JMC 2023

Exactly the same

提示Slide the two cut edges outwards until they sit on the old outline.

为什么The two new edges of the notch are just the missing pieces of the old sides, moved inwards, so the total distance round is unchanged. That means a perimeter given for an L-shape tells you the full rectangle’s length plus width at once — and the area follows from subtracting the notch.

Bigger than 1

提示How many times does the small one fit into the large one?

为什么Dividing a larger number by a smaller always gives more than 1, whatever their size — 0.6 ÷ 0.3 = 2. When comparing several expressions built from two numbers below 1, the quotient with the larger number on top is the only one that can exceed 1, so it wins without any calculating.

72°

提示Every slice at the centre is the same size.

为什么The centre of any regular polygon splits a full turn into equal slices: 72° for a pentagon, 60° for a hexagon, 45° for an octagon. The triangles from the centre are isosceles, so the interior angle of a regular pentagon is 2 × (180 − 72) ÷ 2 = 108°.

One is a multiple of 7

提示A prime cannot be split between two numbers.

为什么A prime factor of a product must come from one of the factors: 6 × 35 = 210 is a multiple of 7 only because 35 is. In a grid where the product of each row is given, the rows whose products are multiples of 5 or of 7 pin down where the 5 and the 7 must sit — that is usually the way in.

odd

提示Think about the parity of two neighbouring numbers.

为什么Consecutive whole numbers are one even and one odd; their squares are too, and even plus odd is odd. So such a sum is never divisible by 2, and it may or may not be prime — 49 + 64 = 113 is prime, 64 + 81 = 145 is not. Check the small primes by hand rather than guessing.

identical

提示There are only nine of them, and they rhyme.

为什么From 11 up to 99 the multiples of 11 are the repeated digits, so a two-digit multiple of 11 always has an even digit sum — a quick way to rule out candidates.

60

提示Clock arithmetic is not decimal.

为什么Count to the next whole hour first, then the rest: from 16:52, eight minutes reaches 17:00 and the remaining minutes go on from there. Delays and doubled journey times are just repeated versions of the same step.

Distance divided by time

提示Which quantity would you divide to compare a fast runner with a slow one?

为什么Speed = distance ÷ time, so a 400 m lap in 50 s is 8 m/s. The same relationship runs backwards: at a steady rate, distance covered in a known time gives the rate, and the rate tells you when a journey began.

(n − 2) × 180°

提示Chop the shape into triangles from one corner and count them.

为什么A quadrilateral gives 360°, a pentagon 540°, a hexagon 720°. Questions about marked angles on the outside of a shape are usually this formula in disguise: each marked angle is 180° minus an interior angle, so the outside total is a fixed number.

8

提示Remember 2 counts and 1 does not — then list them.

为什么They are 2, 3, 5, 7, 11, 13, 17 and 19. With 23 and 29 that is the ten primes below 30 — the working set for prime-square puzzles, where each side must total the same and every entry must be prime.

2

提示Start at 2³ and stop before 100.

为什么Only 27 and 64. A crossnumber clue reading "a cube" with two cells therefore has two candidates, and their digits (2, 7, 6, 4) tell you which squares and primes can fit round them.

99; 1000; 36

提示Where the columns of place value open and close.

为什么Boundary numbers appear in "largest multiple of … below" questions and in digit-sum arguments: knowing that four digits can sum to at most 36 tells you, for instance, that a four-digit number whose digit sum is a multiple of 20 has digit sum exactly 20.