Every card in JMC 2025

Its primes fit inside

HintThink building blocks, not long division.

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.

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.

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.

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.

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.

0

HintWhich digit would make a number shorter than it looks?

WhyA leading nought is simply not written. So when you build the smallest number from given digits, the smallest NON-ZERO digit goes first and 0 comes second — 1023 beats 0123, which is really the three-digit 123.

20

HintPer cent means out of a hundred.

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.

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.

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.

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.

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.

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.