Figure Series Questions: Six Transformations and a Checking Order

1. Turn the figures into something comparable

Comparing two figures by eye is unreliable, because visual weight pulls your attention to the wrong places. Discretise first. On a 3x3 grid, number the nine cells in a fixed order and record which ones are filled. Every rule then becomes a change in the set of filled positions, which is far steadier to reason about than an impression.

• Count the filled cells in each figure. A constant count means a movement rule: rotation, reflection or a shift.
• A count that rises or falls by a fixed amount means an accumulation rule.
• A count that alternates between n and 9 minus n means inversion is involved.

Counting filled cells is the cheapest and most informative first move. It usually eliminates more than half the candidate rules in one pass.

2. The six common transformations

Rotation by a fixed angle

The whole figure turns about its centre by a constant angle, usually 90 or 180 degrees, always in the same direction. The signal is a constant fill count with a congruent outline that has changed orientation. To confirm, pick the most distinctive filled cell, such as a lone corner, and track where it travels between figures.

Reflection

The figure flips across a vertical or horizontal axis. This is easiest to confuse with a 180 degree rotation. Tell them apart by handedness: a reflection turns three points arranged clockwise into the same three arranged anticlockwise, while a rotation does not. Pick an asymmetric group of filled cells and check whether their winding direction has flipped.

Cyclic shift

Every filled cell moves a fixed number of steps in one direction, and anything that leaves one edge reappears on the opposite edge. The signal is that the shape of the fill pattern is identical between consecutive figures and only its placement has moved. Looking at a single row or column in isolation makes the offset obvious.

Perimeter movement

Only the eight outer cells take part. Filled cells advance a fixed number of steps around the ring, while the centre cell either stays put or follows a separate rule of its own. The trap is the centre: counting it as part of the ring makes the step count never line up. Cover the centre and look only at the ring.

Inversion

Filled becomes empty and empty becomes filled. Pure inversion makes the fill count alternate between n and 9 minus n, which is very easy to spot. It often appears as an every-other-figure rule, where figures 1, 3 and 5 share a colour scheme and figures 2 and 4 are their negatives.

Rotation combined with inversion

The hardest family: rotate, then invert. Because the fill count jumps it is easy to misread as an accumulation rule, and because the outline changes it is easy to conclude there is no rule at all. The test is to invert every other figure first. If the sequence then resolves into a clean rotation, you have found a combined rule.

3. A fixed checking order

Do not test rules at random. Each step below has a definite elimination attached, and five steps are enough to settle almost any question.

1. Count filled cells. Constant means a movement rule. Alternating between n and 9 minus n means inversion. Monotonic means accumulation.
2. Within movement rules, check whether the outline is congruent. Congruent means rotation or reflection. Not congruent but the same fill pattern means a shift.
3. Separate rotation from reflection using handedness: take three non-collinear filled cells and see whether their winding direction reverses.
4. If none of that holds, isolate the outer ring and check whether only the perimeter is moving.
5. If it still does not hold, invert every other figure and go back to step 1. That is the entry point for combined rules.

4. The verification you must not skip

Once you have a candidate rule, verify it across every consecutive pair, not just the first two. Distractor options are routinely built to satisfy only the first pair, so skipping the full check walks straight into them.

• Test the rule on 1 to 2, 2 to 3, 3 to 4 and 4 to 5. All of them must hold.
• If any pair fails, discard the rule and move to the next step in the checking order.
• Only after all pairs hold should you apply the rule to the final figure to produce the answer.
• Compare against the options. If two options both fit, your rule is underspecified and needs one more constraint.

The solver on this site checks only the six families above. It states explicitly whether the prediction is unique or whether several rules disagree. If it finds nothing, the question uses a rule outside those families and needs separate analysis; it does not mean the tool has miscalculated.

5. Where marks are lost

• Mistaking a reflection for a 180 degree rotation. The two agree on symmetric figures and disagree completely on asymmetric ones, which makes this the single most common error.
• Counting the centre cell as part of the ring in perimeter questions.
• Choosing an answer after verifying only the first pair.
• Staying too long with one hypothesis. If a rule has not verified within thirty seconds, move to the next family. The whole value of a fixed order is that you never double back.
• Only comparing adjacent figures and missing an every-other-figure rule.

6. How to practise

Progress here comes almost entirely from making the checking order automatic. Start by using the solver in reverse: enter a sequence yourself, see which rule it identifies, and train your own reading to agree with it. Once that is consistent, move to timed questions in the practice bank with a target of forty seconds for identification plus verification.