JMC 2016

The multiplication

HintThe order-of-operations rule, not the left-to-right order.

The whyMultiplication and division come before addition and subtraction, so 7 − 4 × 2 means 7 − 8 = −1, not 3 × 2 = 6. 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.

4

HintFold it so the halves match — count the different folds, not only the obvious ones.

The whyFolding a square exactly in half means folding along a line of symmetry: through the midpoints of opposite sides gives a 2 : 1 rectangle, along a diagonal gives a right-angled isosceles triangle. Every 'fold in half' question starts from these two cases.

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.

Isosceles triangle

HintThe word comes from Greek for equal legs.

The whyThe equal angles sit opposite the equal sides. 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 opposite cells match; the centre cell can be any colour.

total

HintThe whole, not the other share.

The whyIn the ratio 2 : 7 the first share is 2 out of 9, not 2 out of 7. Writing the other share as the denominator is the single most common ratio slip in the paper.

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.

60; 24; 365

HintClock, calendar, planet.

The whyMultiplied together they give the number of minutes in a year — a common target in 'how long is this?' questions, where the trick is to convert step by step rather than all at once.

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.