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.

Thirty-one thirtieths

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 — more than the whole number itself. So if the three shares add to a known total, the number is found by dividing by 31 and multiplying by 30. Finding the common denominator first is the whole move.

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 when the fraction is in its lowest terms the denominator divides the total — nine here. Cancel first: two sixths of the pupils only forces a multiple of three. 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.

54

HintAn even power wipes out the minus sign; an odd power keeps it.

The whyAn even count of negative factors gives a plus and an odd count a minus, so (−3)⁴ = 81 while (−3)³ = −27, and 81 − 27 = 54. Watch the brackets: (−3)² is 9 but −3² is −9, because the power binds more tightly than the minus sign. For a long product you never need the digits — count how many factors are negative and look only at whether that count is odd or even.

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.

Down by 4 cm

HintEach side appears twice in the distance round the outside.

The whyPerimeter is 2 × (length + width), so taking 1 cm off each dimension removes 2 cm from the bracket and 4 cm from the total. Halve both sides and the perimeter halves, but the area falls to a quarter.

A kite

HintFold it along the diagonal joining the two corners where the equal sides meet; 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; 86400; 8760

HintCalendar, then clock times clock, then clock times calendar.

The whyEstimation questions about "how many times since" multiply these together and then round hard: 24 × 365 × 50 = 438 000 hours in fifty years, close enough to 25 × 400 × 50 = 500 000 when the options differ by a factor of two. Know 60 × 60 = 3600 seconds in an hour as the step on the way.

Multiplication

HintThink of the letters in BIDMAS or PEMDAS — division and times both come before plus.

The whyMultiplication and division are worked out before addition and subtraction, working left to right within each level. So 2 + 3 × 4 is 2 + 12 = 14, not 5 × 4 = 20. A multiple-choice question with several similar-looking expressions is testing exactly this rule — work out each one in full rather than glancing at the biggest numbers.

Zero

HintIf it started with this digit, the number would really have one fewer digit than you think.

The whyA number such as 045 is just written 45 — the leading zero carries no value, so it is dropped and the number only has two digits, not three. This matters when counting how many numbers can be built from a limited set of digits: any arrangement that starts with 0 does not count, so a non-zero digit must go in the first place and the remaining places can be filled freely.

12

HintMultiply the two numbers together, then divide out the highest factor they share.

The whyEvery multiple of both 4 and 6 is also a multiple of their lowest common multiple, 12 — found by multiplying the two numbers (24) and dividing by their highest common factor (2). This underlies any "skip these, then skip those too" counting problem: a number that satisfies both skip-rules should be skipped only once, because it is a multiple of the LCM, not of the product.

Four

HintTwo go through opposite sides’ midpoints, and two run corner to corner.

The whyTwo lines run through the midpoints of opposite sides, and two run along the diagonals — four in total, more than any other quadrilateral. A symmetric shape can only lose a piece if every line of symmetry using it is broken, so working through one line of symmetry at a time is how you strip a symmetric arrangement down while it stays symmetric for as long as possible.

10

HintThe units place is worth one; the next place along is worth ten times as much.

The whyPlace value means a digit’s actual worth depends on where it sits: in 47, the 4 is worth 40 (four tens) and the 7 is worth 7 (seven units), so 47 = 4 × 10 + 7. Writing a two-digit number this way — tens digit × 10 + units digit — turns a word problem about its digits into an equation instead of guessing digit by digit.

Systematic listing

HintGo in order — smallest first, or by how each one starts — never at random.

The whyCounting problems with an extra rule (like "never fall behind on a running count") are easy to get wrong by jotting down arrangements at random, because you can never be sure you have found them all. Fixing an order — say, always trying the earliest valid choice first — turns a guessing game into a complete, checkable search.

Equal

HintThe triangle folds perfectly onto itself along the line from the odd vertex to the midpoint facing it.

The whyWhenever two sides of a triangle are equal in length, the angles facing those sides are equal too — often the first substitution in an angle-chasing problem, turning an unknown angle into a repeated label such as x. Spot the two equal sides named in the question before looking for the angles; the angle fact then follows automatically.

180

HintThree copies of any triangle’s corners fit exactly along one straight line.

The whyTear the three corners off any triangle and place them together — they fit exactly along a straight line, which measures 180°. This is the single most-used fact in angle chasing: once two angles in a triangle are known, or written algebraically, the third is 180° minus their sum.

4k and 5k, for some positive number k

HintScale both parts of the ratio by the same unknown multiplier to get real lengths.

The whyA ratio such as 4 : 5 does not fix the actual lengths, only their proportion, so the true lengths are 4k cm and 5k cm for some positive k. Tiling that rectangle exactly with smaller identical rectangles forces k to take only certain values, because the areas must divide evenly — turning a ratio question into a search for the smallest whole-number k that works.

Number of tiles times the size of one tile

HintWork out one piece’s area, then multiply by how many identical pieces make up the whole.

The whyThis is just "total = count × one share", dressed up geometrically: if fifteen identical small rectangles tile a bigger rectangle exactly, the bigger rectangle’s area is fifteen times one small rectangle’s area. Setting up that equation is usually the fastest way into a tiling problem, since area does not care how the pieces are arranged, only how many there are.

Multiply it by how many numbers there are

HintUndo the division that turned the total into an average in the first place.

The whyThe mean is defined as total ÷ count, so reversing it — mean × count = total — recovers the sum whenever you are told an average instead of the numbers themselves. This is the standard first move whenever a question gives two overlapping averages, such as the average of an early group and the average of a later, shifted group, and asks about how the groups differ.

45

HintPair the smallest with the largest, the next smallest with the next largest, and so on.

The whyPairing 1+9, 2+8, 3+7 and 4+6 each gives 10, with 5 left over in the middle, so the total is 4 × 10 + 5 = 45. Knowing this instantly is useful whenever a puzzle fills the numbers 1 to 9 into a grid or diagram exactly once each, because the grand total of every number used is fixed before a single one is placed.

24

HintMultiply the choices for first place, then second, then third — the last spot is forced.

The whyFour choices for the first position, three left for the second, two for the third, and the last item has no choice at all: 4 × 3 × 2 × 1 = 24. This shrinking-choices idea is exactly how you count arrangements of any set of distinct items — five items would instead give 5 × 4 × 3 × 2 × 1, and so on.