JMC 2024

2n + 1

提示Draw a square of dots, then add one more row and one more column — count the new dots.

为什么The gaps between squares are the odd numbers: 1, 3, 5, 7, … so 25 and 36 differ by 11, and 100 and 121 by 21. If someone’s age is a square now, the next square age is 2n + 1 years away.

Left to right

提示Neither one beats the other, so read the line the way you read a sentence.

为什么12 ÷ 3 × 2 = 4 × 2 = 8, not 12 ÷ 6 = 2. The same holds for addition and subtraction: equal rank, left to right. Brackets and indices come first; then the multiply-and-divide pair; then the add-and-subtract pair.

A quarter

提示The two diagonals are lines of symmetry, so every piece matches every other.

为什么Four congruent right-angled isosceles triangles. In a large square split into four small ones, a small square’s diagonal cuts off an eighth of the big square, and pairs of tiny corner triangles can be reassembled into a larger piece. Shaded-fraction questions are solved by cutting everything into the same small triangle.

Pretend they are all one kind

提示Start from an extreme, then swap.

为什么Say every head belongs to a hen and count the legs. Each swap of a hen for a sheep adds exactly two legs, so the shortfall divided by two is the number of sheep — and starting with all sheep works just as well. The same move solves coins, tickets and any two-type mixture where each type has a fixed value.

n + 1

提示The cut crosses the boundary of a convex shape exactly twice, and adds one new edge to each piece.

为什么The cut enters and leaves through two edges, splitting each of them, and contributes one new edge of its own — so a piece can gain at most one side. From a pentagon you can cut off anything from a triangle to a hexagon, but never a heptagon. (A dented, non-convex shape can be crossed more often, which is why the word convex matters.)

against

提示Each goal shows up in two teams’ rows.

为什么A goal scored by one team is a goal conceded by another, so the two columns must balance across the whole table — and the wins must equal the losses. If one team has lost a single match and conceded three more than it scored, that lost match was by three goals. Results tables reward this kind of bookkeeping before any guessing.

half

提示Their two heights add up to one side of the square.

为什么Each triangle’s area is half its base times its perpendicular height. The two bases are opposite sides of the square and the two heights add to the square’s side, so the areas add to half the square — whatever the point. That fact settles "find the other triangle’s area" in one line.

The smallest number in both times tables

提示Not the product — the two may share a factor.

为什么For 4 and 6 it is 12, not 24. Anything that must divide by both 4 and 6 is a multiple of 12, and anything that leaves the same remainder with both repeats in steps of 12 — so the candidates sit just above multiples of 12.

A rhombus

提示Its diagonals cross at right angles, like a square’s.

为什么Because all four sides match, any triangle made from two sides of a rhombus is isosceles, and because opposite sides are parallel, neighbouring angles add to 180°. Both facts get used together in angle chases built on a rhombus.

10000

提示Both edges of the square stretch, not just one of them.

为什么Each side of a square metre is 100 cm, so the area is 100 × 100 = 10 000 cm². Lengths scale by 100, areas by 100 × 100 and volumes by 100 × 100 × 100, which makes a cubic metre 1 000 000 cm³. "How many tiles cover the floor?" and "how many sugar cubes fill the box?" are this rule in disguise — and it runs backwards too, so halving every length quarters the area and leaves an eighth of the volume.

1000000

提示A thousand small ones in a metre, and a thousand metres in the big one.

为什么Milli means a thousandth and kilo means a thousand, so a kilometre is 1000 × 1000 = 1 000 000 mm. Estimation questions about tiny creatures or thin coins hinge on conversions like this, then on rounding: something a few millimetres long fits into a metre a few hundred times, which is enough to pick from widely spaced options.

20; 12.5; 33

提示Share 100 by five, by eight, and by three.

为什么Mixture wording is the trap: one part squash to three parts water is a quarter squash, not a third, because the whole drink is four parts. "What fraction of all the pens" questions are the same idea in reverse.

Simplify every expression first, then count only the distinct results

提示Working each one out often reveals repeats you wouldn't expect from the look of them.

为什么Work out each expression on its own, then look for repeats among the results — two expressions that look unlike each other can land on the same number, and repeats don't add to the count. This trick shows up whenever a question asks 'how many different values/outcomes/answers' rather than 'how many expressions'.

(n − 2) × 180°

提示Split the polygon into triangles from one vertex — count how many you get.

为什么Any polygon can be split into (n − 2) triangles by drawing diagonals from a single vertex, and each triangle contributes 180°. For a regular polygon, dividing this total by n gives the size of each interior angle — the move used to find angles in a regular hexagon, octagon or any other regular shape.

Solve for the sum x + y directly, not for each unknown alone

提示Look for one line in the diagram that already equals that combined length.

为什么Some diagrams are built so that a target quantity (a perimeter, a total) depends only on a COMBINATION of unknowns, never on each one by itself. Hunting for x and y individually can be much harder than spotting the one length in the diagram that already equals x + y — read that off, and the rest follows without ever pinning down x or y alone.

Only the units digits of a and b

提示Ignore the tens, hundreds and anything further left — they play no part in it.

为什么37 × 44 and 7 × 4 both end in 8, because only the last digits (7 and 4) affect the last digit of the product. This is why digit puzzles that ask for the units digit of an unknown product can be solved without knowing the full numbers — checking only a handful of candidate last-digit values instead of every possibility.

It equals their sum

提示No protractor needed — it falls straight out of angles on a straight line plus the angle sum of a triangle.

为什么This is the exterior angle theorem: the exterior angle always equals the sum of the two interior angles that are not next to it. It's a shortcut worth knowing, because angle-chasing puzzles built from isosceles triangles often need this instead of working out every angle from scratch via 180° per triangle.

Whether the result's size rules out most candidates straight away

提示A two-digit answer instantly kills any arrangement that was meant to give a single digit.

为什么If a calculation must land on a single-digit answer, any arrangement that would obviously give a two- or three-digit result can be crossed off without working it out fully. Using size (too big, too small, wrong number of digits) to eliminate candidates first turns a long trial-and-error search into a short one.

Multiply by how many shares make up the whole, then take the leftover

提示Write your first answer as an improper fraction, and convert it to a mixed number instead.

为什么Say a total is split into 4 equal cups and five-eighths of the total has been drunk. That's 5/8 × 4 = 20/8 = 2.5 cups — more than two whole cups. The extra half-cup above the first two is what's been drunk from the THIRD cup. Rescale by the number of shares first; only then does the mixed-number split make sense.

The total is fixed even though the individuals aren't

提示You don't have to identify each person by name to still pin down how many there are of each kind.

为什么In some liar/truth-teller puzzles, more than one consistent assignment of roles exists — but if every one of them gives the same overall count (say, always three liars out of five), the question 'how many are liars' has one firm answer even though 'which ones are liars' does not. Don't abandon a case-split just because the individuals stay ambiguous.

equal

提示They both describe the very same edge, just measured two different ways.

为什么Whenever a diagram gives the same length, area or angle two different ways — say 2p from one route around the shape and q + 3 from another — those two expressions must be equal, and that equation is usually the one that unlocks the whole problem. Spotting which length has been described twice is the real skill.

Scale each ratio so the shared quantity matches, then merge the two

提示If Y appears in both A:Y and Y:Z, make the Y-values equal before joining them.

为什么Say thirty of one kind balance forty of another (30a = 40b), and separately twenty of that second kind balance twenty-five of a third (20b = 25c). Scale both equations until b has the same coefficient in each, then combine them into a single ratio linking a directly to c — without ever working out actual weights.

Divide the whole equation by x to drop it to a linear one

提示Before reaching for heavy machinery, check what factor each single piece on both sides shares.

为什么It is tempting to assume the squared parts must be identical on both sides so they vanish, but often one side ends up with a multiple of the other's squared term and nothing cancels. That need not stop you. If x is a common factor of every term, and the setting guarantees x ≠ 0 (it is a length, say), dividing through by x lowers the degree by one and what looked quadratic collapses into something you finish by gathering terms. So before reaching for the quadratic formula, always ask whether an x can be stripped from the whole equation at once.

Multiply every term by the LCM of all the denominators

提示One multiplication clears all the fractions in one go, instead of tackling them one at a time.

为什么An equation like a/3 + b/4 = c/2 can be cleared in one move by multiplying every term by 12 (the LCM of 3, 4 and 2), turning every fraction into a whole number in a single step rather than combining fractions gradually. The same trick works even when the fractions come from percentages of two different subgroups that make up a total.

9

提示Ask yourself: which number multiplied by itself gives 81?

为什么Area and side length are linked by squaring and its reverse, square-rooting: side² = area, so side = √area. This move often appears as a hidden middle step — once you have the side length, you can go on to find the perimeter (×4) or use it inside a bigger area calculation.