JMC 2016

The multiplication

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

为什么Multiplication 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

提示Pull the shared number out in front of a bracket.

为什么3 + 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

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

为什么Folding 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

提示100 is a multiple of 4, so the hundreds and above never matter.

为什么A 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

提示1900 was not one; 2000 was.

为什么The 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

提示The word comes from Greek for equal legs.

为什么The 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

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

为什么1 + 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

提示Spin the page through 180° and compare.

为什么A 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

提示The whole, not the other share.

为什么In 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

提示Scale the fraction so its denominator becomes 1000.

为什么1/40 = 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

提示Clock, calendar, planet.

为什么Multiplied 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

提示Two options followed by three options gives six routes through the tree.

为什么Independent 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.