9.1 Figure Series and Pattern Progression

9.1 Figure Series and Pattern Progression

The Aptitude Test of the AdUCET — the Adamson University College Entrance Test — includes non-verbal or abstract reasoning items, and the figure series is the workhorse format. You are shown three, four or five frames in a row, each containing a small geometric scene, and asked which frame comes next. Adamson University publishes no content outline, so treat any claim about how many of these you will meet as candidate report rather than fact. What is not in doubt is that the skill involved is narrow and trainable. A figure series is never a test of artistic perception; it is a test of whether you can name the things that are changing and write a rule for each one.

Because AdUCET is a computer-based examination taken on campus, the method has to run largely in your head, and Adamson publishes no policy on scratch material. Practise the tables below mentally until holding four short columns at once feels ordinary. Matrix items, where the pattern runs in two directions at once, belong to section 9.2.

The four parameters to track

Almost every series is built from four independent channels. Learn the list as a checklist and run it in the same order every single time.

Name the parameter before you name the rule. A candidate who looks at a series and thinks it is getting bigger has skipped the naming step and will lose to a distractor that is also getting bigger.

Quantity: addition and deletion

Three increment families cover nearly everything.

• Constant increment. A fixed number of elements joins or leaves each step: 2, 5, 8, 11 is $+3$, so the next term is 14.
• Alternating increment. The step alternates, usually one positive and one negative: 4, 7, 5, 8, 6 is $+3, -2, +3, -2$, and the next step is a $+3$, giving 9. Note the drift: each pair of steps nets $+1$, so the series climbs even while it appears to oscillate.
• Geometric progression. Elements double or halve: 1, 2, 4, 8 shaded cells of a 4-by-4 grid, with 16 — the whole grid — as the next term. Geometric rules are easy to spot precisely because they run out of room fast.

Deletion works the same way in reverse, and a series can delete one element while adding another so that the total never changes. A constant total with a changing composition is one of the commonest ways a series hides its rule from a hurried reader.

Position: movement, wrap-around and acceleration

Movement rules are modular arithmetic in disguise. Number the available slots, then treat movement as addition modulo the number of slots.

• Movement around a frame. Number the corners 0 (top-left), 1 (top-right), 2 (bottom-right), 3 (bottom-left) clockwise. A marker advancing one corner per step sits at corner $(n-1) \bmod 4$ in figure $n$, so figure 5 is back at the top-left. That return is not a new rule; it is the same rule wrapping around.
• Movement around polygon vertices. With five or six vertices the wrap is less obvious, which is exactly why examiners use it. Number the vertices clockwise from the top and add modulo 5 or 6.
• Step-size acceleration. The step itself grows: $+1, +2, +3, +4$. Track the running total, not the individual jumps. A dot starting on vertex 0 of a hexagon sits at vertex $\frac{(n-1)n}{2} \bmod 6$ in figure $n$, giving 0, 1, 3, 0, 4, 3 for figures 1 to 6.

Orientation: rotation by fixed increments

An arrow, a spoke or a notch usually turns by a fixed angle each step. Section 9.3 gives the clock-position shortcut for reading those angles quickly. For a series, though, only two quantities matter — the increment, and the number of intervals, which is always one fewer than the number of figures. Four figures contain three intervals; reaching figure 7 from figure 1 takes six intervals, not seven. That off-by-one is the single most productive trap in the whole item type. Whether an option is a genuine turn or a mirror image of the correct figure is settled by the test in section 9.3.

Shading: cycles, inversion and phase shifts

• Cycles. White, hatched, solid black, back to white is a period-3 cycle; the state in figure $n$ depends only on $n \bmod 3$.
• Inversion. Black and white swap every step — a period-2 cycle in which every element flips together.
• Phase-shifted cycles. An inner and an outer element change on different periods. Suppose the outer square runs the period-3 cycle while the inner circle merely alternates white and black on period 2. The pair repeats only after the least common multiple of 2 and 3, which is 6.

Figure 4 is the trap. The outer square is white again, so a hurried candidate declares the cycle complete. The inner circle says otherwise, and the true repeat is figure 7.

Worked example 1: a four-parameter tracker

Four figures are given; find the fifth.

• Figure 1 — a white triangle; a small solid dot at the top-left corner of the frame; a short arrow inside pointing due north.
• Figure 2 — a hatched square; the dot at the top-right corner; the arrow pointing north-east.
• Figure 3 — a solid black pentagon; the dot at the bottom-right corner; the arrow pointing due east.
• Figure 4 — a white hexagon; the dot at the bottom-left corner; the arrow pointing south-east.

Build one column per parameter and nothing else.

State the rules explicitly: sides $= n + 2$; dot corner $= (n-1) \bmod 4$ clockwise from the top-left; bearing $= 45(n-1)$ degrees clockwise from north; fill cycles white, hatched, black on period 3. The target is a hatched heptagon with a dot at the top-left corner and an arrow pointing due south. Any option that misses even one of those four facts is dead.

Worked example 2: acceleration with a constant total

A regular hexagon has its vertices numbered 0 to 5 clockwise from the top, so 0 is the top vertex, 1 upper-right, 2 lower-right, 3 the bottom, 4 lower-left and 5 upper-left. Figure 1 shows a black dot on vertex 0 and white dots on the other five vertices. In each later figure the black dot has moved clockwise, and the jump grows by one vertex each time.

1. Count the slots. Six vertices, so every calculation is modulo 6.
2. Sum the jumps. After $n-1$ moves the dot has travelled $1 + 2 + \dots + (n-1) = \frac{(n-1)n}{2}$ vertices.
3. Evaluate. Figure 2 gives 1; figure 3 gives 3; figure 4 gives $6 \bmod 6 = 0$; figure 5 gives $10 \bmod 6 = 4$; figure 6 gives $15 \bmod 6 = 3$.
4. Use the frozen parameter. Quantity never changes — one black dot and five white ones in every frame — so any option showing two black dots is eliminated before you check position at all.

Figure 5 is therefore a hexagon with the black dot on vertex 4, the lower-left vertex. Figure 4 sitting back on vertex 0 looks like a reset and is not one.

The five-step extrapolation routine

1. Name the parameters. Run form, position, quantity, shading in that order.
2. Fill one column per parameter across every given figure. Four short columns beat one vague impression.
3. State each rule as a formula with a period. "+1 side" and "clockwise, period 4" are usable; "it moves around" is not.
4. Predict the target before reading the options. Say it to yourself in full: hatched heptagon, dot top-left, arrow south.
5. Eliminate on the cheapest parameter first. Counting beats judging an angle, and reading a fill beats counting. If two options survive, the discriminator is the parameter you never wrote down.

Common traps

• Partial transformation. The distractor obeys three rules and breaks one, usually shading. It feels right because the shape and position are correct.
• Mirror instead of turn. The distractor is the correct figure flipped rather than rotated. Check an asymmetric detail — the shorter arm, a notch, an offset dot.
• Reset traps. A cycling parameter returning to its opening state proves nothing about the series as a whole. Confirm with a parameter whose period is different.
• Off-by-one intervals. Figure 1 to figure 6 is five steps, not six.
• Rules read from two figures. Two figures support several rules at once. Confirm on a third before you trust one.