JMC 2019

The difference

HintNot the ratio — that shrinks every birthday.

The whyIf one person is twice another's age now, they will not be twice it later — but the gap in years is fixed for life. Age puzzles are almost always solved by finding that gap first.

180°

HintThey lie between the parallels on the same side, like the two angles at the ends of a trapezium's slanted side.

The whyIn any parallelogram, rhombus or trapezium, the two angles at the ends of one side add to 180° because the other two sides are parallel. That one fact usually unlocks the angle beside a given 120° or 60°.

right; adjacent; parallel

HintDiamond, toy, table-top shape.

The whyA rhombus is a parallelogram with all four sides equal, so its diagonals bisect each other at right angles; a kite's diagonals also cross at right angles. Diagram questions use these properties to hand you parallel sides (for angle facts) and equal sides (for isosceles triangles).

They are equal

HintWrite both as fractions of 100 and look at the multiplications.

The whya% of b and b% of a are both a × b ÷ 100, so the two are always the same. This turns an ugly-looking calculation — 12% of 25 plus 25% of 12 — into 3 + 3 = 6 in your head.

9

HintThe fewer digits a number has, the smaller it is — so make each digit carry as much as possible.

The whyA number with fewer digits is always smaller than one with more, so pack the digit sum into the largest digit. For a digit sum of 30 that is 3999 — the remainder 3 goes first, the 9s after. Its last digit is 9 whenever the total is 9 or more.

A sum of the form 1 + 2 + 3 + … + n

HintBowling-pin or snooker-rack numbers.

The whyThe triangular numbers run 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91 … Learn the two-digit ones; they get asked for alongside squares and primes, and 36 is the one that is both square and triangular.

8000

HintCube the 2 and the 10 separately.

The whySo there are exactly ten cubes up to 10³ and twenty up to 20³. Knowing 10³ = 1000 and 20³ = 8000 lets you count cubes or squares in a range without listing them.

Exactly two factors needed

HintHow many different numbers divide into 1?

The whyA prime has exactly two factors, itself and 1; the number 1 has only one. So the smallest prime is 2, the smallest two-digit prime is 11, and 'prime digits' means 2, 3, 5 and 7 — never 1.

360

HintOne complete turn.

The whyWhen a square, an equilateral triangle and a regular hexagon share a vertex, their corners use 90° + 60° + 120° = 270°, leaving 90° — a right angle hiding in plain sight. Angles-at-a-point is the fastest route to a missing angle in a tiled diagram.

n(n + 1) ÷ 2

HintPair the first with the last, the second with the second-last…

The why1 + 2 + … + 12 = 12 × 13 ÷ 2 = 78, and 1 + … + 100 = 5050. Number-triangle and arrangement questions lean on this total: knowing the sum of all the pieces lets you say what the edges or corners must add to.

Subtract, working backwards

HintUndo the rule one step at a time from the far end.

The whyIf each term is the sum of the ones before it, then each earlier term is the later one minus the others. Start at the last known term and peel off one unknown per step — no algebra needed.

1000

HintThe prefix is the clue.

The whyMetric prefixes: kilo = 1000, centi = 1/100, milli = 1/1000. Comparing a length in metres with one in kilometres means converting first — the ratio is a thousand times bigger than it looks.