Last reviewed 22 Sept 2026 · 5 min read
How these are built
A setter chooses one rule and applies it to two figures completely, leaving the third with a gap. So the work is:
- Find the rule using the figure that is complete.
- Confirm it on the second complete figure — a rule that fits only one figure is a coincidence.
- Apply it to the third.
Two different rules will usually fit one figure. The second figure is not decoration: it is the test that decides between them.
The order to test relationships
| Try | Looks like |
|---|---|
| Sum of the outer numbers = the middle | round a 18 |
| Difference or product of two numbers | |
| Sum of one pair minus another | |
| Square, cube, or their sum | |
| Each row or column sums to the same total | magic-square style |
| The same operation on each row | multiply the first two, subtract the third |
| Digits: sum, product, reversal | or or |
| Half, twice, or a fixed ratio |
Matrices
A grid usually follows one of three schemes:
- Row rule: the third entry in each row is built from the first two.
- Column rule: the same, down the columns.
- Diagonal or symmetric rule: corners relate to the centre.
Test rows first, then columns. If the third entries are consistently larger, a product or a sum is at work; if they are small, look at differences or ratios.
Circles, triangles and boxes
- Circle divided into four: usually opposite quarters relate, or all four combine into the centre.
- Triangle with numbers at the corners and one in the middle: the middle is built from the three corners — sum, product, or sum of two minus the third.
- Boxes of four with a central value: try and before anything more inventive.
Worked examples
Triangles with corners and a centre: ; ;
Solution. The centre is the sum of the corners → 18.
; ;
Solution. "Product minus sum" gives for the first pair — wrong. Try the difference of squares: ✓, and ✓. So 40. The first rule was rejected by the very first figure; the second survived both.
| 4 | 3 | 24 |
|---|---|---|
| 5 | 2 | 20 |
| 6 | 4 | ? |
Solution. Third first second : ✓, ✓ → 48.
| 9 | 16 | 25 |
|---|---|---|
| 3 | 4 | 5 |
| 27 | 64 | ? |
Solution. Row 1 is the square of row 2 and row 3 is its cube → 125.
; ;
Solution. The product of the digits → 24.
A circle is divided into four quarters with a number in the centre. First: top 8 and 12, bottom 3 and 4, centre 13. Second: top 9 and 15, bottom 5 and 3, centre 16. Third: top 7 and 11, bottom 2 and 5, centre ?
Solution. Centre (sum of the top two) (sum of the bottom two): ✓ and ✓. Third: 11.
Each row, column and diagonal of a square sums to 15. The middle row reads 3, 5, ?.
Solution. 7.
; ;
Solution. ✓ and ✓ → 4.