne of the five options must be right — test them against the question instead of solving it.

WhySubstituting each option is often faster than algebra, and it can never produce an answer that is not on the list. Start with the middle value: when the quantity grows steadily with x, a too-big or too-small result rules out half the options at once.

Draw a diagram

HintMake the shape visible on your rough paper, with every length and angle you are told written on it.

WhyNot drawing is the single most common geometry mistake. A sketch with the known values labelled turns a paragraph into a puzzle you can see — and a construction line such as a diagonal or a height often reveals the answer on its own.

Extreme cases

HintPush everything else to its smallest allowed value and see what is left over.

WhyWhenever a question says largest, smallest, maximum or minimum, go to the edge. If five different positive whole numbers add up to 20, make four of them 1, 2, 3 and 4 and the fifth can be 10 — the conditions in the question do the work for you.

Try small cases and list systematically

HintStart with a simpler version of the same question and keep your counting organised.

WhyCompetition counting questions are rarely solved in one leap. Shrink the problem — digits adding to 3, then to 4 — and the rule for 15 usually shows itself. Listing in a fixed order (all the 1__ numbers, then 2__) is what stops you counting something twice or missing it altogether.

4

HintCount how many different last digits appear before the first one comes round again.

WhyThe last digit of a power always cycles with a short period — four for 2, 3, 7 and 8, two for 4 and 9. To find the units digit of 7 to the power 2026, divide 2026 by 4: the remainder 2 says it matches 7 squared, so the answer is 9.

Add its digits — 15 — and check whether that is a multiple of 3

HintYou never need to do the division itself; the number's own figures tell you.

WhyThe digit-sum test works for 3 and for 9: 4 + 7 + 3 + 1 = 15, a multiple of 3 but not of 9, so 4731 divides by 3 and not by 9. For 4, look only at the last two digits; for 5, only at the last one.

Use 50 × 50 = 2500 and take away 1

HintBoth numbers sit one step either side of a round number.

WhyRounding to 50 gets you close enough to pick between spread-out answer choices, and here it is exact: (50 − 1)(50 + 1) = 2500 − 1. Estimate first, then refine only if the choices are close together.

None — a wrong answer and a blank both score 0

HintThis is the rule that makes guessing on the junior paper a free shot.

WhySo on the JMC you should always put an answer down: a blind guess is right one time in five, and crossing out two options makes it one in three. Be careful — this is NOT true of the Intermediate Challenge, where a wrong answer to the later questions does lose marks.

6 marks

HintOne more than the earlier questions.

WhyQuestions 1 to 15 score 5 each (75 marks) and 16 to 25 score 6 each (60 marks), a total of 135. In 2026 the Gold line was 75, so a perfect run on the first fifteen alone was already Gold-level — secure those before anything else.

Move on and come back later

HintTime spent staring is time taken from questions you can do.

WhySixty minutes for twenty-five questions is under two and a half minutes each. Coming back with a fresh head is far more productive than five minutes of staring, and the early questions you skip past are the cheapest marks on the paper.

3

HintCount how many options are left.

WhyElimination turns a one-in-five guess into one-in-three, and one more cross-out makes it a coin flip. On the junior paper, where a wrong answer costs nothing, that is always worth doing.