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Formation of Figures

Reasoning · Non-Verbal Reasoning

7 Part2 worked examples4 practice questions

Remember the childhood puzzle with a few coloured pieces that you had to join into a shape. In the exam the same puzzle comes on paper: a few pieces are given and four shapes. You have to say which shape can be built from those pieces, or which one can never be built.

The pieces in this chapter are made of squares (squares joined edge to edge) because squares are easy to count and test. The answer of every question was found by the computer trying every possible place and turn, so it is certain; the wrong shapes differ from a real one by just the place of one square — they look right at first sight but cannot be built.

Good news: there is a sure method, no guessing needed — (1) count the squares, (2) place the most unusual piece first, (3) fill the rest with the other pieces. That is the essence of this chapter.

Map of this chapter

PartForm of question
1. FoundationThe rules and the search method
2. Building a shapeWhich can be made / which cannot be made
3. Cutting a shapeWhich pieces make the shape; which piece fills the gap
4. PairsWhich two of five pieces
RememberBefore every question do one thing: count the squares. It often removes half the options at once.
1

Foundation — rules and method

You may turn but not flip; no covering, no gaps

The rules for building a shape and the sure search method.

The rules — what is allowed

Four rules that hold in every question

The rules for joining pieces and the exact meaning of “can be made”.

A shape is said to be makeable when all the given pieces (every one, once each) sit inside it so that the shape is filled completely. In the figure below there are three pieces and, on the right, a shape built from them; each piece is shown in its own colour.

Three pieces (left) and the shape built from them (right), every piece in its own colour
RuleMeaning
1. You may turnA piece can be turned any way
2. You may not flipThe mirror image of a piece counts as a different piece
3. No overlappingTwo pieces cannot occupy the same place
4. No gapsNo square of the shape may stay empty and no piece may stick out
A piece (orange) can be turned into a new position, but its mirror image (teal) is not the same piece
RememberRule 4 gives a direct fact: the total of the squares of the pieces = the squares of the shape. If the totals differ the shape cannot be made.
ABCD
Top: the pieces. Below: four shapes
Example 1 · Basic
Which shape can be made by joining the coloured pieces above — only turning them (no flipping), with no overlap and no gap?
Solution, step by step
  1. 1Count the squares in the pieces: 4 + 4 = 8. A shape without 8 squares is out at once.
  2. 2Every shape has 8 squares, so place the biggest piece in a corner of each shape and try to fill the rest — all turns and places were tried.
  3. 3In C the pieces fit exactly (✓). In A, B, D every way of placing leaves one square out of place (✗).
Answer (C)
In the wrong shapes one square was in another place, so the pieces would not fit.
A✗B✗C✓D✗
Solved: ✓ can be made (colours show where the pieces go), ✗ cannot be made
ABCD
Top: the pieces. Below: four shapes
BasicTry it yourself first
Which shape can be made by joining the coloured pieces above — only turning them (no flipping), with no overlap and no gap?
ABCD
Top: the pieces. Below: four shapes
BasicTry it yourself first
Which shape can be made by joining the coloured pieces above — only turning them (no flipping), with no overlap and no gap?

The sure search method

Count, place the oddest piece first, fill the rest

Four steps to test every shape and clues for testing quickly.

Four steps
1. Count the squares2. Pick the oddest piece3. Put it in a bump or corner of the shape4. Fill the rest
  • The oddest piece is the one whose build is the least common — for example the longest or the bent one. It has the fewest places to go, so place it first.
  • Bumps and corners: a thin bump or corner of the shape must be entered by some piece. See which piece can sit in that bump.
  • Straight runs: if the shape has a long straight run and no piece is that long, the run will be formed by two pieces together.
  • Final check: count the squares of every piece again to see that no square is left empty or in excess.

When a shape cannot be made, in these steps one square will be left over or be short. That is its proof. So in a “cannot be made” question build three of the shapes (check and cross them out) and the one that does not work is the answer.

TrickTo save time: check the shape that looks the most awkward first. The wrong shapes are often the ones with a square hanging in an awkward place.
ABCD
Top: the pieces. Below: four shapes
Example 2 · Basic
Which shape can not be made by joining the coloured pieces above, using turns only?
Solution, step by step
  1. 1Count the squares in the pieces: 4 + 3 = 7. A shape without 7 squares is out at once.
  2. 2Every shape has 7 squares, so place the biggest piece in a corner of each shape and try to fill the rest — all turns and places were tried.
  3. 3A, B, C can all be made from the pieces (✓). In D every way of placing leaves one square out of place (✗) — so that is the one that cannot be made.
Answer (D)
Here three shapes were built and ticked, and for the remaining one no way of building it was found.
A✓B✓C✓D✗
Solved: ✓ can be made (colours show where the pieces go), ✗ cannot be made
ABCD
Top: the pieces. Below: four shapes
BasicTry it yourself first
Which shape can not be made by joining the coloured pieces above, using turns only?
ABCD
Top: the pieces. Below: four shapes
BasicTry it yourself first
Which shape can not be made by joining the coloured pieces above, using turns only?
2

Building a shape

Pieces are given — which shape can be made and which cannot?

Questions on building shapes from two and three pieces and spotting the shape that cannot be made.

Which shape can be made

Four shapes, one that works

The method of picking the right shape and clues to spot the wrong options.

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Which shape cannot be made

Three can be made, one cannot

The method for the reverse question: building three shapes to show that the fourth fails.

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3

Cutting a shape — pieces and the gap

The shape is given — which pieces make it?

From a given shape to the set of pieces, and the piece that fills a gap.

The pieces of a shape

Go from the shape to the pieces

The method of checking which pieces a shape is cut into.

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Filling a gap

The leftover space has the build of the missing piece

Choosing the piece that fits exactly in the space left after one piece is placed.

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4

Pairs — which two pieces

Choosing two out of five pieces

Recognising the pair of pieces that makes a shape.

Two pieces that make the shape together

The total of squares first, then the build

The method of picking the right pair from five pieces.

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5

Strategy guide

See the form of the question, then choose the test

The order for each form of question and common mistakes.

The order of tests

The first move for each form

A table of the first test and the next move for every form of question.

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Common traps and time-savers

The mistakes that keep happening

The commonest reasons for losing marks in shape-formation 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

Two small pieces, clear differences

Two-piece questions in every form.

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Standard practice

Three pieces, fake brothers and mirror images

Three pieces, bigger shapes and mirror images of the gap.

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Advanced practice

Three or four pieces and large shapes

Shapes of ten to twelve squares and fine differences.

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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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