JMC 2016

The squaring

HintIndices come before multiplication in the order of operations, and both come before addition.

The whyIndices come first, then multiplication and division, then addition and subtraction: 2 + 3 × 4² = 2 + 3 × 16 = 50, not (2 + 3) × 4² = 80 and not 2 + (3 × 4)² = 146. Competition papers open with a question like this most years, and the trap is always reading left to right.

Factorise it first

HintPull the shared number out in front of a bracket.

The why3 + 6 + 9 + 12 = 3 × (1 + 2 + 3 + 4). When such a sum sits over another sum in a fraction, the bracket often cancels completely and the whole thing collapses to a single number — no adding required.

6

HintSome pass through opposite corners, others through the midpoints of opposite edges — count both kinds.

The whyA regular polygon with n sides has n lines of symmetry: for a hexagon, three join pairs of opposite vertices and three join the midpoints of opposite sides. Folding a shape exactly in half means folding along a line of symmetry, so every 'fold in half' question starts by finding those lines.

Check its last two digits

Hint100 is a multiple of 4, so the hundreds and above never matter.

The whyA number is a multiple of 4 exactly when the number formed by its final two digits is. 5678 ends in 78, which is not a multiple of 4, so 5678 is not either — even though it is even. The matching test for 8 uses the final three digits.

Multiples of 400

Hint1900 was not one; 2000 was.

The whyThe rule: a leap year is a multiple of 4, unless it is a multiple of 100, unless it is also a multiple of 400. So 1900 and 2100 are ordinary years while 2000 and 2400 are leap years. Time questions that span centuries depend on it.

Opposite the equal sides

HintLook where the two same-length legs point: the matching angles are at their far ends.

The whyThe equal angles are opposite the equal sides, never between them. Spotting an isosceles triangle in a diagram gives you an equal pair of angles for free, and combined with the angle sum of 180° that often solves the whole question.

square

Hint1 + 8 = 9 and 1 + 8 + 27 = 36: what kind of numbers are those?

The why1 + 8 + 27 + 64 = 100 = 10², and in general the sum equals (1 + 2 + … + n)². The pattern is worth knowing as a fact; proving it is a lovely investigation.

Looks the same after a half turn

HintSpin the page through 180° and compare.

The whyA shape can have rotational symmetry without any line of symmetry — the letter S is the classic example. In grid-colouring questions, order-2 symmetry means every cell must match the cell opposite it through the centre. Only a grid with an odd number of rows and an odd number of columns has a middle cell that maps onto itself, so only there is one cell free to be any colour; in a 4 by 4 grid every cell is paired with another.

18

HintThe gap between the shares covers the difference in parts, so work out what a single part is worth first.

The whyRatio questions arrive in three flavours: you are given the total, one share, or the difference. Here 7 − 3 = 4 parts are worth 24, so one part is 6 and the shares are 18 and 42, totalling 60. Naming what one part is worth turns every flavour into one multiplication. Beware also that one share as a fraction of the whole has the TOTAL of the parts underneath: in 2 : 7 the first share is 2 out of 9, not 2 out of 7.

0.025

HintScale the fraction so its denominator becomes 1000.

The why1/40 = 25/1000 = 0.025. Any fraction whose denominator divides a power of ten converts this way: 1/8 = 0.125, 1/20 = 0.05, 3/50 = 0.06.

3600; 1440; 8760

HintMultiply the everyday counts together, one step at a time.

The whyEach is a product of the familiar units: 60 × 60, 24 × 60 and 365 × 24. 'How long is this?' questions reward converting step by step rather than all at once — and a leap year has 8784 hours.

Multiply the counts

HintTwo options followed by three options gives six routes through the tree.

The whyIndependent choices multiply: 2 shirts, 3 pairs of trousers and 4 hats give 2 × 3 × 4 = 24 outfits. Add only when the options are alternatives to each other, never when the choices happen one after another.

Work out just the final digit

HintOnly the ones column decides it — find that one figure and match it to an option.

The whySubtracting or adding within the last column of a sum determines the final digit of the result, and that's often enough to eliminate every wrong option in a multiple-choice list — even before you've worked out the full number. It works best when a carry or borrow only needs tracking one column deep, and it's no substitute for full working when the options share a units digit.

Part of one whole share

HintExpress the leftover using the divisor as the denominator, not by rounding it off.

The whySharing 250 sweets into bags of 40 gives 6.25 bags — the .25 isn't an error, it's exactly a quarter of one more bag, since dividing by 40 always writes the remainder over 40 as the denominator. Competition and exam questions often want that precise fractional value rather than a number rounded up or down, so check what the question is actually asking before you round.

bridge; iterate

HintBreak the big problem into smaller, checkable pieces — one step, or one landmark, at a time.

The whyTwo techniques share this idea: bridging a big subtraction through the nearest round number (rather than subtracting straight across a long span) breaks one hard sum into two easy ones, and building a sequence term-by-term from a repeating rule — each new term made from the ones before it — avoids needing a closed formula. Both trade one big leap for several small, checkable steps.

Add or subtract half the difference from half the total

HintPicture two equal piles, then move enough between them to create the gap.

The whyIf two amounts sum to S and differ by D, the larger is S/2 + D/2 and the smaller is S/2 − D/2 — split the amount evenly then shift each side by half the gap between them. It turns a two-unknown word problem into pure arithmetic, with no equation needed.

Scale the given ratio by the same factor

HintFind what one of the stated units is worth, then multiply by however many you actually have.

The whyTurn '3 kg ≈ 6.6 lb' into a rate — 1 kg ≈ 2.2 lb — then multiply that rate by whatever new mass you're converting. This unitary method works for any pair of quantities that scale together: recipes, exchange rates, speeds.

Write out every case and check it

HintWork through the possibilities systematically instead of guessing which one looks right.

The whyWhen there are only a handful of positions and operations to try, it's faster and safer to write out every case and check it against the question than to search for a shortcut or trust a guess. Skipping just one case can make you wrongly rule out an answer that was actually achievable.

Whichever piece has the fewest valid options

HintFind the item that has the least room to manoeuvre, and lock it in before anything else.

The whySome items in a constraint puzzle can only go in one or two places because of the rules, while others have many choices. Pinning down the most restricted item first usually forces the rest of the puzzle into place with far less trial and error.

A carry is coming in from the column before

HintRemember an extra 1 often slips in beside it, on top of what you already see.

The whyColumn addition sometimes produces a result that looks impossible until you remember a carry from the previous column adds an extra 1 (or more) before the column's own digits are combined. Spotting where a carry MUST occur — because the visible digits alone can't produce the given answer — is often the key move that pins down an unknown digit in these puzzles.

multiple

HintChoose your letter so the fraction divides it exactly, with nothing left over.

The whyIf a problem says a fifth of some amount, let the unknown be 5 times a smaller unit (e.g. 5n objects) rather than n objects itself — then a fifth is exactly n, with no fraction to carry through the rest of the working. Choosing a variable that's a multiple of the denominator is a standard way to keep an algebraic solution fraction-free.

Semi-perimeter

HintIt's simply the total distance around the shape, cut in half.

The whyThe semi-perimeter, usually written s, is half of a shape's total perimeter. It's the key ingredient in formulas like Heron's formula for a triangle's area and Brahmagupta's formula for a cyclic quadrilateral's area — both build the area from s and the shape's side lengths.

Find the simple shape's area and subtract the missing part

HintComplete the awkward outline into a figure you already know how to measure, and take away what shouldn't be there.

The whyComposite-area problems often become easy once you spot that the shape is 'a rectangle minus a triangle' or 'a square minus a corner' rather than something with no known formula. Work out the area of the larger, complete shape, then subtract the piece that was removed — using the triangle-area formula (half the base times the height) for any missing triangular corner.

Whichever edges sit on the inside joins

HintOnly the outer boundary counts — any edge glued to a neighbour disappears from the outline.

The whyEvery edge used to join two copies together becomes an internal seam, not part of the final perimeter. To make the perimeter as large as possible, choose joins that hide the SHORTEST edges internally, leaving the longest edges on the outside boundary — and to make it as small as possible, do the reverse.

It's always an odd number

HintCovered squares come in twos each time, so taking an even amount away from an odd total can't leave an even amount.

The whyWhatever pattern the tiles make, the covered squares always total an even number (two per tile). Subtracting an even number from a fixed total preserves whether that total is odd or even — so on a board with an odd number of squares in total, the leftover uncovered squares must also be odd, ruling out any even option immediately. This kind of parity argument is a powerful shortcut in tiling and covering problems generally.