Every card in Team coding toolkit

Recursion

HintRussian dolls do it; so does a mirror facing another mirror.

WhyIt needs two parts: a base case small enough to answer outright, and a step that shrinks the problem towards that base case. Miss the base case and the program never stops. Term-to-term sequences in maths are the same idea — each term is defined from the one before, and the first term is the base case.

Whole-number division

HintThe kind that keeps what is left over as a separate answer, rather than trailing off into a decimal tail.

WhyMost languages offer two divide operators: one gives 3.5, the other gives 3 with a leftover of 1. Timed rounds lean on the second constantly — filling boxes, counting complete rows, wrapping a clock past 12. Reaching for the wrong one is a favourite way to lose an otherwise correct solution.

The nth term rule

HintThe position-to-value formula — the one written with n in it.

WhyA term-to-term rule ('add 4 each time') needs every step before the one you want; a position-to-value rule ('4n + 1') goes straight there. For a linear pattern the multiplier of n is the common difference, and the constant is whatever makes the first value come out right. In a timed round, replacing a loop of 500 steps with one line of arithmetic is often the whole trick.

triangular

HintThink of the shape a rack of snooker reds makes, then use the adjective form of that shape.

WhyThey are the running totals 1, 1+2, 1+2+3 …, so the nth one is n(n+1)/2. That formula replaces a whole loop, which is why it turns up in timed rounds so often.

Its square root

HintIf a divisor bigger than this existed, its partner would be smaller and you would have met it already.

WhyDivisors come in pairs that multiply to the number: for 36 they are 2 × 18, 3 × 12, 4 × 9, 6 × 6. Past 6 the pairs simply repeat the other way round, so nothing new can appear. Stopping there turns a check on a million from a million steps into a thousand — which is the difference between a solution that finishes inside the round and one that does not.

Their lowest common multiple

HintYou meet it as the partner of HCF; buses at a stop setting off again at the same moment is the classic question.

WhyFlashing lights, buses at a stop, gear teeth lining up — all the same question. For cycles of 4 and 6 the answer is 12, not 24: multiplying the two cycles always gives a time that works, but usually not the earliest one. Divide the product by the HCF to get it right, or in code walk through the multiples of the larger cycle until one fits the smaller.

The hypotenuse

HintGreek for 'stretching under'; it is the side a ladder makes when it leans against a wall.

WhyPythagoras gives its length: square the two shorter sides, add them, take the root. Programs use it constantly for the straight-line distance between two points on a grid, where the difference across and the difference up are the shorter sides. Comparing squared distances instead of rooted ones is a common speed trick, because the ordering comes out the same either way.

close

HintExact matching fails here — what you want is a tolerance, a tiny gap you are willing to accept.

WhyIn most languages 0.1 + 0.2 does not equal 0.3 exactly; it comes out a hair over, because binary cannot write 0.1 any more neatly than decimal can write a third. The fix is to check the difference is smaller than some tiny amount. Whole numbers have no such problem, which is one reason experienced coders stay with integers wherever they can.

BIDMAS

HintThe six-letter word your maths teacher writes on the board; brackets come first in it.

WhyBrackets, Indices, Division and Multiplication, then Addition and Subtraction. Languages agree with the maths, so 2 + 3 * 4 is 14 rather than 20. Where they can differ from each other is what a bare minus sign does in front of a power — when in doubt, add brackets: they cost nothing and settle the argument.

factor; multiple; prime

HintAll three are Year 7 words about how one whole number sits inside another; the last is the special case with the fewest.

WhyThese three words carry most of the number theory a coding round needs. Note the two edge cases a test program will throw at you first: 1 is not prime, because it has only one divisor, and 2 is the only even prime there is.

In pairs on one computer

HintNot alone and not in a big group — and there is only a single keyboard between them.

WhyRound 1 runs for 40 minutes, with entrants working in twos — or solo where a school has to — on a single machine, and a high score carries them through to Round 2, which lasts an hour and allows teams of up to three. The challenge is run by The Perse School in Cambridge, is free to enter, and is open to UK pupils in Year 11 and below. Solutions may be written in Python, C++, C#, Java, JavaScript or Visual Basic.Net.

40

HintIt is shorter than Round 2, which runs for a full hour.

WhyRound 1 is forty minutes, Round 2 sixty. Judging roughly how long a problem will take before you start it is a real part of the skill — an unfinished perfect answer scores nothing at all.