JMC 2015

360°

HintOne full circuit brings you back facing the way you started.

The whyThis is true for a triangle, a hexagon or a fifty-sided shape: the exterior angles sum to 360° whatever the number of sides. It is often quicker than the interior-angle formula — find the missing exterior angle first, then use the straight line (180°) to reach the interior angle beside it.

Exterior angle = sum of the two opposite interior angles

HintExtend one side past a corner: the turn made there matches the pair of corners far away.

The whyIt follows from two facts you already know: the angles of a triangle add to 180°, and the angles on a straight line add to 180°. Competition setters love it because it lets you leap straight to an answer without finding every angle in the diagram.

When either is 1

HintTry a few pairs where one number is tiny.

The why1 + 7 = 8 but 1 × 7 = 7, so a 1 is the one case where the plus sign wins. With 2 and 2 the two results are equal (4 and 4); from 2 and 3 upwards, multiplying is always bigger. So when you may choose + or × to make an expression as large as possible, multiply everything — except that a 1 should be added, not multiplied.

2

HintEvery other even number has this number as a factor, so it cannot be prime itself.

The whyBecause 2 is the only even prime, every other prime is odd. That drives a family of parity arguments: odd + odd is even, so the sum of two odd primes can never be prime, and any question about sums of primes usually turns on whether a 2 is involved.

2520

HintNot 10 factorial — that repeats factors. One highest power per prime.

The why2520 = 8 × 9 × 5 × 7. Any number divisible by everything from 1 to 10 must be a multiple of 2520, so the fast check is: does it have three 2s, two 3s, a 5 and a 7 among its prime factors? Being divisible by 5 needs a last digit of 0 or 5, which alone eliminates most options.

60; 90; 120

HintThe three shapes that tile a flat floor on their own: think how many meet at a point.

The whyThese three appear together constantly: a square sitting on a triangle, a hexagon with a square on one side. Whenever shapes meet at a point, add the known corners and subtract from 360° to find the gap; whenever they meet along a line, subtract from 180°.

Only their units digits

HintThink about what the last column of the working depends on.

The whyThe last digit of a product depends only on the last digits of the factors: 7 × 6 = 42, so 4837 × 2596 ends in 2. Two products with different final digits cannot be equal, which lets you reject answer options without doing any long multiplication.

Total of areas − area covered

HintWhere two tiles sit on the same spot, that spot has been counted twice.

The whyAdding the tiles' areas counts every overlapping patch once per tile, while the covered region counts it once. The difference is exactly the overlap (as long as no spot is covered three times over). It is the same idea as a Venn diagram: add the parts, then take away what was counted twice.

A number that reads the same forwards and backwards

HintLike the words noon and level, but made of digits.

The whySuch a number can never end in 0, because it would then have to begin with 0. So whatever a last-digit test tells you about the units digit — even, or a multiple of 5 — it tells you about the leading digit as well, before you have done any arithmetic.

4

HintNeither 0 nor 1 is prime, and every even digit but one is out.

The whyThe prime digits are 2, 3, 5 and 7. Knowing them by heart speeds up any question about primes built from digits.

equilateral

HintThe angle at the centre is a sixth of a full turn, and two sides of each piece are the same length.

The whyEach central angle is 360° ÷ 6 = 60°, and the two sides from the centre are equal, so every piece is an equilateral triangle with the hexagon's own side length. Shaded-fraction questions on hexagons are usually solved by counting these six triangles — or the twenty-four smaller ones you get by quartering each.

Brick above minus neighbour

HintSums going up mean the reverse operation coming down.

The whyWorking downwards is the inverse of working upwards, so every step down is a subtraction. Fill in only the bricks you actually need: a missing bottom corner often needs just a diagonal chain of subtractions, not the whole pyramid.