Achievers Section · one of four sections

The Achievers Section, and why it decides the rank

Five questions, worth two or three marks each, and the part of the paper that separates a good score from a rank. These are not longer questions. They are questions with a door in them: see the idea and it is short, miss it and there is no route through.

We have written 150 Achievers Section questions by hand across classes 3 to 8.

Try two of them

One from a class 4 paper and one from class 7, so you can see how the same section grows. Have a go before opening the answer.

5 pointsClass 4Question 32

Ria writes the numbers 1 to 6 in the six circles of a triangle, one number in each circle: three circles at the corners and one on the middle of each side. The three numbers along every side add up to 10. Which three numbers are at the corners?

Show the answer and how to see it

Answer: 1, 3 and 5

Each corner number is counted on TWO sides, and each middle number on one. All three sides together make 30, and 1 to 6 add up to 21, so the corners add up to 30 take away 21, which is 9. The only three of the numbers adding to 9 that work out are 1, 3 and 5: the sides are then 1 with 6 and 3, 3 with 2 and 5, and 5 with 4 and 1, each making 10.

Why a child picks 2, 4 and 6: These add up to 12, but the corners must add up to 9 for the sides to make 10 each.

Sit the whole paper this came from

5 pointsClass 7Question 46

For how many pairs of whole numbers a and b, where a is not larger than b, is one over a plus one over b equal to one sixth?

Show the answer and how to see it

Answer: 5

Clear the fractions and the equation rearranges into (a minus 6) times (b minus 6) equals 36. Every way of writing 36 as a product of two whole numbers gives one pair, so the count is the number of factor pairs of 36.

Why a child picks 9: This is how many factors 36 has. Each PAIR of factors is one solution, and pairs are counted once because a is not larger than b.

Sit the whole paper this came from

What this section actually tests

1Invariants: something that does not change no matter what you do, which turns a long search into one observation.
2Extremal cases: the largest or smallest something can be, found by pushing everything else to its limit.
3Impossibility: showing a thing cannot be done at all, usually by a parity or counting argument.
4Working backwards from the end state, when going forwards branches too fast.
5Counting cleverly: counting the same thing two ways, or counting what is hidden instead of what shows.

The method, in order

Worth saying out loud the first few times. The order matters more than the speed.

1. Look for what stays the same
If a move changes the situation, ask what it leaves alone. A quantity that never changes usually answers the question by itself.
2. Try the smallest case
Do the same problem with 3 instead of 20. The pattern that appears is nearly always the whole answer.
3. Ask whether it is even possible
When a question asks for a number and nothing works, check parity or a colouring. Some Achievers questions have no answer by design, and knowing that is the answer.
4. Count the other side
If what you want is hard to count, count what is left over and subtract. This is the whole trick in a surprising number of these.

The four ways children lose marks here

These are not guesses. Each one is a wrong option we deliberately built into real questions, which is why every one of them feels reasonable at the time.

Treating a long question as a hard one

The fix: Length is not difficulty. Look for the one observation that collapses it before starting any calculation.

Starting at the beginning when the end is fixed

The fix: If the final state is given and the start is not, work backwards. Forwards branches, backwards usually does not.

Assuming an answer exists

The fix: Some of these ask whether something can be done, and the honest answer is no. Trying every case and failing is evidence, not defeat.

Spending the same time per mark as elsewhere

The fix: These are worth two or three times an ordinary question. Leave time for them, and do not leave them until the last five minutes.

How many Achievers Section questions are on the paper

Class	Achievers Section	Questions on the paper
Class 3	5	35
Class 4	5	35
Class 5	5	50
Class 6	5	50
Class 7	5	50
Class 8	5	50

Section names and the shape of the paper follow the Science Olympiad Foundation olympiad blueprint. We are not affiliated with them, these questions are our own, and the official pattern is theirs to confirm.

Questions parents ask

How many marks is the Achievers Section worth?

Five questions in every class. In our classes 3 and 4 papers they carry 2 marks each, 10 of the 40 marks on the paper. In classes 5 to 8 they carry 3 marks each, 15 of 60. Either way it is a quarter of the paper from a seventh of the questions.

Is it worth attempting if my child finds them hard?

Yes, and earlier than feels comfortable. A child who attempts none of them has capped their score at three quarters before the paper starts.

What makes an Achievers question different from a hard ordinary one?

An ordinary hard question takes several correct steps. An Achievers question needs one idea, and without it no number of steps gets there. That is why practice on them looks different: the aim is to collect ideas, not speed.

How do you decide whether a question really belongs here?

We put every one of ours to two independent language models with the stem alone and ask whether there is an insight a solver could miss and then be stuck. Questions that come back as routine get replaced. That check has rewritten 34 of ours so far.

Can a child be taught these, or is it talent?

They can be taught, but as ideas rather than methods. Invariance, parity, extremal cases and working backwards are perhaps eight ideas in total, and a child who has met each one a few times starts recognising which is in play.

Practise it inside a whole paper

One section at a time is how you fix a weakness. A full timed paper is how you find out whether it is fixed.

Sit a full paper SOF IMO preparation

The other three sections: Logical Reasoning, Mathematical Reasoning, Everyday Mathematics.