Count the structure, not the things: combinations and logic
The Short Answer
These questions ask how many combinations are possible, or use clues to fill in a grid. Listing everything stops being practical once the numbers grow, so write down enough of the list to see how it is built — then count the structure instead of the items. Fix one thing and hold it still while you vary the others.
What are these questions?
Two kinds. Counting — how many outfits, how many fixtures, how many routes. And grid logic — a set of clues, and a grid to fill in from them.
Both were doable in Year 4 by writing everything out. That is the thing that changes.
Why listing stops working
A question about forty-two fixtures is not going to let your child list forty-two things in the time available.
So the skill changes. Write down enough of the list to see how it is built — then count the structure instead of the things.
Fix one thing and hold it still
When to halve
This is the distinction that decides the harder counting questions.
Seven teams, home and away. Each of the seven plays the other six, twice — once at each ground. 7 × 6 = 42. A playing B at home and B playing A at home are genuinely different fixtures, so nothing is double counted.
Seven teams, meeting once only. Now A-versus-B and B-versus-A are the same match. Every fixture in the 42 has been counted twice. 42 ÷ 2 = 21.
Grid logic
A set of clues and a grid — who owns which pet, who sat in which seat.
The move that matters: mark what each clue rules out, not only what it confirms. Sam does not have the dog eliminates one box and confirms nothing, and children who are hunting for confirmations skip past it.
Then go through the clues a second time. The second pass frequently resolves things the first could not, because the grid has more in it by then — exactly as in a word grid.
Where do pupils lose marks?
Trying to list everything and running out of time.
Listing at random, which produces repeats and gaps.
Forgetting to halve when order does not matter.
Using clues only to confirm, never to eliminate.
Try it yourself
Write the first few of the list in a fixed order, see how it is built, then count the structure.
- Three tops and four pairs of trousers. How many outfits?
- Seven teams each play each other once, home and away. How many fixtures?
- Seven teams each play each other once only. How many fixtures?
- Six people each shake hands with everyone else once. How many handshakes?
- Why does the answer halve between questions 2 and 3?
- 12. Each of the three tops goes with each of the four trousers, so 3 × 4.
- 42. Each of the seven teams plays the other six twice — once at home and once away — which is 7 × 6.
- 21. Now each pair meets only once, so every fixture in the previous answer has been counted twice. 42 ÷ 2.
- 15. Six people, each shaking hands with five others, is 30 — but every handshake involves two people and has been counted twice, so 15.
- Because in the home-and-away version, A playing B and B playing A are two different fixtures. When they meet only once, those two entries describe the same match, so every fixture was counted twice and the total halves.
How much practice does this type need at home?
Ask counting questions about real things. How many different lunches from three sandwiches and two drinks? How many matches if everyone in the family plays everyone else at cards?
The second one is the good question, because your child will list them, count some twice, and get it wrong — and then work out why. That discovery is worth considerably more than being told about halving.
Free: a Maths workbook
Questions across the Maths question types, with the method printed at the top and full answers at the back. One workbook for the whole strand, not a sheet per question type.
Frequently Asked Questions
When do you multiply and when do you halve?
Multiply when each item of one set pairs with each item of another and the two sets are different — three tops with four trousers is 3 × 4. Halve when the items are pairing within one set and the order does not matter, because A with B and B with A are the same thing counted twice. Handshakes halve; outfits do not.
What does 'fix one thing and hold it still' mean?
List everything that goes with the first item before moving on. All the outfits with the red top, then all with the blue, then all with the green. That produces a list in a fixed order, which is what makes it complete — and after two groups your child can usually see the pattern and count the rest without writing them.
How do the grid logic questions work?
You get a set of clues and a grid to fill — who owns which pet, who sat where. The method is to mark what each clue rules out rather than what it confirms, because a clue such as 'Sam does not have the dog' eliminates one box and confirms nothing. Work through the clues twice: the second pass often resolves things the first could not, because the grid has more in it.
Why not just list everything?
Because a question about forty-two fixtures will not let you list forty-two things in the time available. Listing works in Year 4 and stops working here, which is the actual change this topic represents. Writing the first five or six is still worth doing — but to see how the list is built, not to finish it.
Sources
- GL Assessment — 11+ familiarisation materials and question type descriptions Published 2025.
- Clever Clogs Academy — our own Year 4 and Year 5 lesson records, Winslow Published September 2025 to July 2026.
We are not affiliated with, endorsed by or connected to GL Assessment or Buckinghamshire Council. Test arrangements change; always check the council's own pages for the current year's dates.
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