Figure Series, Analogy and Classification

In a movement series, the shapes do not change; they only change place. A symbol steps around the boundary of a square frame, or from cell to cell in a grid, and each symbol has its own speed and direction.

The boundary has eight places: four corners and four mid-points. Name them A to H clockwise from the top-left corner and number them 1 to 8. Corners and mid-points are equal steps of 45° about the centre, so one step means one place, whether you land on a corner or a mid-point.

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The method has four short moves:
1. Write the place number of each symbol in every frame.
2. Find each symbol's step: the difference between neighbouring numbers. If the number would pass 8, subtract 8 (so 8 + 1 = 1).
3. See if the step is fixed (+2, +2, +2), alternating (+2, +1, +2, +1), growing (+1, +2, +3) or opposite for two symbols (+1 and −1).
4. Extend each symbol on its own, then pick the option that matches all symbols.

Each symbol is a separate little series; never mix their steps. In a grid, the same idea works with rows and columns: write each symbol's cell as (row, column) and find the move. A symbol that does not move is an anchor, so do not waste time on it.

A rotation series asks one question: by how many degrees, and in which direction, does the figure turn from one frame to the next? Look for a part that shows the turn clearly, such as an arrow, a flag, or the long arm of an L. A plain circle or a square looks the same after turning, so it tells you nothing; ignore it and watch the part that moves.

All the angles in these questions are multiples of 45°. So count the turns in steps of 45°: one step is 45°, two steps 90°, three steps 135° and four steps 180°. Clockwise is positive and anticlockwise is negative. A turn of 270° clockwise is the same as 90° anticlockwise.

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Now write the turn between frames 1 and 2, between 2 and 3, and so on, as numbers of steps. Four patterns cover almost every question:

• Equal turns: the same turn each time (+1, +1, +1).
• Alternating turns: two turns repeating (+1, +2, +1, +2).
• Growing turns: each turn one step bigger (+1, +2, +3, +4).
• Reversing turns: direction flips, for example clockwise then anticlockwise.

Read the turn between two neighbouring frames, never from the first frame alone. Then apply the pattern once more to the last frame and compare with the options. Check direction first, then angle. If another part of the frame (a dot, a fixed shape) never changes, it is an anchor: ignore it.

In these questions the shapes stay recognisable; what changes is a number. It may be the number of sides of a polygon, of dots, of lines or of small shapes. One count may rise by a fixed amount while another falls, and a frame can carry two or three such counts at once.

Treat every count as its own number series. Write the count under each frame and find the difference between neighbours. A difference of +1 means one more each frame, −2 means two fewer. Because counts are whole numbers, they cannot go below zero, and a polygon cannot have fewer than three sides, so the series has a natural end.

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Follow these moves:
1. List what can be counted in the frame: sides, dots, lines, shapes, shaded parts.
2. Write each count for all the given frames.
3. Find the difference for each count and carry it one step further.
4. Build the missing frame with all the counts together and match it with the options.

Sometimes a pair of counts is linked, for example the total is always the same (as one kind of shape rises, another falls by the same amount). When two counts change, an option that gets one right and the other wrong is a trap. Always confirm the shape as well as the number. Polygons with many sides look alike, so count their vertices carefully.

Also in this type: nested figures.

A nested figure is a set of shapes drawn one inside another, like Russian dolls. Two kinds of series are built from them. In the first, the outer shape drops away at every step and a new shape appears innermost, in a repeating order. In the second, a small shape grows (or a large one shrinks) a little each step.

For the first kind, write each frame as a list from outside to inside, for example [square, circle, hexagon]. Then watch two things: which shape leaves from the front of the list, and which shape arrives at the back. The shapes that arrive follow their own order, which repeats.

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For the second kind, compare sizes by eye against the frame around them: equal growth each step means equal differences in size. Check that the new size still fits inside the frame.

Each shape must be smaller than the one outside it, so an option that breaks this can be rejected at once. When every frame has the same number of shapes, a list is enough; when the number changes, count the shapes first, as in a count series.

Sometimes one pattern for the whole row simply does not fit. The reason is that two separate patterns are woven together: odd-numbered frames follow one pattern and even-numbered frames follow another. This is called an interleaved series. It is easy to spot, because the shapes in neighbouring frames look unrelated, but frames 1, 3, 5 look like a family and frames 2, 4, 6 look like another.

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The method is to split the row. Write the odd frames in one line and the even frames in another, and solve each line as an ordinary series: a turn, a count, a move. Then look at the number of the missing frame. An odd frame belongs to the odd line, an even frame to the even line, so you only need one of the two patterns.

The second family in this type is the flip and shading series. Here a shape alternates between upright and inverted, or between white and black, at every step. These are simple two-step cycles: the same state returns after two frames. Look at frames 1, 3, 5, which should be identical, and at frames 2, 4, 6.

If a single pattern fails at the first step, split into odd and even before you give up. Check that each separate line has at least two steps that agree before trusting it. If the lines are very short, look at the options: only one will fit the line the missing frame belongs to.

A figure analogy has the form figure 1 : figure 2 :: figure 3 : ?. It says: whatever happened to figure 1 to give figure 2 must also happen to figure 3. The change is usually a rotation, a change of shading, a change of size, or a mix of two or three of these.

The method we use is ABC:

• Analyse the first pair and list what is different (position, rotation, count, shading, shape).
• Break the change into parts, each stated in words with its direction and amount: 'turns 180°', 'inner becomes black'.
• Copy each part onto figure 3, one at a time.

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Keep the direction and the angle exactly as they were. A turn of 90° clockwise is not the same as 90° anticlockwise. If figure 3 is a different shape from figure 1, the pattern still copies: the part that changed in figure 1 is the part that changes in figure 3, even when its look is new.

There is also an options-first scan that saves time. Place the four options side by side and find the one feature in which they differ (direction, shading, size). Decide that feature from the first pair, and the answer appears. Count the changes in the first pair; the answer must show all of them, not just the one you noticed first. Options almost always carry the same change minus one part, to catch exactly this slip.

Also in this type: swap analogy.

In a swap analogy, the parts of a figure exchange roles. The commonest swap is inner and outer: the shape that was outside moves inside and the shape that was inside comes outside. Two other forms appear often: the shapes are replaced by shapes with one more or one fewer side, and two parts exchange shading.

The tool is a bracket. Write each figure as [outer, inner]. Then the pair shows the pattern at once: [square, circle] → [circle, square] is a swap; [triangle, hexagon] → [square, pentagon] is 'outer +1 side, inner −1 side'.

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Decide whether the pattern is 'exchange', 'sides change', or both, before you look at the options. Options often include the unchanged figure, a half-done swap and the correct swap, so a quick bracket for each option removes most of them in seconds. If the pair gives two different patterns for outer and inner, apply each pattern to its own part.

In a position-shift analogy, the symbols in a grid do not change; they move to new cells. The grid is usually 2 by 2, but it may be 3 by 3 or a ring of boundary places. Figure 1 shows the starting positions and figure 2 the new ones. Figure 3 has different symbols, and you must move them in the same way.

First learn the common moves of a 2 by 2 grid:

• Left-right swap: each row swaps its two symbols.
• Up-down swap: each column swaps its two symbols.
• Diagonal swap: top-left changes with bottom-right and top-right with bottom-left.
• Cyclic shift: every symbol moves one corner clockwise (or anticlockwise).

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To find the move, follow one symbol from figure 1 to figure 2, such as the star going from the top-left corner to the top-right. Then confirm with a second symbol. Several moves send one symbol to the same cell, so a single symbol is never enough: for example, the star's journey fits both 'left-right swap' and 'cyclic shift'. The second symbol separates them.

Apply the move to figure 3 by cell, not by symbol. Ask: 'what goes into the top-left now?' and fill it from the cell that sends something there. Test the pattern with two symbols before you trust it. For a ring of eight places, number the places from 1 to 8 clockwise, find the shift as a number, and wrap round after 8.

This is a newer pattern. The outer shape is a plain frame, and what changes is one small element: a dot that moves to another corner, a small shape that jumps to the opposite side, or the figure at the centre that turns, changes shading or changes into another shape.

The first step is to separate the parts. In figure 1 and figure 2 write down what stays the same (usually the outer shape), and what changes:

• A moving element: note its place in both figures. Is it one corner clockwise, one corner anticlockwise, or the opposite corner (a diagonal jump)?
• A centre element: is it turned, shaded or replaced?

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Then copy only those changes onto figure 3. The outer shape of figure 3 may be new; do not copy the outer shape of figure 2.

Options in this type usually look almost identical: they differ in the one slot that changes, such as the corner of the dot. So use the options-first scan: look at the four options, find the one slot where they differ, and decide that slot alone from the first pair. Compare the same slot in figure 1 and figure 2; the corner order is clockwise: top-left, top-right, bottom-right, bottom-left. If two parts change (a dot moves and the centre darkens), check each part separately and reject any option that gets only one of them right.

Some analogies use drawings of everyday objects instead of shapes. Here the change from figure 1 to figure 2 is not a turn or a shading change. It is a relation between two real things, and you must find the same relation for figure 3.

The relations that appear in exams are few:

• Part : whole. A brick and a wall, a petal and a flower, a spoke and a wheel, a page and a book.
• State change. The same object in two states: a closed umbrella and an open umbrella, a folded fan and an open fan.
• Maker or user : product. A tool and what it makes, if the drawing is clear.
• Group : member. One object and many objects of the same kind.

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The method is to name each drawing in one word and say the relation aloud: 'a brick is a part of a wall'. Then say the same sentence for figure 3 and look for the figure that completes it. Never choose by outline alone. A ladder may look like a fence, but the relation is what matters.

Say the relation in words first; it works like a key that opens only one option. Mixed drawings often tempt you with a similar-looking decoy that stands in a different relation (a page is part of a book; a book is not part of a page). Check the direction too: part to whole is not the same as whole to part.

Also in this type: figure to number.

In a mixed analogy the first pair connects a figure with a number, and you must find the number that belongs to the third figure. It looks different from the other analogies, but the pattern is hidden in what you can count in the figure: sides, dots, lines, closed regions or parts.

The method has three small steps.
1. Count everything countable in figure 1 and write it down: sides, dots, lines, regions.
2. Find the link to the number. Try the simplest ones first: the number equals one count; the number is the sum of two counts; the number is the product of two counts; the number is a square or double of a count.
3. Check on every given pair, then apply to figure 3.

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Take care with decoys: options often include the answer that a wrong pattern would give, such as 'sides only' or 'dots only'. Write out your pattern in one line (for example, number = sides × dots) and verify it on the first pair before you calculate for the third figure.

In an odd-one-out question, four figures look different from each other, but three of them obey one pattern that the fourth breaks. Your job is to discover the common pattern. In this type the pattern is a count: the number of sides, parts, lines or dots, or a relation between two counts.

Do not look at appearance at all. Take a sheet of rough paper and write a count under every figure: sides of the outer shape, sides of the inner shape, number of dots, number of lines. Then read the numbers across.

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Common relations to look for:

• the inner shape has one side more (or one fewer) than the outer shape;
• the number of dots equals the number of sides;
• the total of sides (or lines) is the same in each figure;
• one count is double or half of another.

Often the pattern can be found in two figures and then checked on the third. The odd figure is the only one that does not match the relation the other three share. If two different patterns each pick a different odd figure, prefer the one that is simpler and uses a count that is easy to see. The exam setters like to place a figure that looks very different from the rest as a decoy; ignore its looks and keep counting.

Also in this type: symbols inside and outside.

In this type every figure has a main shape and some small symbols (circles, dots, small squares). The pattern is about how many symbols are inside and how many are outside the main shape, or in which region of two overlapping shapes they sit, or what colour they are inside versus outside.

The method is a small table. For every figure write the counts in the form (inside, outside). If there are two overlapping shapes, write a count for each region: for example (square only, both, circle only). If symbols come in two colours, write the counts for each colour as well.

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Also in this type: letters by number of lines.

This type asks you to treat capital letters as drawings. Each letter is built from a certain number of straight strokes, and the odd letter is the one that needs a different number, or has a different kind of stroke. Questions may give four letters, or groups of letters, and ask for the odd one out.

The standard way is to count the minimum number of straight lines needed to draw the capital letter exactly as it is printed. Learn the common ones:

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Letters such as C, G, J, O, S, U, Q, R, B, D and P contain curves. They are separate from the straight-line letters, so a question that mixes them is really asking about curved versus straight letters, and the odd one is found by that property. Before counting, check whether any letter is curved.

In an odd-pair question, each option is not one figure but a pair of figures joined by an arrow, the left figure turning into the right one. In three pairs, the same change happens; in one pair, a different change happens. This type is a close cousin of the analogy: the first figure of each pair is changed in the same way.

The method is to write the change of each pair, from left to right, in one short phrase: 'white to black', '90° clockwise', 'inner shape grows', 'dots double'. Then see which phrase is different from the other three.

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Look for the change that the three common pairs share, and mark the pair that disagrees. Pairs can differ in:

• direction of the change (black to white instead of white to black);
• direction of turn (clockwise against anticlockwise);
• amount (a 90° turn against a 180° turn);
• number of changes (one pair does only one of the two changes the others do).

Read every pair from left to right; reading one pair from right to left gives the opposite change. When the pairs show two changes together (a turn and a shading), split them into two columns, and check each pair on both. A pair that gets one right and misses the other is the odd one. If all four pairs seem different, you have probably mixed two attributes; separate them, and a common pattern will appear in the remaining ones.

Also in this type: rotation versus mirror image.

Here the four figures are almost the same drawing, so counting does not help. Three of them are the same figure turned to different angles, and the odd one is its mirror image. A mirror image has the same parts, but their left-right order is reversed, so no amount of turning can bring it back.

The test is handedness. Choose one clear landmark in the figure: for example the long arm and its dot, the flag at the end of a pole, or the black cell that sits alone. Then walk round the figure in a fixed direction, say clockwise, and write down the order in which you meet the parts. In a rotation the order stays the same. In a mirror image it is reversed.

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A figure that is symmetric (has a mirror line itself) cannot be used for this test, because a symmetric figure and its mirror image are the same. Questions use figures with no symmetry. A rotation keeps the handedness; a mirror image reverses it. Never call two turned figures 'different' until you have checked the handedness.

This is a grouping question. Six figures are numbered 1 to 6, and exactly three of them are identical. The four options each list three numbers, and you must choose the option that lists the identical three. Nothing needs to be calculated; it is a matching job that rewards neat habits.

The method is to sort the six figures into piles. Take figure 1 and look through the other five for copies of it. Note the numbers. If it has two copies, you have found the triple. If it has none, or only one, move to the next figure that has not been placed in a pile.

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What does identical mean? Two figures are identical when every feature matches: the shape, its shading, its size, its direction (and the position of every small part). A white circle and a black circle are not identical. A figure and the same figure turned by 90° are not identical in this type, unless the question says that turned figures count as the same. Read the instruction line for this.

Also in this type: grouping objects by class.

In this grouping question, the six figures are pictures of recognisable objects, and the options show different ways of dividing the six into groups. Your task is to find which grouping puts objects of the same class together. The class comes from the way the objects are used: stationery, clothing, tools, furniture, vehicles, fruits, musical instruments and so on.

The method has three steps. First, name every figure in one word. Do not move on until each is named. Second, decide the class of each by asking what the object is for: a pencil is for writing, so it is stationery; a saw is for cutting, so it is a tool. Third, make the groups and look for the option that matches.

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Be careful when figures are abstract shapes (circle, triangle, star). They have no use, so they cannot be grouped by class. In that case look for a feature of the drawing: curved or straight outline, number of sides, closed or open. Group by use for objects, and by a feature of the drawing for abstract shapes. When a drawing is small, use the options to decide what it most likely shows.

Also in this type: grouping by a count.

When the six figures are abstract, they cannot be grouped by use, so the grouping must come from a feature that can be counted: the number of dots, the number of lines, the number of closed regions, the number of sides or parts. Three figures share one value of the feature and three share another.

The method is a table. Number the columns 1 to 6, and for each figure write the count of one feature. Then see whether the counts fall into two groups of three. If not, try the next feature.

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The features to try, in order:
1. Number of dots or small symbols inside the figure.
2. Number of closed regions (parts into which the lines divide the figure).
3. Number of straight line segments or curves.
4. Number of sides of the outer shape.