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Embedded Figures

Reasoning · Non-Verbal Reasoning

5 Part2 worked examples10 practice questions

In the exam a simple figure X is given first — say a triangle, a staircase or an arrow. Then four complex figures follow, and one of them hides X completely inside it — the other lines were added on top to disguise it. You must pick that option.

This is not a guessing game. Every time you make a list of the lines of X and see in which option the list is found complete. In this chapter we learn to make and check exactly that list.

Question typeWhat it asksKey rule
In which option is X hiddenPick one optionEvery line of X, same size and direction
In which option is X not foundThe reverse questionFind it in three and cross them out
Turned copies allowedA turned X also countsTurn it, never flip it
How many placesHow many times X occursCount each separate place

How to use this chapter

  1. In Part 1 learn the rules and the search method.
  2. In Part 2 take the four question types one by one.
  3. In every question X is in the orange frame at the top and the options A, B, C, D are below. Trace the lines of X on the figure with a finger or pencil, then choose the option label below.
AssuranceFigures and answers come in different orders in each question, and every option was checked by computer — X is really in the right option and really not in the other three.
1

Foundation — the rules and the search method

List the lines of X and check them

The rules of a hidden figure, reading X as unit lines, and the checks that cut options.

What “hidden” means — four rules

Same size, same direction, all lines, extra lines allowed

What must hold when X is said to be hidden in a figure.

X is hidden in a figure Y when picking out some of Y’s lines gives exactly X. There are four rules:

  1. Same size — every line of X must lie on a line of Y at the same length. (It may sit on a part of a longer line, but a long line can never sit on a short one.)
  2. Same direction — X may not be turned or flipped (unless the question says so). A vertical line stays vertical, a slanting line keeps its slope.
  3. All the lines — if even one line of X is missing, X is not hidden, however much of the rest is found.
  4. Extra lines are fine — Y may have more lines besides X, even passing through the inside of X.
TrapThe commonest trap: an option shows almost all of X, just one line short or missing. That single line makes the option wrong. In the exam two options are often such near-misses.
(A)(B)(C)(D)
Question figure and options
Example 1 · Basic
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
Solution, step by step
  1. 1Figure X has unit lines: horizontal, vertical and slanting.
  2. 2First pick the most distinctive part of X (the longest line or the slanting corner) and look for it in each option.
  3. 3Option (C) holds X completely — every line is there in the same size and direction.
  4. 4(A): even the best fit leaves 1 line(s) of X missing — incomplete.
  5. 5(B): even the best fit leaves 1 line(s) of X missing — incomplete.
  6. 6(D): even the best fit leaves 1 line(s) of X missing — incomplete.
Answer Figure C
Look for every line of X in an option; if even one is missing, cross the option out.
X✔
X (left) and where it hides — the orange lines
(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?

The four-step search method

List → anchor → trace → cut options

Turning X into a list of lines, choosing an “anchor” and chasing it in every option.

Now the method. Four steps, the same in every question:

Four steps to find a hidden figure
1. List X: how many lines — horizontal, vertical, slanting?2. Anchor: pick the most unusual part of X (a slanting line, a sharp corner, the longest line)3. Trace: find the anchor in each option and check the other lines from there4. Cut: if even one line is missing, cross the option out

Quick cuts — before searching an option in full, cut some options by counting:

CheckIf this happensthen
Does X have slanting lines?The option has fewer / noneCross it out
How long is X’s longest line?No line of the option is that longCross it out
Is X a closed figure (loop)?The option has no loopCross it out
TrickWithout an anchor your eyes wander. Choose as anchor the part that appears least often in the options — such as X’s only slanting line.
(A)(B)(C)(D)
Question figure and options
Example 2 · Standard
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
Solution, step by step
  1. 1Figure X has unit lines: horizontal, vertical and slanting.
  2. 2First pick the most distinctive part of X (the longest line or the slanting corner) and look for it in each option.
  3. 3Option (B) holds X completely — every line is there in the same size and direction.
  4. 4(A): even the best fit leaves 1 line(s) of X missing — incomplete.
  5. 5(C): even the best fit leaves 1 line(s) of X missing — incomplete.
  6. 6(D): even the best fit leaves 1 line(s) of X missing — incomplete.
Answer Figure B
Look for every line of X in an option; if even one is missing, cross the option out.
X✔
X (left) and where it hides — the orange lines
(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
(A)(B)(C)(D)
Question figure and options
StandardTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
(A)(B)(C)(D)
Question figure and options
StandardTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
(A)(B)(C)(D)
Question figure and options
StandardTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?

Counting unit lines and checking lengths

A short line can sit inside a long one

Seeing figures as small units and matching lengths.

Think of every drawing as made on graph paper. The shortest distance between two points is one unit. Every line of X is then some number of units, and so is every line of the option. A line of X can sit on a line of the option only if the option’s line is as long or longer and has the same direction.

Horizontal / vertical line
1 unit = the side of one cell
Slanting line
1 unit = the diagonal of one cell
Matching
length ≥ and direction =
X✔
X’s vertical and horizontal lines sit on parts of longer lines (orange)
  • X’s long line may be made of two short lines in the option and still count — provided there is no gap in between.
  • The ends of X’s line need not be at the ends of the option’s line; it may sit in the middle of it.
  • But the corners of X must be real meeting points in the option. X’s square counts only if its four lines meet in the right places.
(A)(B)(C)(D)
Question figure and options
BasicTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
(A)(B)(C)(D)
Question figure and options
StandardTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
(A)(B)(C)(D)
Question figure and options
StandardTry it yourself first
The figure X is in the orange frame above. X is hidden in one of the four options — same size and same direction (not turned or flipped). In which one?
2

The four question types

Find · reverse · turned · count

The direct question, the reverse question, the question that allows turning and the counting question.

Which option hides X

The commonest type — a full ladder of practice

Choosing the one right option: list, anchor, trace, cut.

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In which option is X not found

The reverse question: find it in three, the fourth is the answer

How to check when you are asked where X is not.

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Turning allowed, and the mirror trap

Try the four turns; never flip

The difference between a turned X and a mirror image of X.

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How many places hold X

Test every place from the anchor

Handling repeated and overlapping places in a counting question.

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3

Strategy guide

The order of checks and the slips to avoid

The order of checks in any hidden-figure question, common traps and time-savers.

The order of checks

List → anchor → count checks → trace

What to look at as soon as you read the question and how the options fall.

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Common traps and time-savers

The mistakes that keep happening

The commonest reasons for losing marks in hidden-figure questions.

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4

Mixed practice

All types mixed — first spot the type of question

All four types of question in three levels.

Basic practice

Small figures, few extra lines

Easy direct, reverse and counting questions.

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Standard practice

More lines, slanting lines

X with five or six lines and options with more extra lines.

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Advanced practice

Complex X, crowded options

X with seven or eight lines, denser options, turns and counting.

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5

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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