JMC 2021

A multiple of 9

HintExact division and the times table are the same statement.

The whyDividing exactly by 9 means the number sits in the 9 times table — it is 9 × something. Pair that with the digit-sum test (a multiple of 9 has a digit sum that is a multiple of 9) and with the boundaries: the three-digit multiples of 9 run from 108 upwards in steps of 9. A "the answer is also a whole number" clue is usually this fact in disguise.

Alternating sum of digits

HintThe digit-sum idea, with a twist.

The whyStarting from the right, add and subtract the figures in turn. If the result is 0 or another multiple of 11, the number is a multiple of 11. For 2475: 5 − 7 + 4 − 2 = 0, and indeed 2475 = 11 × 225. The test lets you check five candidates in the time it takes to do one long division.

Two equal base angles

HintFold the triangle along its line of symmetry and see which corners land on each other.

The whyAn isosceles triangle has a mirror line through its apex, so the two angles opposite the equal sides match. Nearly every JMC angle-chasing question is a chain of isosceles triangles: mark each pair of equal sides, write the matching angles as the same letter, and the unknown falls out of the 180° in a triangle.

Length plus width

HintRun your eye along one outer edge of the picture: it is made of one long side and one short side.

The whyThe classic windmill arrangement. Each outer edge is a long side followed by a short side, so the outer square has side length + width, while the inner square has side length − width. Those two facts turn an area question into a two-line calculation.

Negatives and zero

HintThe word "positive" has been left out on purpose.

The whyThe integers are the whole numbers in both directions: …, −3, −2, −1, 0, 1, 2, 3, …. When several different integers multiply to a positive result, an even number of them must be negative — with a small product, exactly two. Setters use this exact wording to reward careful readers.

four

HintSome numbers cannot be done with three.

The whyZero counts as a square, so 12 = 4 + 4 + 4 and 30 = 25 + 4 + 1 use fewer. Some numbers genuinely need the full set — 23 = 9 + 9 + 4 + 1 cannot be made with three squares. Lagrange proved the theorem in 1770; JMC setters like to ask which small numbers manage with fewer.

8

HintWork out the last digits of 1², 2², 3² … 9² and see which figures never show up.

The whySquaring the digits 0 to 9 gives last digits 0, 1, 4, 9, 6, 5, 6, 9, 4, 1 — so a square ends in 0, 1, 4, 5, 6 or 9 and never in 2, 3, 7 or 8. In a crossnumber whose clue says "a square", this rules out most cells before you write anything.

V + F = E + 2

HintCheck it on a cube before you trust it.

The whyA cube has 8 vertices and 6 faces, 14 altogether, which is its 12 edges plus 2. The formula holds for every prism and pyramid, so if a question gives you two of the counts you can find the third without drawing. Named after Leonhard Euler, the Swiss mathematician who found it in the 1700s.

3n

HintTop, bottom, and the uprights joining them.

The whyn edges round the top, n round the bottom and n joining the two ends. A prism also has n + 2 faces and 2n vertices: a pentagonal prism has 15 edges, 7 faces and 10 vertices, and Euler agrees — 10 + 7 = 15 + 2. So "a prism with seven faces" is telling you the base is a pentagon.

45

HintPair the smallest with the largest, the next smallest with the next largest, and count the pairs.

The whyFour pairs of ten plus the 5 left in the middle. Every grid puzzle that places the digits 1 to 9 once each rests on this total: three rows of equal sum must each come to a third of it, which is why the classic magic-square line is 15.

6

HintSquares, not roots — write them out.

The whyThey are 16, 25, 36, 49, 64 and 81. Knowing the list by heart, along with which of them are odd, makes crossnumber clues such as "a square" or "an odd square" instant.

6; 12; 3

HintCount the squares on a die, the lines where two squares meet, and the lines running into one corner.

The whyThree numbers to know cold. Counting dots or markers along a cube’s edges, or edges left after a corner is sliced off, always starts from these. The vertex count, 8, then follows from Euler.