How to solve Figure Matrices questions

Figure Matrices questions show a grid of figures with one position missing. Your task is to choose the figure that completes the relationships in the grid. The official TACHS handbook includes this family in its Ability samples. The examples below teach a method; they are original TACHS Prep examples, not copied official questions.

Begin by looking at how figures change across a complete row. Then test that idea on another row and check the columns. A rule that describes only one pair may be a coincidence. A useful rule explains the displayed figures and gives a definite prediction for the missing position.

Separate the features before proposing a rule

Inspect shape, number, position, fill, and internal lines separately. If several features change at once, describe each one. Avoid saying that a figure gets “more complicated” when you could say that one vertical mark is added on the right.

Also check what stays constant. If every cell has the same outside square, that square may define the space rather than supply the changing feature. If an internal mark moves while its shape stays the same, position may be the important information.

Do not choose a rule because it has an interesting name. You need a repeatable action you can test. “Combine the marks in the first two cells” is testable. “The pictures balance” is too vague until you explain what balance means in that particular grid.

Worked example 1: Combine the marks

Choose the figure that belongs in the missing cell. In these diagrams, the large boxes separate cells. The small black circles are the marks that matter. The answer choices are shown beneath the matrix.

The answer is C. In each row, the third cell contains the marks from the first two cells together. In the top row, the left mark and the center mark produce two marks. In the middle row, the right mark and the center mark produce two marks.

The bottom row begins with two outside marks and a center mark. Combining them gives all three marks shown in C. The columns confirm the result: the bottom cell combines the positions in the two cells above it, with an overlapping position shown once.

A loses the right mark. B loses the left mark. D loses the center mark. E keeps only the center mark and loses both outside marks. Only C keeps exactly the required set.

Notice that counting alone is insufficient. Both A and B contain two marks, but their marks occupy different places. After finding a count rule, check the positions. An option can have the right number of parts and still be wrong.

Worked example 2: Keep both kinds of line

Here the changing features are vertical and horizontal strokes. Treat them as separate features and find the missing figure. The cell borders are not counted as strokes.

The answer is B. Across each row, the number of vertical strokes increases from one to two to three. Down each column, the number of horizontal strokes increases from zero to one to two. The blank is in the third column and third row, so it needs three vertical strokes and two horizontal strokes.

A has the correct three vertical strokes but only one horizontal stroke. C has the correct two horizontal strokes but only two vertical strokes. D has three horizontal strokes, one too many. E has only one vertical stroke. B is the only choice that satisfies both requirements.

You can verify the result in two directions. Read across the bottom row to check the vertical counts. Then read down the right column to check the horizontal counts. Neither direction requires a separate exception for the missing cell.

This is why you should describe a candidate in full before picking it. Saying “three lines” would hide which orientation you mean. Saying “three vertical and two horizontal strokes” gives a clear prediction and makes the wrong choices easier to reject.

Use the choices to test, not invent, a rule

It can help to glance at the choices after you have inspected the grid. Their differences may remind you to check a feature you overlooked. But do not begin by deciding which option looks most finished or most symmetrical.

Try to state your prediction before studying each answer. For example, you might expect two outside marks, or a particular combination of horizontal and vertical strokes. Then compare each option with that prediction. If none fits, return to the grid and test the rule again.

If two options appear to fit, find the feature that distinguishes them and check that feature throughout the grid. Do not quietly add a special rule that applies only to the last cell. A correction to your rule should explain the completed cells as well.

Review errors by naming the missed relationship

After a wrong answer, write a short explanation of the missed feature. Perhaps you combined counts but ignored location, checked a row but not a column, or confused a reflection with a rotation. These are different errors and need different practice.

For a location error, use a few figures with marks placed near corners or edges. For a count error, compare sets with the same outside shape. For a direction error, describe movement using a fixed page orientation, such as toward the top or toward the right.

During a learning session, take enough time to explain the relationship. Then use unfamiliar items to see whether you can find it independently. Repeating a solved diagram can strengthen memory of the answer without showing whether the reasoning method transfers.

Move from learning to pacing

Start without a timer if the format is unfamiliar. Once you can explain your answers, try a timed set and review where you spent time testing a weak idea. A short, complete rule is often more useful than several half-tested guesses.

TACHS Prep’s Figure Matrices practice uses an estimated 22 questions in ten standard-time minutes. That is a practice setting based on public information, not an official per-part specification. Use the six-part guide to see how that setting fits the rest of practice.

For the other Ability families, learn to distinguish finding a relationship in a matrix from reversing paper folds and classifying figures. The same careful attention to features helps, but each task asks a different question.