JMC 2022

Half the total

HintGoing round one way from the first to the second covers exactly the same distance as going the other way.

The whyIf 20 people stand in a ring, number 4 faces number 14. Turn it round and the gap between two opposite numbers tells you the whole ring: opposite numbers 5 and 17 differ by 12, so there are 24 people. Draw a small ring of six or eight to convince yourself before trusting the rule.

Remainder on dividing by 7

HintWhole weeks change nothing — only the days left over matter.

The whyDays of the week cycle every seven days, so 50 days on is the same weekday as 1 day on, because 50 = 7 × 7 + 1. The same idea handles units digits, coloured beads on a repeating string and "what is the 100th letter of this pattern?" — find the cycle length, then the remainder.

Count each size separately, then add

HintThe tiny ones, the biggest one and everything in between — one group at a time.

The whyA grid hides more squares than the little cells: in a 3 by 3 grid there are 9 unit squares, 4 of side two and 1 of side three — 14 altogether. Fix a size, count that size, move up. The systematic count is the whole answer.

The combined total

HintLook for something that moves the same way every round, win or lose.

The whyLook for what changes predictably. If a win adds 5 and a loss removes 2, the two piles together rise by exactly 3 each round, so the change in the grand total tells you how many rounds were played before you know who won any of them. Setters build many "how many rounds?" questions on this.

Exactly one

HintTry both being honest, then both being dishonest — neither works.

The whyIf both were truthful, each would be calling a truthful person a liar. If both were lying, each would be correctly describing a liar — so telling the truth. Either way it collapses, leaving one of each. That is often enough to test every other statement in the puzzle without knowing which of the two it is.

vertical

HintIs the mirror line lying flat like the floor, or standing like a door?

The whyA vertical mirror reverses the order of things across it: the digits of 4763 read 3674 in the mirror, with each digit itself flipped. A horizontal mirror leaves the order alone and turns each digit upside down instead. Deciding which line the mirror is on comes before anything else.

shared

HintScale each ratio until the middle matches.

The whyScale each ratio up until the middle quantity matches — usually to the lowest common multiple of its two values. With a : b = 2 : 3 and b : c = 4 : 5, take b to 12: a : b : c = 8 : 12 : 15. A total that must be a whole number is then a multiple of the sum of the parts.

The weight of a quarter of the paint

HintThe pot itself is in both weighings.

The whyThe container appears in both readings, so subtracting removes it and leaves pure paint. If the pot is 7 kg half full and 4 kg a quarter full, a quarter of the paint weighs 3 kg, the empty pot weighs 1 kg, and the full pot is 1 + 4 × 3 = 13 kg. Two readings of a container are almost always this move.

Multiplying by its reciprocal

HintDividing by a half doubles — which multiplication does the same job?

The whyDividing by 2/3 is the same as multiplying by 3/2. Nested divisions such as a ÷ (b ÷ c) become a × c ÷ b once you apply the rule from the inside out. Work from the innermost bracket and the fractions tidy themselves.

0.375

HintBetween a quarter and a half.

The whyThe eighths as decimals should be automatic: 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875 — and the quarters 0.25 and 0.75 sit between them. Adding a decimal to a fraction is only easy once both are in the same form.

0.75

HintHow much of the original is left after a quarter has been taken away?

The whyA percentage change is a multiplier: down 25% is × 0.75, up 10% is × 1.1. Two changes in a row multiply together — up 10% then down 25% is × 1.1 × 0.75 = × 0.825, a 17.5% fall overall — and that single multiplier is what links the final price to the original.

1000; 1440; 168

HintScales first, then multiply the clock by the calendar.

The whyUnit facts sit under many early JMC questions: 24 × 60 = 1440 minutes in a day, 24 × 7 = 168 hours in a week, so a week holds 1440 × 7 = 10 080 minutes. Write the conversion down first.

Multiplication

HintThink BIDMAS — one of the two operations always outranks the other, whatever order they appear in the expression.

The whyTwo expressions built from the same digits can give very different totals depending on where the × sign sits, because BIDMAS always evaluates multiplication before addition or subtraction. 3 + 4 × 2 = 11, not 14 — always resolve the × first, then add. This is why comparing several similar-looking expressions needs real evaluation, not a glance at the digits.

The unknown number cancels out of the working

HintTry it with algebra: call the mystery value x and see what happens to the x terms as you go.

The whyRepresent the mystery number as x and carry it through every step. If the steps are 'add six, treble it, subtract nine, divide by three, subtract the original number', you get x → x+6 → 3x+18 → 3x+9 → x+3 → 3 — the x has vanished by the last step, leaving a fixed number whatever x started as. Setting up algebra like this is the fastest way to see why a puzzle's answer never changes.

0 or 5

HintCheck only the units digit — nothing else about the number matters for this test.

The whyQuick divisibility checks save time: divisible by 2 when the last digit is even, by 5 when the last digit is 0 or 5, and by 10 when it's 0. These tests only ever look at the final digit, never the whole number, which is why they're so fast to apply when checking several long numbers under time pressure.

360

HintIt's the same total as one full turn.

The whyAngles around a single point always sum to 360°, whether they come from two shapes, three, or a mix of shapes and straight lines. Combined with an isosceles triangle's equal base angles, this single fact is often enough to find an unlabelled angle buried inside a diagram of overlapping shapes.

The same as multiplying all three fractions together first

HintTurn the words 'of' into the matching arithmetic symbol, each time you see it.

The whyApplying three fraction operations in a row — say two-thirds of a half of three-quarters of a number — means (2/3) × (1/2) × (3/4) × the number; multiply the fractions first, then apply the single combined fraction to the number, or work through them one step at a time. Either order gives the same result, but combining first is usually quicker.

It must itself be a multiple of n

HintWhatever you leave out, the numbers you keep have to add up to something that divides evenly by n.

The whySuppose ten numbers must be split into two groups that are each a multiple of 3, after quietly dropping one or two of them first. The numbers you keep have to sum to a multiple of 3 — so instead of testing every possible split, check which numbers can be dropped so the leftover total is already a multiple of 3, then split what's left however you like. That turns a huge search into a quick divisibility check.

The big square's area minus the small square's area

HintWhatever's left over is what remains after removing the missing piece.

The whyComposite shapes made by removing one shape from inside another always work the same way: find the area of the whole outline, find the area of the missing piece, and subtract. This is quicker than trying to work out the remaining shape’s area directly by splitting it into rectangles, especially once several pieces are involved.

Add the two, then subtract the overlap's perimeter once

HintEdges along the shared boundary get counted twice if you simply add the two big shapes' totals together — take that extra length back off.

The whyWhen two shapes overlap, adding their two perimeters counts the shared boundary's edges twice — so subtract the overlapping region's own perimeter once to correct for that. This composition trick turns an oddly-shaped combined outline into simple addition and subtraction of known perimeters.

180°

HintIt's the same total as angles on a straight line.

The whyWhenever a line crosses a pair of parallel lines, the angles trapped between the parallels on the same side of the crossing line are co-interior (allied) angles, and they always sum to 180°. Diagonally opposite angles inside a parallelogram are equal for a different reason — the shape’s rotational symmetry — so a single diagram can use both facts at once to work out several unknown angles.

As 3n and n, using the same n for both

HintPick one letter for the smaller share, then scale it up for the larger one.

The whyWriting the two quantities as 3n and n (rather than two separate unknowns) keeps their ratio locked in throughout the algebra, which matters when a word problem changes one quantity but not the other — for example, some of one amount is used up while the other stays fixed. The single unknown n then falls out of one equation built from that change.

They differ by exactly half the original gap between the totals

HintWork out how far apart the two sums start, then think about what one swap needs to close that gap.

The whyIf group A's total is 10 more than group B's, swapping a number out of A for a number into A must lower A by 5 and raise B by 5 to meet in the middle — so the number leaving A has to be exactly 5 more than the number entering it. In general the swapped numbers must differ by half of the starting gap between the two totals.

Three edges vanish at that vertex and the new triangle supplies three replacements

HintCount what disappears at the corner, then count what the brand new flat surface adds — compare the two totals.

The whyThe cut plane here runs through vertices the cube already had, so the three edges meeting at that corner disappear whole and the new triangular face contributes exactly three fresh edges: 12 − 3 + 3 = 12. Don't turn that into a general rule. The usual 'trim a corner' cut goes through brand new points partway along those three edges, so the original edges survive (just shorter) and the triangular cut face adds three more — a cube then jumps from 12 edges to 15. On any 'shape after a cut' question, first ask whether the cut passes through existing vertices or makes new ones; that single detail decides whether the total holds steady or climbs.

Pascal's triangle: 1, 4, 6, 4, 1

HintAdd the two directly below, one row after another — the multipliers grow into a very famous number pattern.

The whyBecause each cell adds its two neighbours from the row below, the very top of a 5-row pyramid ends up equal to a weighted sum of the bottom row using the weights 1, 4, 6, 4 and 1 — exactly the coefficients you'd get from expanding (a + b)⁴. Spotting this shortcut avoids grinding through every intermediate row of a tall pyramid by hand.