Figure Completion and Pattern

Not every cell holds a drawing. In many questions the cells of a grid, a circle or a 2 × 2 block hold numbers or letters, and one cell holds a question mark. The idea is the same as before: the missing cell is tied to the visible cells by a simple relation. The only new skill is to know what kind of change a number or a letter can make.

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In the figure every row follows one pattern: first number + second number = third number. A pattern is real only when it works on two complete rows, not on one.

Follow the method in this order.

1. Kind. Decide whether the "?" cell must hold a letter or a number. Letter cells and number cells follow different series, and a mixed grid often hides two series in one picture.
2. Reading order. Pick out only the cells of that kind and read them left to right, top to bottom. This gives one clean series.
3. Test. For numbers test the row sum, then the difference, then the product, then the column. For letters write the places and look at the gaps.
4. Turned letters. If a letter is turned from cell to cell, treat it like a shape and follow its stem, as you do in a 2 × 2 block. A mirror image of F is never a turned F.

At the end, put your answer back and check it on a second row before you choose.

A 2 × 2 block is built the way a designer lays a tile floor: draw one small cell, then make the other three from it. Only four tools are used: a left-right mirror, a top-bottom mirror (water image), a 90° turn, or a diagonal copy (a 180° turn). Your job is to find the tool that joins the cells you can see, and to use the same tool on the blank cell.

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Why does one tool decide everything? Because the designer used it for the whole block. If TL → TR is a left-right mirror, then BL → BR is a left-right mirror too. If each cell is the previous cell turned, the turn simply goes on round the block. So the blank cell is never a new idea; it is the same idea once more.

Run the tests in this order.

One matching neighbour is not enough. Trust a relation only when two given cells confirm it. A mirror and a quarter turn can look alike for one pair of cells, but they disagree on the third.

To judge quickly, follow one clearly lopsided mark (a dot, the thick corner of a triangle, the cloth of a flag) and write its clock position in each cell, as taught in the foundation. The hours of the given cells show the relation, and the next hour is the position in the blank cell.

Finally, the answer is not turned: choose the option exactly as it would sit in the blank cell.

This type has two faces. In both of them the hidden part is fixed by what you can see.

A. A piece missing from a figure. One region of a bigger figure (a square, circle, hexagon or triangle) is replaced by a question mark, and the options are tiles for that region. Think of a jigsaw with one piece missing: the piece must continue every line that reaches its border and keep the fill (black, white or shaded). Most figures of this kind are built from one quarter repeated by mirror or by turning, so the other quarters tell you what the hidden quarter looks like.

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1. Mark the points where lines meet the border of the hidden region (the red dots in the figure).
2. Keep only the options whose lines leave the tile at those same points.
3. Check the fill of each part: black, white, shaded.
4. Check the size and spacing of inner parts against a visible quarter. A ring a little too thick is the favourite distractor.

B. Half a figure, completed by reflection. Only one half of a figure is drawn beside a dotted split line, and the options are complete figures. The completed figure is symmetric about the split line, so the missing half is the mirror image of the drawn half: a left-right mirror for a vertical line, a top-bottom mirror (water image) for a horizontal line.

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A point at some distance from the split line goes to the same distance on the other side, along the same horizontal line (vertical split) or the same vertical line (horizontal split). A corner close to the line stays close, and a corner far from it stays far. Place the mirror image of one distinctive feature, cross out the options that do not have it there, and then check where the two halves join on the line. A line that lies on the split line belongs to the whole figure and stays in it.

No turning, no sliding: only a flip across the line. A slide gives two copies facing the same way, and a 180° turn flips both ways. Both are common distractors. In both faces of this type the piece or the figure is placed as it is drawn; you do not turn it unless the question says so.

A 3 × 3 grid has eight cells filled and one marked "?". The options are single cells. Every row (or every column) follows the same pattern, and that pattern tells you the missing cell.

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Why is it solvable? Because the examiner builds each cell from a few simple patterns, one pattern for each feature (shape, fill, count, position). A cell looks hard only because it carries many features together. Take the features apart, and each one follows a simple pattern of its own. A wrong option usually breaks just one of them.

The patterns that come again and again are these.

For the "combine" pattern look at which parts are in either cell (union), in both cells (common), or in exactly one of the two (only the different parts remain). The last one is the usual pattern when the question says that common parts vanish.

Solve one feature at a time: shape, then fill, then count. Check the rows first. If the pattern fails, try the columns. When a pattern seems to fit, test it on a second row before you use it.

You see a strip of frames: three drawn and the fourth marked "?". Each frame holds two or three small elements, and each element has its own habit. The examiner does not change everything at once; he gives every element one simple habit, and the frame looks complex only because the habits are mixed. So the way in is to follow one element at a time.

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Every element behaves in one of these ways.

In the figure the dot walks round the corners anticlockwise, the bar with a bead turns 45° clockwise at each step, and the small square alternates between top and bottom.

Write each element's clock hour in frames 1, 2 and 3. A fixed step in hours (for example −3 for a dot walking anticlockwise round the corners, +1.5 for a bead turning 45° clockwise) gives frame 4 by one more step. Ignore the fixed element, solve the alternating one first because it has only two choices, then the turning ones.

When an element is symmetric (a plain bar without a bead) it looks the same after 180°, so a 180° turn can seem like no change. A bead or an arrow head makes the direction visible. Also, turn only the element that turns. The frame does not rotate unless the whole figure turns.

Some figures have a gap in the middle of a line, a curve or an outline. The missing stretch is not a new design. It is the continuation of what you can already see, so the visible ends tell you everything. A closed shape must end up closed, with no loose ends.

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Stand at the gap and run three checks. They are the red dot, the orange arrow and the green dashed curve in the figure.

Many options pass the first check, because the examiner gives them the same end points. So the real work is direction and curvature. A circle keeps bending away from its centre. A straight line keeps its slope. A curve never continues as a straight line, and a straight line never continues as a curve.

When two strands cross the gap (a line and a curve, say), check each strand on its own. An option can be right for one strand and wrong for the other. A piece that is shown as a shaded region, not as a line, belongs to the missing-piece type (missing piece inside a figure).