JMC 2019

The difference

提示Not the ratio — that shrinks every birthday.

为什么If 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°

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

为什么In 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

提示Diamond, toy, table-top shape.

为什么A 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

提示Write both as fractions of 100 and look at the multiplications.

为什么a% 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

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

为什么A 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

提示Bowling-pin or snooker-rack numbers.

为什么The 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

提示Cube the 2 and the 10 separately.

为什么So 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

提示How many different numbers divide into 1?

为什么A 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

提示One complete turn.

为什么When 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

提示Pair the first with the last, the second with the second-last…

为什么1 + 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

提示Undo the rule one step at a time from the far end.

为什么If 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

提示The prefix is the clue.

为什么Metric 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.