JMC 2026

Down by 2

HintOne edge from each of the touching squares disappears from the outline.

The whyFive separate squares have perimeter 20. Join them in a straight row (four shared edges) and it is 12; the plus shape also has four shared edges and perimeter 12; a 2 by 2 block with one square tacked on has five shared edges and perimeter 10. Shapes made from the same squares can have different perimeters — count the shared edges, not the squares.

A common denominator

HintMake all the pieces the same size before you count them.

The whyHalves, thirds and fifths all convert to thirtieths: 15/30 + 10/30 + 6/30 = 31/30. So "a half plus a third plus a fifth of a number" is thirty-one thirtieths of it, and the number is found by dividing by 31 and multiplying by 30.

Always odd

HintPair them up — one of them is left over.

The whyTwo odds make an even, so an even count of odd numbers sums to an even total and an odd count to an odd one. So an ODD number of primes with an even total must include 2 — and an odd count of odd numbers can never make an even total, which is worth checking before you search.

two

HintFold a paper strip into a loop and see which faces never touch.

The whyIn a strip of four, the first and third are opposite, and so are the second and fourth — while the two ends of the strip meet. Any square hanging off the side of the strip is opposite the other one hanging off. Work out the three opposite pairs and every "adjacent face" question answers itself.

A multiple of 9

HintThe class must split into equal groups, one for each part of the fraction.

The whyA fraction of a count of people or animals must itself be a whole number, so the denominator divides the total. Two such fractions in one question make the total a sum of a multiple of one number and a multiple of another — and some totals simply cannot be built that way.

positive

HintThink of −6 ÷ −2: how many lots of −2 make −6?

The whySame signs give a plus, different signs give a minus — for both multiplying and dividing. A fraction whose top and bottom are both negative after simplifying is therefore a plain positive fraction.

No — at least one is false

HintFollow the claims all the way round and back to the start.

The whyA cycle of "more than" claims can never all hold: following them round would make someone have more than themselves. Break the cycle at one link and the remaining claims can all be satisfied by lining people up in order — so the most that can be true is all but one.

2 × (length + width)

HintGo all the way round — each side appears twice.

The whyShorten both the length and the width by the same amount and the perimeter drops by four times that amount. Halve both sides and the perimeter halves, but the area falls to a quarter.

A kite

HintFold it along its long diagonal and the two halves match.

The whyThe equal sides meet at the two ends of the fold line, and the angles at the other two corners — between an unequal pair — are equal. Together with the 360° inside any quadrilateral, that pair of equal angles is usually the key to the missing one.

One odd, one even

HintPrimes above 2 are all the same kind of number.

The whyEvery prime above 2 is odd, and two odds or two evens both add to an even. So when neighbours in a ring must sum to a prime, odds and evens have to alternate — which halves the search at once.

360

HintSame total whether it is a square, a kite or a dented dart.

The whyA square’s four right angles show it, and it stays true for kites, trapeziums and even a dart with a reflex angle. In a kite, the two equal angles plus one known angle leave the fourth by subtraction — most "find the missing angle" questions on a four-sided shape end with this step.

12; 52; 3600

HintCalendar, clock, stopwatch.

The whyEstimation questions about "how many times since" multiply these together and then round hard: 24 × 365 × 50 is close to 25 × 400 × 50 = 500 000, and a rough answer is enough when the options differ by a factor of two.