JMC 2024

2n + 1

HintDraw a square of dots, then add one more row and one more column — count the new dots.

The whyThe gaps between squares are the odd numbers: 1, 3, 5, 7, … so 25 and 36 differ by 11, and 100 and 121 by 21. If someone’s age is a square now, the next square age is 2n + 1 years away.

Left to right

HintNeither one beats the other, so read the line the way you read a sentence.

The why12 ÷ 3 × 2 = 4 × 2 = 8, not 12 ÷ 6 = 2. The same holds for addition and subtraction: equal rank, left to right. Brackets and indices come first; then the multiply-and-divide pair; then the add-and-subtract pair.

A quarter

HintThe two diagonals are lines of symmetry, so every piece matches every other.

The whyFour congruent right-angled isosceles triangles. In a large square split into four small ones, a small square’s diagonal cuts off an eighth of the big square, and pairs of tiny corner triangles can be reassembled into a larger piece. Shaded-fraction questions are solved by cutting everything into the same small triangle.

Pretend they are all one kind

HintStart from an extreme, then swap.

The whySay every head belongs to a hen and count the legs. Each swap of a hen for a sheep adds exactly two legs, so the shortfall divided by two is the number of sheep — and starting with all sheep works just as well. The same move solves coins, tickets and any two-type mixture where each type has a fixed value.

n + 1

HintThe cut crosses the boundary of a convex shape exactly twice, and adds one new edge to each piece.

The whyThe cut enters and leaves through two edges, splitting each of them, and contributes one new edge of its own — so a piece can gain at most one side. From a pentagon you can cut off anything from a triangle to a hexagon, but never a heptagon. (A dented, non-convex shape can be crossed more often, which is why the word convex matters.)

against

HintEach goal shows up in two teams’ rows.

The whyA goal scored by one team is a goal conceded by another, so the two columns must balance across the whole table — and the wins must equal the losses. If one team has lost a single match and conceded three more than it scored, that lost match was by three goals. Results tables reward this kind of bookkeeping before any guessing.

half

HintCompare it with the rectangle that just encloses it.

The whyTwo triangles standing on opposite sides of a square, sharing a point inside it, have heights that add to the square’s side — so their areas add to half the square, whatever the point. That fact settles "find the other triangle’s area" in one line.

The smallest number in both times tables

HintNot the product — the two may share a factor.

The whyFor 4 and 6 it is 12, not 24. Anything that must divide by both 4 and 6 is a multiple of 12, and anything that leaves the same remainder with both repeats in steps of 12 — so the candidates sit just above multiples of 12.

A rhombus

HintIts diagonals cross at right angles, like a square’s.

The whyBecause all four sides match, any triangle made from two sides of a rhombus is isosceles, and because opposite sides are parallel, neighbouring angles add to 180°. Both facts get used together in angle chases built on a rhombus.

5

HintThink medicine doses.

The whyA sense of scale for capacity: a teaspoon is about 5 ml, a mug about 250 ml, a large bottle of milk about 2 litres, a bath well over 100 litres. "Which of these could hold …?" questions reward knowing a few anchors like these rather than calculating anything.

1000

HintThink of the tiniest marks on a ruler.

The whyMilli means a thousandth. Estimation questions about tiny creatures or thin coins hinge on this conversion, then on rounding: something a few millimetres long fits into a metre a few hundred times, which is enough to pick from widely spaced options.

20; 12.5; 33

HintShare 100 by five, by eight, and by three.

The whyMixture wording is the trap: one part squash to three parts water is a quarter squash, not a third, because the whole drink is four parts. "What fraction of all the pens" questions are the same idea in reverse.