How to Pass Logical Reasoning Tests: Questions and Answers for the 11+

What Logical Reasoning Really Tests

Logical reasoning questions look intimidating because they arrive without instructions. There is no formula to recall and no topic to revise. Your child is simply shown some numbers, shapes or symbols, and asked what comes next.

That is precisely the point. Grammar schools use these questions because they measure how a child thinks when the path is not laid out for them — and because they are difficult to coach in the traditional sense. What can be taught is a reliable method for finding the hidden rule.

Below we work through the question types that appear most often, using genuine examples of the kind found in 11+ and general aptitude papers.

Did you know? Most logical reasoning questions have only one rule — but the questions children lose marks on usually have two or three rules operating at once. Learning to keep looking after finding the first rule is one of the highest-value habits you can build.

The Core Method: Find, Test, Apply

Every logical reasoning question yields to the same three-step routine.

The middle step is the one children skip. A rule that explains the first example but not the second is not a rule — it is a coincidence.

Number Pattern Questions

Triangles and Circles With Hidden Sums

A common format shows three triangles, each with four numbers inside. In the first: 2, 15, 10 with 5 in the centre. In the second: 3, 16, 12 with 4 in the centre. The third has 14, 7 and 2 in the centre — and one number missing.

Label the positions a, b, c and d. The rule is:

• a × b = c
• c + b = d

Checking it: 2 × 5 = 10, then 10 + 5 = 15. ✓ And 3 × 4 = 12, then 12 + 4 = 16. ✓ So the missing number is 7 × 2 = 14, then 14 + 2 = 16.

Digits, Not Numbers

Some of the hardest questions ask children to stop treating a number as a single quantity and start treating it as a string of digits.

Take an octagon containing 48, 76, 59, 98, 12, 13, 14 and a gap. The rule connects each number to the one directly opposite: 4 + 8 = 12, 7 + 6 = 13, 5 + 9 = 14. So the missing value is 9 + 8 = 17.

The same trick appears in “impossible” equations:

• 23 × 23 = 25
• 25 × 25 = 49
• 27 × 27 = ?

Traditional multiplication gives nonsense. But sum the digits first: (2+3) × (2+3) = 5 × 5 = 25. ✓ Then (2+5) × (2+5) = 7 × 7 = 49. ✓ So (2+7) × (2+7) = 9 × 9 = 81.

And in a circle of eight sections — 72, 52, 43, 23, 49, 25, 64, ? — each left-hand number becomes the right-hand one via first digit to the power of the second: 7² = 49, 5² = 25, 4³ = 64, so 2³ = 8.

Tip: When a number question looks unsolvable, tell your child to try splitting the digits. It is one of the single most productive moves in logical reasoning, and it rarely occurs to children unless they have been shown it.

Sequences With Two Patterns Running at Once

Consider: 23, 11, 20, 12, 18, 14, ?

Nothing sensible connects consecutive terms. That is the clue. Split the sequence into alternating positions:

• Odd positions: 23, 20, 18, ? — subtract 3, then 2, then 1 → 17
• Even positions: 11, 12, 14, ? — add 1, then 2, then 3 → 17

Both interleaved strands are internally consistent, and the answer is 17.

A related idea appears in: 21, 35, 41, 59, 61, 83, ? Every term is built from a multiple of 11 with an alternating adjustment — 11×2−1 = 21, 11×3+2 = 35, 11×4−3 = 41, 11×5+4 = 59, 11×6−5 = 61, 11×7+6 = 83, and so 11×8−7 = 81.

This alternating-strand technique carries straight over to letter sequences, which we cover in our guide to solving alphabet sequence questions.

Matrix Questions: Check the Columns

Here is a mistake worth inoculating your child against. Look at this 3×3 grid:

Because the data is presented in rows, children look for row rules. The calculation actually runs down the columns, and it is layered: sum the digits of the number above, then square the result.

• Column 1: (3+2)² = 25, then (2+5)² = 49
• Column 2: (1+7)² = 64, then (6+4)² = 100
• Column 3: (6+3)² = 81, then (8+1)² = 81

Tip: Teach your child to check rows, then columns, then diagonals before concluding a matrix has no pattern. Presentation order is not the same as calculation order — and examiners know it.

Another grid rewards a different insight entirely:

Each row multiplies left to right by a fixed increment: ×2 in the top row, ×3 in the middle, ×4 in the bottom. The unusual part is that the missing number here is a starting value, not a calculated one: since ? × 4 = 20, the answer is 5. Children who assume the gap must always be the output of a calculation get stuck.

Shape and Spatial Questions

Visual logic questions follow the same find–test–apply routine, but several rules typically operate together.

Tracking Multiple Moving Elements

A four-square sequence might contain a smiley face alternating between dark and light backgrounds, a hollow circle moving alternately clockwise and anticlockwise with an increasing step size, and a triangle moving clockwise with doubling steps. Handle them one element at a time:

1. Start with the simplest element and use it to eliminate options.
2. Move to the next element and eliminate again.
3. Use the hardest element only to separate the final two candidates.

Even when a rule eliminates nothing (all four options showed a dark smiley face), confirming it costs seconds and builds confidence in the rules that do matter.

Merging Two Grids Into a Third

Where column three is formed by combining columns one and two, the merge rules are usually: same colour cancels to white, different colours produce dark. Establish the rule from the completed row, then apply it to the incomplete one. Our guide to odd one out questions uses a similar systematic feature-by-feature approach.

Shape Fitting — Count First

When asked which set of three pieces fills a target shape, the professional move is arithmetic, not visual. Count the squares in the target — say 13 — then count the squares in each option. Any option totalling 12 is eliminated instantly, without a single mental rotation. Only then do you rotate and flip the survivors.

Mirrors, Rotations and Opposite Views

Two spatial traps are worth naming:

• Reflections signalled by arrows. An arrow pointing top-to-bottom can indicate a horizontal reflection — the reflection axis is perpendicular to the arrow. Always confirm the axis from the completed examples rather than assuming.
• Opposite views. To picture an object from the other side, pick one asymmetrical object (a duck, a hammer) and track it. Avoid symmetrical objects such as a ball — they tell you nothing. If the duck faces left and sits left of the ball, from behind it faces right and sits right of the ball.

For more on this, see our non-verbal reasoning tips and practice strategies.

Word Logic and Word-Free Maths

Not every logical reasoning question is abstract. Two examples worth practising:

Percentages in disguise. An item discounted by 20% now costs £72. A 20% discount means the price is 80% of the original, so 0.8x = 72 and x = £90. Always substitute back to check: 90 − 18 = 72. ✓

Ignore the operators. Given 1×2+3, 2×3+4, 3×4+5, what comes next? Strip out the symbols and read only the digits: 1,2,3 → 2,3,4 → 3,4,5. Each row begins with the previous row’s middle number, so the answer is 4×5+6.

A Practical Weekly Routine

The Friday review matters more than the practice itself. Ask your child to explain the rule aloud for each question they missed — verbalising the rule is what makes it transferable to the next unfamiliar question.

Tip: Keep a “rules notebook”. Every time your child meets a new trick — split the digits, check the columns, count the squares — it goes on the list. Within a few weeks they will have a personal checklist to run through whenever a question looks impossible.

Practise Logical Reasoning With Our Apps

Logical reasoning is a skill built through varied, repeated exposure — which is exactly what structured practice provides.

Our 11+ Non-Verbal Reasoning app covers the visual side with 540 visual pattern questions across 18 topics, including analogy, series, odd one out, cube nets, block counting, hidden shapes, jigsaw, paper folding, 3D rotation, and GL-style triangle and star matrix questions — the matrix, rotation and shape-fitting types described above.

For the verbal and numerical side, the 11+ Verbal Reasoning Methods & Techniques app provides 1,050 questions with detailed study notes across 22 topics, including word codes, letter series and analogies. The study notes are the key part: rather than simply marking an answer wrong, they walk your child through the reasoning, so each mistake becomes a method they keep.

Both sit within our suite of 8100+ questions across 7 apps covering maths, English, verbal reasoning, vocabulary and non-verbal reasoning. If your child is starting from scratch, our practical 11+ guide is a sensible place to plan the wider preparation timetable.

Start early, keep sessions short, and treat every unfamiliar question as a puzzle rather than a test. That mindset — curiosity instead of panic — is what carries a child through the questions no one prepared them for.