Dot situation and figure completion for AFCAT: read the regions, then test each option

Dot situation questions look fiddly at first: a few overlapping shapes, two or three dots, and four option figures with the same shapes arranged differently. You must find the option in which dots could be placed so that each one has exactly the same conditions as in the question figure: inside the same shapes and outside the others.

Figure completion is the second half of this guide. A pattern has a piece missing, and you pick the piece that completes it. The two types share a habit: describe precisely what you see before looking at the options.

Dot situation: the idea

Each dot is defined by a list of shapes it is inside and shapes it is outside. That list is all that matters. Its exact position, the size of the shapes and how they are drawn do not matter.

Notation. Write each dot as the set of shapes containing it. With a circle (C), a square (S) and a triangle (T):

The question then becomes: does each option have a region matching each code?

The routine

1. Write the code for every dot in the question figure.
2. Take the rarest code first. A region inside all three shapes, or inside one shape but outside two that overlap it, is the most likely to be missing.
3. Check each option for that region. Strike out any option that lacks it.
4. Check the remaining codes in the surviving options.
5. Watch for containment. If one shape lies wholly inside another, no region can be inside the small one but outside the large one.

Worked example 1

Question figure. A circle, a square and a triangle. One dot is inside the circle and the square but outside the triangle (code CS). Another dot is inside all three (code CST).

Options.

• (a) The circle and square overlap. The triangle lies entirely inside the square and does not touch the circle.
• (b) The square lies entirely inside the circle. The triangle overlaps the square, so part of the square is inside the triangle and part is outside.
• (c) The triangle lies entirely inside the circle. The square overlaps the circle but does not touch the triangle.
• (d) The square lies entirely inside the triangle. The circle overlaps the square.

Solution. Start with CST, which needs all three shapes to share a region.

• (a) The triangle never meets the circle, so there is no CST. Out.
• (c) The square never meets the triangle, so there is no CST. Out.
• (b) The part of the square inside the triangle is also inside the circle, so CST exists. The part of the square outside the triangle is still inside the circle, so CS exists. Passes.
• (d) The circle overlaps the square, and the square is inside the triangle, so CST exists. But every point of the square is inside the triangle, so CS cannot exist. Out.

Answer: (b). Notice that (d) failed only on the containment rule. That is the classic trap.

Worked example 2: three dots

Question figure. A circle (C) and a rectangle (R) overlap, and a small triangle (T) sits across the edge of the circle, away from the rectangle. Three dots: one in C only, one in CR, one in CT.

To match, an option needs the circle to overlap both other shapes, with neither of them swallowing the circle. It must also leave some part of the circle outside both. An option in which the triangle sits entirely inside the rectangle fails on CT, because the triangle's part inside the circle would then also be inside the rectangle, giving CRT, not CT.

Figure completion

A figure, often a square pattern, has one part missing, and you pick the option that completes it. Three rules cover most questions.

Mirror and rotational symmetry are easy to confuse, and options often include both answers. Decide which one the visible parts obey before you predict the missing part.

Grid notation. The pattern is a 4×4 grid. Rows 1–4 run top to bottom, columns 1–4 left to right, and (r, c) is row r, column c. Each quarter is a 2×2 block: top-left is rows 1–2, columns 1–2, and so on.

Worked example 3: rotation, not mirror. The shaded cells are:

• Top-left quarter: (1, 1) and (1, 2).
• Top-right quarter: (1, 4) and (2, 4).
• Bottom-left quarter: (3, 1) and (4, 1).
• Bottom-right quarter: missing.

Test mirror symmetry first. Reflecting the top-left across the vertical centre line would give (1, 4) and (1, 3) in the top-right, but the top-right actually has (1, 4) and (2, 4). So it is not a mirror pattern.

Test rotation. Each quarter is a bar of two cells along the outer edge, and it moves round clockwise: along the top of the top-left quarter, down the right side of the top-right quarter, then (skipping the missing quarter) up the left side of the bottom-left quarter. The bottom-right must have its bar along the bottom edge.

Missing quarter: (4, 3) and (4, 4). An option showing (3, 4) and (4, 4), the mirror of the bottom-left quarter, is the trap.

Worked example 4: a row rule. A 3 × 3 matrix of small squares, each with some of its sides drawn thick.

• In each row, a side thick in both figures 1 and 2 disappears. A side thick in only one of them stays. (Row 1: left is common and goes; top and right remain.)
• Row 3: bottom is common and goes; left and top remain.
• Missing figure: thick left and top sides.
• A "combine everything" rule would give three thick sides in row 1 figure 3, so it is ruled out by the rows you can see.

Practice set

Dot situation. The question figure has a circle (C), a square (S) and a triangle (T). One dot is inside the circle and the triangle but outside the square (CT). Another dot is inside the square only (S).

1. Option (a): the square lies entirely inside the circle; the triangle overlaps the circle but not the square. Does it work?
2. Option (b): the circle, triangle and square sit in a row; the circle overlaps the triangle, the triangle overlaps the square, and the circle and square do not meet. Does it work?
3. Option (c): the triangle lies entirely inside the square; the circle overlaps both. Does it work?
4. Figure completion, using the 4×4 grid notation: a pattern is symmetric about its vertical centre line. The left half has shaded cells (1, 1), (2, 2), (3, 2) and (4, 1). The right half is missing. Which cells are shaded in it?
5. In a 3 × 3 matrix using the row rule of Example 4, row 1 is: "top, bottom", then "bottom, left", then "?". What is the missing figure?

Answers:

1. No. CT exists, but the square is wholly inside the circle, so there is no region inside the square only.
2. Yes. The circle–triangle overlap gives CT (the square does not reach it), and the part of the square outside the triangle gives S.
3. No. Every point of the triangle is inside the square, so CT, which must be outside the square, cannot exist.
4. (1, 4), (2, 3), (3, 3) and (4, 4). Reflecting across the vertical centre swaps column 1 with 4 and column 2 with 3.
5. Thick top and left sides. Bottom is in both figures and disappears; top and left appear once each and remain.

What to do next

• For the next 10 dot-situation questions, write the code of each dot before looking at the options.
• In every figure-completion question, test mirror and rotation separately and note which one the visible parts obey.
• Revise embedded figures for careful searching, then move to spatial ability.