JMC 2020

Its square root

HintAny factor above that point pairs with a smaller one you would already have found.

The why143 needs only 2, 3, 5, 7 and 11 tested (11 × 11 = 121, 13 × 13 = 169), and 11 × 13 = 143 catches it. Odd numbers between 120 and 170 are a classic trap — 143, 161 and 169 all look prime and none is. Try 3 (digit sum), then 7, 11 and 13 before declaring one prime.

Its order of rotational symmetry

HintSpin it and count the matches, including the one at the end.

The whyOrder 2 means a half turn works; order 4, a quarter turn. Every shape has order at least 1. A shape can have rotational symmetry with no line of symmetry, and a rectangle's centre is the point everything turns about.

5

HintOne hop for every factor of ten in the power.

The why10⁵ = 100 000, five factors of ten, so 4.25 × 10⁵ = 425 000. Digits move; the decimal point stays put. Converting kilometres to millimetres is a six-hop version of the same move.

1

HintNot prime, not composite — and easy to forget that it counts as a factor.

The why1 is not prime, so it never appears in a prime factorisation, but it is a perfectly good positive whole number. 30 = 2 × 3 × 5 = 1 × 2 × 3 × 5. Watch for the difference between 'prime factors' and 'positive whole numbers' in the wording.

multiply

HintThe word 'of' between two fractions is an operation sign.

The whyTwo thirds of three fifths is 6/15 = 2/5. When a chain of "of"s ends in a known value, reverse it by dividing by each fraction in turn — or, faster, by multiplying by each one turned upside down.

n + 1

HintA 1 by 1 grid already needs a line on both sides.

The whyCells and lines differ by one, like fence panels and fence posts. A 5 × 5 grid has 6 lines each way; each line is as long as the grid, so the wire needed is 2 × (n + 1) × n.

Mean × number of values

HintThe average is fair shares — reverse the sharing.

The whyMean = total ÷ count, so total = mean × count. Most mean questions are really about the total: find it, subtract the parts you know, and what is left belongs to the missing values.

20

HintTwo angles side by side on a straight line make a half turn — add the expressions, then solve.

The whyAngles on a straight line add to 180°, so 4x + 5x = 180 gives 9x = 180 and x = 20. The four angles are then 80°, 100°, 80° and 100°: vertically opposite angles are equal, and each neighbouring pair adds to 180°. Once a diagram labels angles with expressions, turning it into an equation is the standard route — and with the exterior angle theorem for a triangle, those facts settle almost every angle chase in the paper.

1/8; 1/5; 3/8

HintWrite each as thousandths or ten-thousandths, then cancel.

The why0.375 is 375/1000, which cancels to 3/8. Moving the decimal point one place divides by 10, so 1.25 = 5/4 and 0.0125 = 1/80 follow at once.

It would be a multiple of 5

HintThink about which times table those last digits always belong to.

The whyA number ending in 0 or 5 divides by 5, and one ending in an even digit divides by 2. So beyond the single digits every prime ends in 1, 3, 7 or 9 — the first filter to apply before testing 3 (digit sum), 7 and 11.

It is multiplied by 8

HintEach of the three directions doubles the count of unit cubes in turn.

The whyVolume = length × width × height, so doubling one dimension doubles the volume and doubling all three multiplies it by 2 × 2 × 2 = 8. A cuboid that is 2, 2 and 5 times the size of a cube in its three directions holds 2 × 2 × 5 = 20 times as many unit cubes.

210

HintSmall primes, each used once.

The whyAny number that is a multiple of 2, 3, 5 and 7 is a multiple of 210. Adding 1 to a product of primes is the heart of Euclid's proof that primes never run out: the result cannot be divided by any prime you multiplied, so some prime is still missing from your list.

Leaves the quotient (the answer) unchanged

HintTry it with a simple pair of numbers first, like halving both before you divide.

The whyFor example, 480 ÷ 40 gives the same answer as 48 ÷ 4, because both numbers were divided by 10 first. Spotting a shared factor like this turns an awkward division into a much simpler one before you calculate — the same idea works for multiplying too.

Collect all the unknown terms onto one side

HintTreat it like tidying — gather every x together before dealing with the plain numbers.

The whyFor example, if 2n + 5 = 3n − 1, subtract 2n from both sides to get 5 = n − 1, then add 1 to find n = 6. Puzzles phrased as a chain of operations on "a number" almost always collapse to an equation exactly like this once you write out each step algebraically.

numerators

HintThink of it as one big fraction — only the top parts need adding or subtracting.

The why3/7 + 2/7 − 4/7 = (3 + 2 − 4)/7 = 1/7. Once every fraction in the calculation shares a denominator, treat the whole thing as ordinary addition and subtraction of the numerators, then simplify the result at the end if you can.

List every case in a fixed, systematic order

HintWork through the picture in the same direction each time, ticking options off as you go.

The whyJumping between different parts of a diagram makes it easy to count a pair twice or miss one. Working left-to-right and top-to-bottom (or by size, or alphabetically) keeps every case visible exactly once, and is far more reliable under time pressure than trusting you've spotted them all.

Substitute

HintPop the number straight into the equation and see if both sides balance.

The whySolving x + 3 = 15 from scratch takes longer than simply putting a candidate value (say 12) in place of x and checking 12 + 3 = 15 is true. When a paper offers several equations and one specific number, substitution is almost always faster than solving each equation in full.

Creates four congruent triangles, each a quarter of the area

HintPicture cutting a shape into identical smaller copies that all look the same as the whole.

The whyThis split (the medial triangle construction) always produces four smaller triangles congruent to each other and similar to the original, each holding exactly one quarter of its area — a pattern behind many area-fraction and dissection puzzles.

Give each distinct group its own letter, then link them with equations

HintPicture three separate boxes — only-A, only-B, and both — before writing anything down.

The whyFor instance, if some children like only reading, some only drawing, and some both, give each group its own letter (say r, d, b), turn a phrase like 'twice as many do both as do only drawing' into an equation in those letters, then combine the equations to leave one unknown to solve.

Add or subtract the lengths of the squares that touch it

HintWork outward from the smallest piece, matching up where edges meet.

The whyA 'squared rectangle' dissection is a classic puzzle: start from the smallest square, and each neighbouring square's side is fixed by the sizes of the squares beside it. Chaining these relationships outward eventually gives every side length, and multiplying the two outer sides gives the whole rectangle's area.

add; cancel

HintCombining two rows or two columns together sometimes makes the unhelpful letters disappear, leaving one solvable equation.

The whyIf two straight-line facts each total a known amount and both contain the same troublesome letter, adding the two equations together can make that letter's coefficients combine in a way that cancels or simplifies — sometimes solving for one unknown without ever isolating the others individually.

Depends only on the units digits of the two numbers

HintIgnore the rest of each number and multiply just their final digit.

The whyFor example, 23 × 47 and 43 × 27 both end in the same digit, because 3 × 7 = 21 ends in 1 either way — swapping the other digits around does not change that. This is why comparing just the final digits of two products can quickly rule out most options in a puzzle.

Write each length as a multiple of one unknown, then solve for it

HintPick a single letter for the smallest part, and express the rest of the ratio using that same letter.

The whyIf two sides are always in the ratio 3:1, call the smaller one x and the larger 3x. Once you have an equation for the total — say a perimeter equal to 24 — solving it gives x, and every length follows immediately by substitution.

empty

HintTwo occupied in a row always needs a gap before the pattern can repeat safely.

The whyThis is a classic 'no three in a row' packing constraint: with a repeating unit of two filled places followed by one empty place, exactly two-thirds of a long row can be filled while still satisfying the rule — the same idea behind maximising seating or parking under a no-three-together restriction.

Cancel the unwanted terms, leaving a solvable combination

HintAdd the two equations first and simplify before trying to solve for anything individually.

The whyIf straight-line angle facts give two equations each containing the same unhelpful letter, adding the equations can make that letter combine with the angle-sum fact from a nearby triangle, cancelling it out entirely and leaving one clean equation for the angle you actually want.