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.

Each size of square

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.

swapped

HintLook at a word in a mirror.

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; 60; 7

HintScales, clock, calendar.

The whyUnit facts sit underneath many early JMC questions: grams before dividing a kilogram, weeks before counting days. Write the conversion down first and the arithmetic stays honest.