A diagrammatic reasoning example usually shows a short flowchart. Letters, numbers or shapes go in, pass through a few symbols called operators, and come out changed. For example, if ABCD goes through a star and comes out as DCBA, the star means “reverse the order”. You decode every operator like this, then use them to find a missing output, input or operator.
Below are six worked examples covering the question types you are most likely to meet, written out in text so you can follow each step. When you have worked through them, try the same formats with real diagrams and a timer in a free diagrammatic reasoning test.
The operators used in examples 1 to 3
Examples 1 to 3 share one set of operators. In a real test you would decode them from completed paths like these:
| Path | Input | Operators | Output |
|---|---|---|---|
| 1 | ABCD | ★ | DCBA |
| 2 | JKLM | ★ then ● | LKJM |
| 3 | PQRS | ▲ then ● | RSTQ |
Decoding. Path 1 has only one operator, so start there: ★ reverses the order. In path 2, ★ turns JKLM into MLKJ. To get from MLKJ to LKJM, the first letter has moved to the end, so ● means “move the first letter to the end”. In path 3, work backwards from the output: undoing ● on RSTQ gives QRST. So ▲ turned PQRS into QRST, which means it moves each letter one place forward in the alphabet.
| Operator | Rule |
|---|---|
| ★ | Reverse the order |
| ● | Move the first letter to the end |
| ▲ | Move each letter one place forward in the alphabet |
Writing the rules down like this, in a few words each, is the habit that saves most time in the real test.
Example 1: find the output
Question. EFGH passes through ● then ★ then ▲. What comes out?
Options: A) FIHG. B) HGFI. C) EHGF. D) FGHI.
Working. Apply each operator in order and write down the result after every step.
- ● moves the first letter to the end: EFGH becomes FGHE.
- ★ reverses it: FGHE becomes EHGF.
- ▲ moves each letter forward one place: EHGF becomes FIHG.
Answer: A. Option B is what you get if you apply ★ before ●. Option C is the answer with the last operator forgotten. Both are classic distractors, which is why you work the answer out before you look at the list.
Example 2: find the input
Question. A string passes through ★ then ▲ and comes out as TSRQ. What went in?
Working. Work backwards, undoing the operators in reverse order.
- Undo ▲ by moving each letter back one place: TSRQ becomes SRQP.
- Undo ★ by reversing again: SRQP becomes PQRS.
Answer: PQRS. Check it forwards: ★ gives SRQP, then ▲ gives TSRQ. When you work backwards, always run your answer forwards once to confirm it.
Example 3: find the missing operator
Question. KLMN passes through an unknown operator, then ★, and comes out as KNML. Is the unknown operator ●, ▲ or ★?
Working. Test each candidate.
- ● first gives LMNK. Then ★ gives KNML. This matches.
- ▲ first gives LMNO. Then ★ gives ONML. No match.
- ★ first gives NMLK. Then ★ again reverses it back to KLMN. No match.
Answer: ●. With three or four candidates, testing each is quick. You can also save time by noticing that the output still contains K, L, M and N, so ▲, which changes the letters, cannot be involved.
Example 4: shape operators
Operators work on shapes in the same way. Here are three completed paths:
| Path | Input | Operators | Output |
|---|---|---|---|
| 1 | Black arrow pointing up | ■ | White arrow pointing up |
| 2 | White arrow pointing left | ◆ | White arrow pointing up |
| 3 | Black arrow pointing down | ■ then ◆ | White arrow pointing left |
Decoding. Path 1 shows ■ swaps black and white. Path 2 shows ◆ turns the arrow 90 degrees clockwise, from left to up. Check both on path 3: ■ makes the arrow white, still pointing down, then ◆ turns it clockwise from down to left. Both rules hold.
Question. A white arrow pointing right passes through ◆ then ◆ then ■. What comes out?
Working. The first ◆ turns it from right to down. The second ◆ turns it from down to left. ■ makes it black.
Answer: a black arrow pointing left.
Example 5: a shape sequence
Some diagrammatic tests include sequence questions. Each frame below holds a row of triangles, a row of stars and one circle in a corner.
| Frame | Triangles | Stars | Circle |
|---|---|---|---|
| 1 | 1 | 5 | Top left |
| 2 | 2 | 4 | Top right |
| 3 | 3 | 3 | Bottom right |
| 4 | ? | ? | ? |
| 5 | 5 | 1 | Top left |
Working. Track one feature at a time. Triangles go up by one each frame and stars go down by one, so there are always 6 shapes in the rows. The missing frame has 4 triangles and 2 stars. The circle moves one corner clockwise each step, so after bottom right it goes to bottom left, and then on to top left in frame 5, which confirms the rule.
Answer: 4 triangles, 2 stars and the circle in the bottom left. Using the frame after the gap to check your answer is a good habit whenever the missing frame is in the middle.
Example 6: which set does it belong to?
Sorting questions show two sets of diagrams and ask where a new one belongs.
- Set A. Every diagram has an arrow pointing at a circle. The other shapes vary in number, size and shading.
- Set B. Every diagram has an arrow pointing away from a circle. The other shapes vary in the same way.
Question. A new diagram has three black squares, a white triangle and an arrow pointing at a circle. Which set does it belong to?
Answer: Set A. The squares and triangle are decoration. Only the arrow and circle carry the rule. Check your rule against every diagram in both sets before you trust it.
What the examples have in common
Every example was solved the same way: decode one operator or feature at a time, write the rule down, apply the rules strictly in order, and work out the answer before reading the options. Our guide to diagrammatic reasoning question types explains the method for each format in more detail. For timing and test-day advice, read how to pass a diagrammatic reasoning test, and for the background, see what diagrammatic reasoning is.
Practise with real diagrams
Text examples teach the method, but the real test is visual and timed. Work through a full diagrammatic reasoning practice test and read the solution for every question you miss.
