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Grouping of Identical Figures

Reasoning · Non-Verbal Reasoning

7 Part2 worked examples4 practice questions

Five shoes lie in a cupboard. Two come from the same pair, but one is for the right foot and one for the left. Are they “identical”? They look it, but you cannot turn one around and wear it in place of the other. Figure questions test exactly this fine point: which figures are really identical and which only look alike.

The questions of this chapter come in four forms: (1) choosing the turned copy of a given figure; (2) finding the odd one out of five; (3) naming the two identical figures or how many are identical; (4) splitting six figures into groups. For each form you first learn the method and then practise.

All the figures were drawn by computer and every answer was found by really turning and testing the figures, so the answers are certain. The wrong options are the ones students usually fall for: a flipped figure, a figure with one square moved, and a different figure.

Which figures count as “the same”?

ChangeIs it still the same figure?
Turning (90°, 180°, 270° or any angle)Yes
Moving it elsewhereYes
Making it bigger or smallerSame build — “similar”; but not equal in size
Flipping (a mirror image)No, unless the figure is itself symmetric
Moving one square or sideNo
RememberYou may turn, you may not flip. The whole chapter rests on this one sentence.
1

Foundation — what “identical” means

Turning yes, flipping no

The definition of an identical figure and sure ways to test it.

Identical, similar and different

Three relations two figures can have

The difference between identical (congruent), similar and different figures.

Two figures can have three relations. Identical (congruent): they sit exactly on each other after turning. Similar: the size differs, but the build is the same — like a square and a bigger square. Different: the build itself differs. In the first parts we use pieces made of squares, where “identical” means fitting exactly after turning; in a later part we learn to recognise the build ignoring size.

0°90°180°270°↔
One piece in four turns (orange) — all identical. On the right its mirror image (teal) — this one is different

In the figure the piece is turned by 0°, 90°, 180° and 270°. All four are the same piece. But the mirror image on the right will never fit the other four however you turn it — you would have to flip it.

ABCD
Top: the given figure. Below: four options
Example 1 · Basic
Which option becomes exactly the given figure (orange) when it is only turned?
Solution, step by step
  1. 1The given piece has 4 squares. Each option must also have 4 squares, and must match the piece after turning.
  2. 2(A) the figure has one square in a different place
  3. 3(B) the figure has one square in a different place
  4. 4(C) the figure matches by turning ✔
  5. 5(D) the figure has one square in a different place
  6. 6So only (C) is the given piece turned.
Answer (C)
In an easy question the wrong options are clear; even so it is worth stating the reason for each.
A≠B≠C=D≠
Solved: = matches by turning, ↔ mirror image, ≠ different figure
ABCD
Top: the given figure. Below: four options
BasicTry it yourself first
Which option becomes exactly the given figure (orange) when it is only turned?
ABCD
Top: the given figure. Below: four options
BasicTry it yourself first
Which option becomes exactly the given figure (orange) when it is only turned?

Sure ways to test

Count, pick a mark, turn

A three-step test of whether two figures are identical and the quick tricks.

The three-step test
1. Count the squares / sides2. Pick a mark3. Turn and match
  • Count: if two pieces have a different number of squares they cannot be identical — this check removes options in five seconds.
  • A mark: pick the longest straight run or the most unusual corner of the piece, such as “one square sticking out”.
  • Turn: turn the book (or tilt your head) so that the mark points the same way in both, and then compare the rest of the build.

Now the most important trap: the flipped figure. If after turning you find the mark but the other squares sit “the wrong way round”, it is a mirror image. Such pieces have no line of symmetry — pieces with a line of symmetry (like a straight row, a square, a T) stay the same when flipped.

TrapAnother trap: an option with one square moved. It looks almost the same at first sight. So after matching the mark, check the place of every square.
ABCDE
Figures A to E
Example 2 · Basic
Which of the five figures above does not belong with the other four?
Solution, step by step
  1. 1Take one figure and turn the others to match it (no flipping).
  2. 2Four figures match each other after turning.
  3. 3In A one square sits in another place, so the build has changed — it is the odd one.
Answer A
Four pieces matched each other after turning; in the fifth one square sat elsewhere.
A≠B=C=D=E=
Solved: = like the rest, ≠ different; the odd figure has the orange frame
ABCDE
Figures A to E
BasicTry it yourself first
Which of the five figures above does not belong with the other four?
ABCDE
Figures A to E
BasicTry it yourself first
Which of the five figures above does not belong with the other four?
2

The same piece — turns and flips

Recognising pieces made of squares

Choosing the turned copy and finding the odd one and the identical pair.

Choosing the turned copy

Four options, one right turn

The version of the given figure that comes only from turning.

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The odd one and the identical pair

Five figures — four identical, or only two

Two more forms: one different among four, and two identical among five.

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3

Grouping and counting

Splitting six figures into teams of identical figures

Splitting into two and three groups and counting identical figures.

Two groups

Find the partners of every figure

The method of splitting six figures into two groups.

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Three groups and flipped look-alikes

When two teams are mirror images of each other

Splitting into three teams and the trap of the flipped figure.

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How many figures are identical?

Test every figure against the given one

Care and traps in counting identical figures.

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4

Build without the size

Small, big and tilted figures can share one build

When size and tilt are ignored and only the build is compared.

Similar figures — recognising the build

The angles and the proportion of sides are the build

Clues for recognising the build among bigger and smaller figures, and finding the odd one.

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Groups by build

Make teams leaving out size and tilt

Splitting six polygons into groups by their build.

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5

Strategy guide

Which test first and which traps

The order of tests for each form of question and common mistakes.

The order of tests

See the form of the question, then choose the test

A firm order for each form of question and tricks for cutting options.

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Common traps and time-savers

The mistakes that keep happening

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

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6

Mixed practice

All forms mixed — first spot the form, then solve

Questions of different forms in three levels.

Basic practice

Small pieces, clear differences

Pieces of four or five squares and clearly different figures.

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Standard practice

Flipped and moved traps

Pieces of five or six squares, mirror images and three groups.

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Advanced practice

Six or seven squares, fine differences, near builds

Bigger pieces, polygons of near builds and hard groupings.

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7

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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