JMC 2025

Its primes fit inside

HintThink building blocks, not long division.

The whyEvery prime in the smaller number must appear in the larger at least as many times. 27 = 3³ is a factor of 3⁴ × 5² because three 3s are available; 50 = 2 × 5² is not, because there is no 2. One factorisation checks a whole list of candidates at once.

Three quarters

HintThe named fraction and the fixed part together make the whole.

The why"Plus a quarter of itself" leaves the other three quarters as the fixed part, so the whole rope is 21 ÷ 3 × 4 = 28 cm. Whenever a quantity is described as "a fixed amount plus a fraction of itself", the fixed amount is the remaining fraction of the whole.

Round and estimate

HintThe options are spaced widely enough that exact working is wasted time.

The why97 × 31.8 is close to 100 × 30 = 3000, which is all you need to pick between choices that differ by a thousand. Look at the spread of the options before you decide how precisely to calculate — the spread is a hint the setter has given you.

A square number

HintFactors usually come in pairs, unless one pair is a repeat of itself.

The whyFactors pair up (2 with 18 for 36) except when a factor pairs with itself (6 × 6), so only squares have an odd count. 36 has nine factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. A question saying "an odd number of factors" is quietly saying "a square".

Smaller than N

HintIf a whole extra N were still left over, the division would not be finished.

The whyA remainder of 3 on dividing by N means N is bigger than 3 — and N is a factor of the number minus 3. So "which N leave remainder 3?" means: list the factors of (number − 3) and keep only those above 3.

27

HintWrite it as ten lots of the first figure plus the second, then form an equation.

The whyPlace value turns digit puzzles into algebra: a two-digit number is 10a + b, so 10a + b = 3(a + b) tidies to 7a = 2b. With single figures and a leading one that cannot be nought, only a = 2 with b = 7 fits, giving 27. The same set-up cracks "reverse the digits" questions — the reversed number is 10b + a, so the difference is 9(a − b), always a multiple of 9.

20

HintPer cent means out of a hundred.

The whyOnce 10% is known, 5% is half of it, 20% is double, 30% is triple. Nested percentages multiply: 30% of 50% of an amount is 0.3 × 0.5 = 0.15 of it, so chains of "of" reduce to one multiplier.

The digit sum is a multiple of 3

HintYou never need to divide — the answer hides in the digits.

The why741 has digit sum 12, so it divides by 3; 742 does not. The same test with 9 finds multiples of 9. Contrast the test for 4, which looks only at the last two digits — 916 divides by 4 because 16 does.

a² + b² = c²

HintNamed after a Greek; a right angle is required.

The whyA tilted square drawn inside a bigger square has its side as the hypotenuse of a corner triangle, so its area is the sum of two squares — no square root needed. With legs 2 and 3 the tilted square has area 13, whatever unit you use.

45

HintIts square ends in 25, so it ends in 5 — and it is under 50.

The whySetters love the year: 2025 = 45² = 3⁴ × 5², so its factors include 9, 27, 81 and 405. Before any Challenge, know whether that year’s number is prime, a square, or a multiple of small primes — the first question often plays with it.

12

HintThink about one face first, then count the faces.

The whyAll twelve are the same length — each is the hypotenuse of a right-angled triangle whose legs are two edges. Space diagonals, from a corner straight through the middle to the opposite corner, are different: there are only four, and they are longer.

8; 27; 64; 125

HintNot the squares — one power higher.

The whyKnow these as well as the squares. Among the one-digit numbers only 1 and 8 are cubes; 27 and 64 are the two-digit ones; and 27 + 64 + 125 = 216 = 6³ is a famous coincidence worth recognising on sight.

Factor out the common factor first

HintSpot what every term shares before you add anything up.

The whyFor example, 12 + 18 both share a factor of 6, giving 6 × (2 + 3); and 4 + 6 both share a factor of 2, giving 2 × (2 + 3). The (2 + 3) cancels, leaving 6 ÷ 2 = 3 without adding anything first. The same shortcut works even when the sums involve much bigger numbers.

Subtract just the shortfall, not the whole time

HintThink about the small gap to the next full hour, in minutes and seconds only.

The whyFor 3 h 58 min 50 s, short of 4 hours: the 50 s needs 10 more to complete that minute, and — because that partial minute is already spoken for — the 58 min then needs only one more full minute (not 60 − 58 = 2) to reach the hour. So the shortfall is 1 min 10 s, that is 70 seconds — far quicker than converting both times fully into seconds first.

45°

HintThink about the two 45-45-90 triangles the diagonal creates.

The whyA square’s diagonal cuts each 90° corner exactly in half, because the two triangles it creates are congruent right-angled isosceles triangles. That 45° often combines with other known angles — such as one from an attached equilateral triangle — by simple addition or subtraction to find a shaded or marked angle.

Multiply by the ratio’s scale factor

HintThink of it as a recipe — keep the two amounts in step with each other.

The whyIf one quantity is always 2 units for every 5 of another (a fixed ratio), and the first reaches, say, 20, then the second is 20 ÷ 2 × 5 = 50. The same one-line scaling — divide by the known side of the ratio, then multiply by the other side — works whatever the two paired amounts turn out to be.

1000

HintKilo- means a thousand of the base unit.

The whyThe metric system scales by powers of ten: 1 kg = 1000 g, and 1 g = 1000 mg. Converting to a smaller unit lets you count how many equal-sized portions — such as 100 g bars — fit inside a larger total, by dividing the total grams by the portion size.

Add every section, set the sum equal to the known total

HintWrite the whole length as an addition of its parts, then rearrange to isolate the missing piece.

The whyIf a strip has length x, and two other named sections make up the rest of the total, write total = x + x + (other known parts), simplify, and solve for x. This turns a picture with an unlabeled length into a one-line equation — the general move whenever a diagram’s markings must add to something you already know.

Track how the gap between them changes each step

HintFollow how far apart the two values stay after every stage — not their totals.

The whyIf the gap starts at, say, 5 and shrinks by 2 every step, it never hits exactly zero — after 2 steps it’s 1, after 3 it’s −1, so the two measurements pass without ever matching. Only when the starting gap divides evenly by the per-step change is there a single point where they’re exactly equal.

Count direction by direction, then add the totals

HintGroup the triangles by which line they sit along, one group at a time.

The whyIn the diagram, triangles whose base runs one way share one set of possible bases, and triangles whose base runs the other way share a different set; count how many segments in each set can form a base, then add the two counts. Working direction by direction stops you missing a whole class of triangles or double-counting the same one twice.

Fix the most constrained items first, then see what remains

HintStart where the choices are tightest, cross off each pairing as you assign it, and look at whichever option survives.

The whyIf one group has the most members, pair those first since they have no spare options; each pairing used removes both an item and a partner from what’s still available. Working outward from the tightest constraints pins down every pairing that IS made — and the candidate that never appears on that finished list is the one that is never made.

2

HintThink ‘average minus half the gap’ — same idea as splitting a difference evenly.

The whyPicture the shorter part as x and the longer as x + d, where d is the known gap. Their sum is 2x + d, which must equal the total, so x = (total − d) ÷ 2 — the same ‘sum and difference’ trick used for age-difference or coin-value puzzles: average the two knowns, then adjust for the gap.

It narrows the unknowns to a short, testable list

HintTry each candidate value in turn and keep only the one that also satisfies everything else.

The whyIf a quantity must always work out to a whole number, or specifically a multiple of some fixed factor, you can list the handful of values that qualify and check each against the other conditions in the problem — usually only one survives. This turns ‘too many unknowns for one equation’ into a short trial rather than a dead end.

lowest common multiple

HintThe smallest number that both of the original numbers divide into evenly.

The whyMultiples of 6 are 6, 12, 18, 24, 30, 36…; multiples of 8 are 8, 16, 24, 32…. The smallest number in both lists is 24, so anything required to be a multiple of both 6 and 8 must be a multiple of 24 — not just 6 × 8 = 48, which is often bigger than necessary.

(a + 1)(b + 1)

HintAdd one to each exponent in the prime factorisation, then multiply those together.

The why200 = 2³ × 5² has (3+1)(2+1) = 12 factors — matching the direct list 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200. This ‘exponent-plus-one, then multiply’ rule beats listing factors by hand once a number has more than a couple of prime factors, and explains why only squares (all exponents even) give an odd factor count.