IMC 2016

The digit sum

HintNot the final column — all the columns together.

The whyAdd the digits (and, if the total is still large, add again): a total that is a multiple of 3 means the number is, and a multiple of 9 means the number is. Because sums and differences of multiples of 9 are multiples of 9, the test also catches arithmetic slips.

Lowest common multiple

HintTwo events repeat on different cycles; when do they first happen together?

The whySomething that happens every 4 days and a weekday that comes round every 7 days next coincide after the lowest common multiple of 4 and 7, which is 28 days. When the two numbers share no factor, the LCM is simply their product; when they do, it is smaller.

3600

HintThink of it as minutes, then go one level finer.

The whySpeed questions that mix metres per second with kilometres per hour need this number: multiply a speed in m/s by 3600 to get metres per hour, then divide by 1000 for km/h — so the whole conversion is × 3.6. For an approximate answer, round the awkward numbers first.

720

HintCut it into triangles from one corner and count them.

The whyFour triangles from one vertex, each 180°, gives 720° — the formula 180(n − 2) with n = 6. When an angle is marked round the outside of a vertex, swap it for 360° minus it (an exterior angle proper is 180° minus the interior), then use this total.

They are parallel

HintCo-interior angles behave this way for exactly one kind of pair of lines.

The whyAngles that add to 180° between two lines and a transversal mean the lines are parallel — and if the other pair does not add to 180°, those sides are not. One pair parallel and the other not is the definition of a trapezium, which is how a quadrilateral can be classified from its angles alone.

Contains 2³ × 3

HintFactorise the divisor first, then look for every one of its prime powers.

The whyA number is a multiple of d exactly when its prime factorisation contains every prime power in d's. For 24 that means at least three 2s and at least one 3; 2² × 3² × 5 fails, however large the other powers are. The same check works for any divisor you can factorise.

Big circle minus small circle

HintTwo discs, one take-away.

The whyRing areas are a subtraction of two circle areas: π(R² − r²), which factorises as π(R + r)(R − r). Keep the π as a common factor to the end — it usually cancels or stays in the answer.

mode; median; mean

HintMost common, middle, and fair share — three different words.

The whyQuestions that give a mode, a median and a mean and ask for the smallest or largest possible value are logic puzzles built on these definitions: a mode must appear at least twice, a median splits the ordered list in half, and a mean fixes the total. Each fact forces something about the list.

Perpendicular bisectors of each other

HintTwo lines through the centre; think about the symmetry of the shape.

The whyThe diagonals of a rhombus cross at right angles and each cuts the other in half, so they carve the shape into four congruent right-angled triangles. Two circles of equal radius that cross: their centres and crossing points form a rhombus, so the line of centres and the common chord bisect each other at right angles.

Their own speed plus the walkway's speed

HintBoth motions carry the person forward at once.

The whySpeeds add, times do not: convert both to speeds (distance ÷ time), add, convert back.

k²

HintLength counts once, area twice.

The whyDoubling every length quadruples the area — for a hexagon, a triangle, or any shape at all.

Take out the common factor

HintShift a power of ten so both products share a piece, then factorise.

The whyNon-calculator papers plant these on purpose: multiplying out is slow and error-prone, but rewriting one number as the other times a power of ten turns the whole thing into (common factor) × (a simple subtraction). If a calculation looks brutal, look for what the two halves share.