What you see: the figure turns. Behind it: progression on orientation.
Start here — one guess
Before you read anything, look at this for 20 seconds and decide what goes in the empty cell.
Don't worry about being right. Guessing first makes the rest of the chapter stick — your brain remembers what it worked for. If you'd rather just be told the rule, skip ahead. No problem.
What you probably noticed: the same shape keeps appearing, and only its angle changes. That noticing is the whole skill. By the end of this chapter you'll see "rotation" the moment a matrix opens.
The family in one sentence
Rotation: the shape stays the same; only its orientation changes.
If a shape turns to point in different directions across the cells, you are looking at Rotation. If the colour also changes, or the size, or the position — that's a different family (or a compound). Pure Rotation has one moving part: orientation. Everything else is decoration.
The canonical Rotation example: the same shape, three orientations, nothing else moving — a +90° turn each cell.
What Rotation looks like
The same shape sits in every cell. Only its angle changes. Colour stays the same. Size stays the same. Position stays the same.
That makes Rotation one of the easiest families to recognise. The work is naming the pattern inside it — because that decides how you project the missing cell:
- Reading-order — the turn continues smoothly cell to cell, left to right, top to bottom, with no break between rows.
- Row-reset — each row starts fresh; rows don't connect.
- Anti-diagonal — cells on the same diagonal share the same angle.
- Snake-path — the turn reads zig-zag (row 0 left-to-right, row 1 right-to-left), but the direction stays the same.
- Counter-clockwise (CCW) — the same idea, but the angle decreases each cell.
- Row-alternating direction — the angle is the same, but the direction flips between rows (row 0 clockwise, row 1 counter-clockwise).
- Variable step — the step itself grows (+90°, then +180°, then +270°).
- Whole-grid symmetry — there's no per-cell step at all; cells on opposite sides of the centre are rotations of each other.
You'll recognise the family in two seconds. Naming the pattern takes a few more.
How to think about it
A reliable sequence for every Rotation question:
- 1Confirm it's rotation. Same shape every cell? No size, colour, or position change? If yes — rotation.
- 2Measure the step. Find the angle from cell 0 to cell 1. Confirm it with cell 2. Two cells are not enough — three are.
- 3Find the traversal. Does the turn carry across the row break (reading-order), or reset each row?
- 4Project. Apply the step to the missing cell.
- 5Eliminate. Drop options with the wrong direction, the wrong step, or that just copy a visible cell.
The whole solve takes 10–15 seconds once you trust the family.
The tactics
Every tactic uses five steps: Spot the signal · Name the rule · Execute · Eliminate the traps · Takeaway.
Tactic R-1 · Fixed-angle rotation
Covers reading-order, row-reset, anti-diagonal, and snake-path. All four use a fixed angular step — only the path differs.
Spot the signal
- Same shape every cell.
- Only the orientation changes.
- No colour, size, or position shift.
Name the rule
"The shape rotates by a fixed angle per cell."
Usually 90° clockwise. Sometimes 45° or 180°.
Execute
- 1Compare cell 0 and cell 1. Measure the angle — that's your step.
- 2Compare cell 1 and cell 2. Same step? Confirmed.
- 3Check the row break. If cell 3 continues the step from cell 2, it's reading-order. If cell 3 starts fresh, it's row-reset.
- 4Apply the step (and the traversal) to the missing cell.
Eliminate the traps
- Off-by-one — the option repeats a visible cell or jumps two steps.
- Wrong direction — the option turns the other way.
- Row-break confusion — if the pattern is row-reset, the option that "continues from cell 2" fits the wrong rule. Check cells 3 and 6 before you commit.
Takeaway — Three cells confirm step and direction. The row break is where most people misread the pattern.
*How do you know not to use this tactic?* — If the colour or size also changes between cells, this isn't pure rotation any more — it's a compound. Use R-1 for the turn, then look for the second rule.
Tactic R-2 · Counter-clockwise rotation
Same as R-1, but the shape turns backwards. Rare, and used on harder items to catch solvers who default to clockwise.
The same L-shape, turning counter-clockwise — the arrows curl the other way.
Spot the signal
- The first cells seem to "retreat" — the shape turns in a way that feels backwards.
- Cell 2 confirms the direction, not cell 1 alone.
Name the rule
"The shape rotates by a fixed angle per cell, counter-clockwise."
Execute — Same as R-1, but the step is negative. If you measure cell 0 → cell 1 as +270° clockwise, it is more likely −90° (the shorter description usually wins).
Eliminate the traps
- Clockwise-predicted answer — the most common trap on a CCW item. Always confirm direction with three cells.
- Off-by-one — as in R-1.
Takeaway — If a clockwise step looks too big (270° or more), suspect counter-clockwise. Confirm with cell 2 before committing.
*How do you know not to use this tactic?* — If cells 1 and 2 both confirm a clean clockwise reading, it's clockwise. Don't reach for CCW just because the item looks hard.
Tactic R-3 · Advanced variants
Three variants where the simple "measure and project" model breaks: variable step, row-alternating direction, and whole-grid symmetry.
Left: a step that grows (+90°, then +180°). Right: whole-grid 2-fold symmetry — cells pair across the centre, not cell to cell.
Spot the signal
- The shape is the same in every cell (rotation confirmed).
- But measuring cell 0 → cell 1 → cell 2 doesn't give one clean step. The rule is hiding somewhere other than the per-cell step.
Name the rule
"The rotation rule is operating on something other than the per-cell step."
You have to find which thing.
Execute
- 1Try R-1 first. Measure cell 0 → cell 1, then cell 1 → cell 2.
- 2Row-alternating — if the step is equal within a row but row 1 reads in the opposite direction, project the missing cell using its own row's direction.
- 3Variable step — if the step grows inside a row, work out how much it grows, then project the next step.
- 4Whole-grid symmetry — if per-cell steps never confirm but cells pair up across the centre, stop reading in sequence. Find the missing cell's partner across the grid centre — cells (r, c) and (2−r, 2−c) — and rotate it.
Eliminate the traps
- Auto-pilot R-1 — "I assumed a fixed step." The most common trap on all three variants. Read the next row before projecting.
- Wrong direction on row-alternating — use the missing cell's own row's direction, not the grid's overall direction.
- Wrong pivot on whole-grid — the centre is the middle cell, not a corner.
Takeaway — When R-1 fails, the rule has moved up a level: to a meta-step, a per-row direction, or a whole-grid structure. Name which, then project.
*How do you know not to use this tactic? — If R-1 measured cleanly and held across the row break, the rule is a fixed step and you're done. Don't reach for R-3 because the item looks* hard.
Worked questions
Three questions, easy to hard. Try each before reading the walkthrough.
Worked question 1 · Easy
Try it. 20 seconds. Apply R-1.
Answer & walkthrough
- Spot. The same F-shape sits in every cell. Only its angle changes — colour, size, and position all stay the same. One moving part: orientation.
- Name. The F rotates 90° clockwise each cell, in reading order — and the turn carries across the row breaks; it doesn't reset.
- Execute. Measure cell 1 → cell 2: a quarter-turn clockwise. Cell 2 → cell 3 confirms it. Because the turn carries across the rows, keep stepping — the missing cell is one more 90° clockwise turn on from cell 8.
- Eliminate. Each wrong option encodes a named mistake:
| Option | What it encodes |
|---|---|
| A | Turned the figure one quarter-turn too far. |
| B | Mirrored the figure instead of rotating it — the handedness is flipped. |
| C ✓ | The answer. |
| D | Turned the figure a half-turn off the correct orientation. |
| E | Stopped one quarter-turn short (or turned the wrong way). |
- Takeaway. Confirm the step with three cells, then carry it across the row breaks. The answer is just one more clockwise turn.
Worked question 2 · Medium
Apply R-1 — but watch the wrap. What happens to the angle after 270°? 30 seconds.
Answer & walkthrough
- Spot. The same F-shape sits in every cell; only its angle changes. Colour, size, and position all stay put. That's pure rotation.
- Name. The F turns 90° clockwise each cell, in reading order: 0° → 90° → 180° → 270°, then wraps back to 0°.
- Execute. Confirm the 90° step from the first two cells, then follow it across the grid. Cell 8 shows the F at 270°. One more clockwise quarter-turn brings it back to 0° — the upright F. So cell 9 is the plain, upright shape.
- Eliminate. Each wrong option encodes a named mistake:
| Option | What it encodes |
|---|---|
| A ✓ | The answer. |
| B | Counted the reading-order turn wrong — this is 1 quarter-turn away from where the pattern lands. |
| C | Counted the reading-order turn wrong — this is 2 quarter-turns away from where the pattern lands. |
| D | Counted the reading-order turn wrong — this is 3 quarter-turns away from where the pattern lands. |
- Takeaway. After 270° the cycle wraps to 0°. The trap is forgetting the wrap and turning one step too far.
Worked question 3 · Hard
Apply R-1 first. It explains the turn — but not everything. When one rule isn't enough, look for a second. 40 seconds.
Answer & walkthrough
- Spot. Same shape every cell, and it clearly turns — but the size also changes. Two rules are running at once.
- Name. Rule one: the figure turns 90° clockwise each step. Rule two: the size follows a Latin square — each row and each column shows all three sizes (small, medium, large) once.
- Execute. Project the rotation to the missing cell with R-1. Then read the size: its row and its column each already show two sizes, so the missing cell takes the third. The answer must match both the angle and the size.
- Eliminate. Each wrong option gets one rule right and the other wrong:
| Option | What it encodes |
|---|---|
| A | Followed the size rule but ignored the rotation rule. |
| B | Followed the rotation rule but ignored the size rule. |
| C ✓ | The answer. |
| D | Breaks the 2 rules it should follow. |
- Takeaway. When R-1 explains the turn but the cells still differ, a second rule is stacked on top. Solve the turn, then the size, and match both.
Discrimination quiz — Rotation, or not?
Five matrices. Don't solve them — just say whether each one is Rotation or not Rotation. If it's not, you don't need to name what it is; that's the rest of the book. Five seconds each.
(quiz slugs provisional — final five chosen from cleared, non-Rotation items at assembly, one per the answer-key families below)
Answer key — none of the five are Rotation.
| # | Family | What rules out Rotation |
|---|---|---|
| 1 | Distribution (each value once) | Three different shapes cycle per row and column — the shape changes, not just the angle. |
| 2 | Combining & overlap (XOR) | Row 2 keeps the dots in exactly one of rows 0 and 1 — that's set arithmetic, not a turn. |
| 3 | Movement | A marker steps around an anchor — its position changes, its orientation doesn't. |
| 4 | Mirror / reflection | The grid mirrors across an axis — reflection looks like rotation on symmetric shapes, but the rule is a mirror. |
| 5 | Counting & arithmetic | Every cell holds the same mix of shapes in different spots — the counts are conserved; nothing turns. |
Score 4–5: the Rotation signal is in your memory — move on. Score 3: re-read "What Rotation looks like." Score 0–2: spend five minutes on the practice page below, naming the family for ten items before solving. Recognition is the skill, not solving.
Pattern cheat sheet — Rotation family
| Pattern | Signal | Step | Direction | Traversal | Tactic |
|---|---|---|---|---|---|
| Reading-order | Smooth turn, no row break | Fixed | CW | Reading order | R-1 |
| Row-reset | Each row resets | Fixed | CW or CCW | Per-row | R-1 |
| Anti-diagonal | Same diagonal, same angle | Fixed | CW | Diagonal | R-1 |
| Snake-path | Row 0 →, row 1 ←, row 2 → | Fixed | CW | Snake | R-1 |
| Counter-clockwise | Cells "retreat" | Fixed | CCW | Any | R-2 |
| Variable step | Step grows or shrinks | Variable | CW | Reading order | R-3 |
| Row-alternating | Row 0 CW, row 1 CCW | Fixed | Flips per row | Per-row | R-3 |
| Whole-grid symmetry | Cells pair across centre | n/a | Structural | Diagonal partners | R-3 |
Family reference card
For day-of-test recall — three lines:
- 1Signal: same shape, only the angle changes.
- 2Ask: what's the step (size and direction), and the traversal (reading-order or row-reset)?
- 3First move: measure cell 0 → cell 1, confirm with cell 2.