Embedded Figures and Figure Completion

1. What this chapter tests

These visual-reasoning problems test whether you can separate a useful shape from distracting strokes and reconstruct a missing part using a stated rule. Work from visible geometry rather than guessing what an object resembles.

Task	What must be found	Main check
Embedded figure	The complete target inside a larger drawing	Every target stroke and junction is present
Figure completion	A missing part of a given drawing	Boundaries and the stated pattern rule agree
Symmetry completion	The piece required by one or more reflection axes	Corresponding points are equally far from the axis
Line continuation	The hidden section of straight or curved paths	Entry, exit and direction remain consistent

In an embedded-figure problem, extra lines in the larger figure are normally permitted. In a completion problem, extra lines in a proposed tile can make it wrong. Always read the exact instruction.

2. How to use the diagrams

The top drawing is target T or the incomplete figure. The four lower panels are options A–D. Panel borders and dashed guides are aids to layout, not strokes that may be used to build the answer. Compare actual dark strokes only. In the solutions, green strokes identify a match or fill a gap.

The first practice family uses a hidden unit grid with equally spaced vertices. Coordinates in explanations begin at the top-left of each option: x increases to the right and y downward. They provide a precise way to verify a trace; you do not need to calculate coordinates to solve visually.

A worked embedded-figure example; green strokes identify the target

3. Translation, rotation and reflection

Translation moves the whole target without changing its orientation or size. Rotation turns it about a point. Reflection reverses its handedness across a mirror line. These operations are different permissions.

If rotation is forbidden, a right-facing hook cannot become a downward-facing hook.
If rotation through multiples of 90° is allowed, try 0°, 90°, 180° and 270°.
Rotation permission does not automatically permit a mirror image.
If resizing is forbidden, preserve the relative and absolute segment lengths shown on the common grid.
If an exam supplies its own orientation rule, use that rule. Do not assume a universal convention for every publisher.

For example, a path that moves right, down, then right retains that turn sequence when translated. A mirrored path changes handedness. An asymmetric shape is a better anchor than a square because a square can conceal an orientation error.

4. A reliable embedded-figure method

Identify one distinctive feature: a long arm, a three-way branch, an unequal pair of sides, or a sequence of two bends.
Locate a possible copy of that feature in an option.
Trace the connected target from that anchor without lifting your imagined pen across a missing stroke.
Compare the direction and length of each next segment.
Check the remaining branches or disconnected components in their correct relative positions.
Reject the option as soon as one required piece is absent.
Confirm the complete target in the surviving option.

Do not merely count lines. Two figures with six strokes can have different connections. Do not build half the target in one area and the other half elsewhere unless the target itself has those disconnected components in exactly that relationship.

5. Crossings, junctions and distracting extensions

A target endpoint may lie on a longer line in the candidate; an added extension does not erase the target. Similarly, an extra line may cross a required stroke. You may ignore the extra line when the target is still present.

However, a visible gap cannot be bridged mentally. A crossing is not evidence that the line bends there. Follow the actual target's geometry. In this chapter, each embedded target is made of connected horizontal and vertical unit strokes. Longer displayed runs can contain several such strokes. Only strokes in the drawing count; box borders are excluded.

Fast rejection checks: a necessary vertical arm is missing; a bend turns the wrong way; the long side is too short; a branch appears at the wrong distance; only a reflected copy exists when reflection is forbidden.

6. Embedded figures with permitted rotation

Use the same tracing method, but fix one orientation at a time. Mentally rotate the entire target, not individual arms independently. After choosing an orientation, all lengths, bends and branches must agree together.

For a local coordinate point (x,y), a quarter-turn can be represented by (-y,x), followed by a translation to place it in the option. This is an optional checking device. Rotating every point of a figure by the same transformation preserves distances and connections; rotating only one branch changes the figure.

A frequent hard-level trap is an option with the correct number of corners but the wrong order of clockwise and anticlockwise turns. Trace from the same endpoint to compare fairly.

7. Figure completion: start at the boundary

Inspect where each visible stroke meets the empty region. Note its position, direction, thickness and any stated colour or shading rule. Then examine the candidate edges before its attractive central design.

A piece that joins at one boundary but creates a jump at another is wrong. A matching number of strokes is only a preliminary check. In a general completion question, continuity alone may not determine a unique answer; a symmetry or repetition rule may also be supplied. Use all the stated constraints rather than inventing a rule.

8. One-axis and two-axis symmetry

With a vertical symmetry axis, left and right exchange places while height stays the same. With a horizontal axis, top and bottom exchange places while horizontal position stays the same.

If a square must be symmetric about both centre lines and its bottom-right quadrant is missing:

Reflect the top-right quadrant across the horizontal centre line.
Independently reflect the bottom-left quadrant across the vertical centre line.
Both constructions must give the same piece.
Equivalently, rotate the top-left quadrant by 180° about the centre of the full square.

Do not place an unchanged copy of the top-left tile into the gap. That works only for special shapes with the relevant internal symmetry. Our options include plausible one-axis and unchanged-copy distractors.

9. Straight-line continuation

A straight stroke keeps the same direction through a covered area. Extend the outside fragments using an imaginary ruler. Check that the opposite fragment lies on the same straight line.

Suppose a line enters the left edge of a tile at (0,1) and exits its right edge at (4,3). It slopes downward to the right in the displayed coordinate system. A horizontal segment from (0,1) to (4,1) shares the entry point but fails the exit and slope tests.

When lines cross inside a missing piece, continue each original line straight through the crossing. Do not switch onto the other line. All relevant fragments must be accounted for and no additional stroke may be introduced when the question forbids it.

10. Curves, shading and repeated motifs

Other examinations may use arcs, circles, shaded sectors, dots or repeating ornaments. Check the local tangent of a curve at the edge, the centre and radius of an arc, and the orientation of shaded regions. A curved piece can join at the right endpoint while bending the wrong way.

For repeated motifs, compare spacing as well as shape. If shading alternates, verify the alternation across the boundary of the missing tile. Do not assume symmetry solely because the figure looks decorative. The 80 online questions here focus on exact straight-stroke embedding, two-axis symmetry and straight-line continuity; use these principles when practising curved and shaded variants elsewhere.

Related topic	Difference
Mirror image	Transform the entire given figure by reflection
Paper folding	Track reflections caused by physical folds
Figure series	Infer a rule across successive complete figures
Counting figures	Count all qualifying shapes rather than locate one target
Embedded figure	Find an existing copy inside additional strokes
Figure completion	Supply a missing part that satisfies the constraints

12. Typical mistakes and corrections

Mistake	Correction
Choosing by overall appearance	Verify each required stroke
Allowing rotation without permission	Read the orientation instruction first
Confusing rotation with reflection	Follow an asymmetric bend sequence
Ignoring a small branch	Mark every target component mentally
Using the option panel border	Ignore layout frames and dashed guides
Checking just one symmetry axis	Construct the answer independently from both adjacent tiles
Counting lines without checking endpoints	Match position, direction and connectivity
Choosing an unnecessarily complex rule	Prefer the explicitly stated rule

13. A practical exam routine

Read permissions → identify the anchor or gap boundaries → eliminate definite mismatches → trace the entire surviving figure → confirm the option letter. If two options appear valid, recheck whether you accidentally used a panel border, reflected the target or ignored an extra stroke in a completion tile.

Practise slowly before adding a timer. Record the type of mistake, not just the score. After solving a question, explain which exact stroke disqualifies each distractor. On a small screen, enlarge the diagram so that short gaps are visible.

14. Revision and practice plan

Embedding checklist: full target, correct orientation, correct lengths, correct branches, no imagined gaps. Completion checklist: all boundaries, correct axes, line directions, no missing or forbidden extra strokes.

The online bank contains 80 illustrated questions: 20 fixed-orientation embeddings, 20 rotation-permitted embeddings, 20 two-axis symmetry completions and 20 straight-line completions. Four focused tests contain 20 questions each; the complete practice test contains all 80. The following 30 exercises sample all four groups. Attempt them before opening the answer section.

15. Illustrated practice

Practice 01

Which option contains every stroke of target T? Extra lines are allowed. Keep the target upright: translation only; no rotation, reflection or resizing.

Question 1