Team Maths Challenge toolkit

720

HintDraw every diagonal from one corner and the shape falls into four triangles.

The whyAny polygon splits into triangles from a single corner: a shape with n sides gives n − 2 of them, and each carries 180 degrees. That makes 360 for a quadrilateral, 540 for a pentagon, 720 for a hexagon. Under Relay pressure it is far quicker to recall the total for the common shapes than to rebuild it each time.

360 degrees

HintWalk right round the outside of the shape and you finish facing the way you set off — one whole turn.

The whyIt does not matter whether the shape has five sides or fifty: going all the way round is one complete turn. That gives a regular polygon its exterior angle in one step — divide the turn by the number of sides — and the interior angle follows, because the two sit on a straight line together. Teams that know this route beat teams that go the long way round through the interior total.

The Shuttle

HintThe name is borrowed from what a weaving loom does, carrying something back and forth across the cloth.

The whyQuestions come in linked sets, so a later one cannot be finished until the earlier answer is right — a single slip near the start takes the whole set down with it. Checking beats hurrying here, which is the opposite of what the clock is telling you to do.

A palindromic number

HintWord puzzlers use the same term for a phrase like "never odd or even".

The whyThey matter in a crossnumber grid because they cost so little to fill: a five-digit one is completely fixed by its first three digits, so two of the five cells come free once the others are known.

Nothing at all

HintThey are there to warm the team up before the marks begin, and the scorers simply do not write them down.

The whyThe Team Maths Challenge is a UKMT competition for teams of four: Year 8 and Year 9 in England and Wales, with at most two Year 9 students. Regional Finals run from February to the Easter break, the day opening with starters and then four scored rounds — the Group, the Crossnumber, the Shuttle and the Relay. The winning team of each region is invited to the National Final in June, and registration costs £50 per team. Because the starters carry no marks, they are the safest place in the day to settle nerves and agree how the team is going to split the work.

two

HintYou never have to look at the hundreds column, because 100 itself splits evenly into fours.

The whySo 3,516 is a multiple of 4 because 16 is, and 3,514 is not because 14 is not. The test for 8 works the same way on the last three digits, since 1000 divides by 8. Neither has anything to do with adding the digits up — that trick belongs to 3 and 9 alone, and using it on 4 is one of the fastest ways to lose a Crossnumber cell.

composite

HintThink of a photograph built by layering several pictures, or a material made from more than one substance.

The whyEvery whole number above 1 is either prime or built out of primes, and 1 is neither — it sits outside both groups because it has only one factor. That split is the reason prime factorisation works at all: each of these numbers breaks down into exactly one set of prime building blocks, no matter which route you take through the factor tree.

3.14

HintIt is a shade more than three, and the two figures after the point are the ones almost everyone memorises first.

The whyIt is the number of times a circle's diameter fits round its own edge, and it is the same for every circle in the world. It never stops and never repeats, so any written version is an approximation — 22/7 is close but not exact. Two decimal places is plenty for an estimate that tells you whether an answer is in the right region.

height; squared; length

HintEach of the three needs one extra measurement: the upright one for the flat pointed shape, what you do to r, and how far the solid stretches.

The whyThe prism fact is the most useful of the three and the least often recalled: any solid with the same cross-section all the way along — a cuboid, a cylinder, a triangular prism, an L-shaped block — has its volume found the same way, so there is one fact to remember rather than four. The triangle fact needs the perpendicular height, not a slanted side, and that is the slip a rushed team makes most.

The hypotenuse

HintIt is always the longest of the three sides, and it is the only one that does not touch the right angle.

The whyThe square on it equals the two other squares added together, so finding it means adding, and finding a shorter side means subtracting. Knowing the small whole-number triangles by sight — 3, 4, 5 and 5, 12, 13 — turns several Crossnumber and Relay questions into recall rather than arithmetic.

The total

HintA mean is what everybody would get if the whole lot were shared out fairly, so undo the sharing.

The whyFive scores averaging 12 must add to 60, and from there a missing score falls out in one subtraction. Reading an average backwards like this is one of the highest-yield moves in a timed round, because the question almost never asks for the average itself — it asks for the value that has gone missing.

The mean

HintThe one that uses every single value in its working is also the one a rogue reading can pull about.

The whyIt uses every value, so an extreme one drags it; the middle value barely notices, because a single huge reading changes the order of the list but not much else. When a question offers you a choice of average, or asks which one is misleading, this is the fact it is testing — and the honest answer to "which average should we quote?" depends entirely on whether that odd reading is a mistake or is real.