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Inserting the Missing Character

Reasoning · Verbal Reasoning

6 Part2 worked examples12 practice questions

The questions in this chapter show a figure — a grid of numbers, triangles, circles or a row of pairs — with a ‘?’ in one place. You must name the number (or letter) that belongs in the ‘?’.

Every figure has a hidden rule that works the same way in all its parts. The parts shown in full are your evidence: find the rule on them, check it, and then apply it to the incomplete part. This chapter teaches the order of searching for the rule and which rules come again and again.

If you feel weak in maths, do not worry: we start from the table of squares and cubes, show every rule with a full check and give the full solution of every question.

How to use this chapter

  1. Read — the idea first, then the method.
  2. Study the solved example — the “Example” right below a figure shows the full solution.
  3. Try it yourself — look at the figure, find the rule on paper, then choose an option. If stuck take the hint; open the solution last.
  4. Know the level — each question is marked Basic, Standard or Advanced.
NoteThe figure of each question stands right above it. The questions you get right are remembered in this browser.
1

Foundation — how to find the rule

Evidence, test, check, apply

The method of finding the rule from the complete parts, the table of squares and cubes and the rule of number pairs.

The four-step method of finding the rule

Look at the complete parts → test → check → apply to the incomplete one

One method for every figure, the list of usual rules to test and the table of squares and cubes.

Whatever the figure, the four steps stay the same:

Four steps
1. Pick the complete parts (without the ‘?’)2. Test easy rules: add, subtract, multiply3. Check the rule that fits on ALL complete parts4. Apply it to the incomplete part

Step 3 is the most important. A rule may fit one line and still be wrong — it becomes the rule only when it fits all the complete parts.

The list of rules to test (try them in this order)

OrderRuleWhen to think of it
1Add, subtractResult near the given numbers
2MultiplyResult much bigger
3Twice / thrice of one, then add / subtractUnequal effect of two numbers
4Square or cubeResult near 4, 9, 16, 25, 36 … or 8, 27, 64
5Product or square of a sum / differenceResult very big

The table of squares and cubes — remember it

Number12345678910
Square149162536496481100
Cube1827641252163435127291000
Number11121314151617181920
Square121144169196225256289324361400
TrickUse the result as a clue: if all results are perfect squares (36, 49, 64…) a square rule is at work; if they are much bigger than the given numbers it is multiplication; and if they hover near the sum of the given numbers it is addition and subtraction.

Below is the first solved example. In the figure you must find how the third number of each row is linked to the first two.

13619861426?
Figure: the same rule works in every row.
Example 1 · Basic
Which number replaces the ? in the figure?
Solution, step by step
  1. 1Find the rule in the complete rows. Row 1: 13, 6 → 19; Row 2: 8, 6 → 14. (a, b = first, second number; c = third)
  2. 2Simple tries: a − b = 7 ≠ 19; a × b = 78 ≠ 19 — these do not fit. Looking closer gives the rule: c = a + b.
  3. 3Check: in row 1, 13 + 6 = 19 ✔; in row 2, 8 + 6 = 14 ✔.
  4. 4Now the same rule on the incomplete row: 2 + 6 = 8.
Answer 8
Adding is the first rule to try.
14112691424614?
Figure: the same rule works in every row.
BasicTry it yourself first
Which number replaces the ? in the figure?
639811531?
Figure: the same rule works in every row.
BasicTry it yourself first
Which number replaces the ? in the figure?
11491213?342119
Figure: the same rule works in every column.
BasicTry it yourself first
Which number replaces the ? in the figure?

The rule of pairs (x → f(x))

How the right number is made from the left

In a row of pairs, the rule that makes the right number from the left: multiples, squares, cubes and ± a number.

The simplest figure is a row of pairs: 3 → 10, 4 → 17 … where in every pair the right number is made from the left number (call it x) by the same rule. The ? is sometimes the right number and sometimes the left.

Kind of ruleExampleHow to spot it
Multiple ± number3 → 10, 4 → 13, 5 → 16 (3x + 1)The right number rises by equal steps
Square ± number3 → 10, 4 → 17, 5 → 26 (x² + 1)Steps 7, 9, … themselves grow (by 2)
Cube2 → 8, 3 → 27, 4 → 64 (x³)Results 8, 27, 64…
x × (x + 1)2 → 6, 3 → 12, 4 → 20Gaps 6, 8 — rising by 2

Take the gaps of the right numbers for a first guess: equal gaps mean a multiple of x (the gap is the multiplier); gaps that themselves rise by equal steps mean a square-type rule; and results like 8, 27, 64 mean a cube.

TrickIf the ? is on the left (the right number is given), find the rule and run it backwards, or put each option into the rule in turn to see which one makes the right number.
3 → 136 → 258 → 33? → 37
Figure: in every pair the right number is made from the left number by the same rule.
Example 2 · Basic
Which number replaces the ? in the figure?
Solution, step by step
  1. 1Find how the right number is made from the left number (x) in each complete pair: 3 → 13, 6 → 25, 8 → 33.
  2. 2To spot it, see whether the result is close to a square, a cube or a multiple of the left number. Here the rule is: 4x + 1.
  3. 3Check: 4 × 3 + 1 = 13 ✔; 4 × 6 + 1 = 25 ✔.
  4. 4Now the right number is 37 and x is unknown: find the number for which 4x + 1 = 37. At x = 9, 4 × 9 + 1 = 37 ✔ → 9.
Answer 9
First the gaps of the right numbers were looked at, then the rule was checked.
2 → 53 → 105 → 26? → 50
Figure: in every pair the right number is made from the left number by the same rule.
BasicTry it yourself first
Which number replaces the ? in the figure?
3 → 75 → 116 → 13? → 17
Figure: in every pair the right number is made from the left number by the same rule.
BasicTry it yourself first
Which number replaces the ? in the figure?
4 → 95 → 118 → 17? → 19
Figure: in every pair the right number is made from the left number by the same rule.
BasicTry it yourself first
Which number replaces the ? in the figure?
2 → 65 → 306 → 42? → 56
Figure: in every pair the right number is made from the left number by the same rule.
StandardTry it yourself first
Which number replaces the ? in the figure?
2 → 95 → 517 → 999 → ?
Figure: in every pair the right number is made from the left number by the same rule.
StandardTry it yourself first
Which number replaces the ? in the figure?
4 → 125 → 208 → 56? → 72
Figure: in every pair the right number is made from the left number by the same rule.
StandardTry it yourself first
Which number replaces the ? in the figure?
4 → 645 → 1257 → 3438 → ?
Figure: in every pair the right number is made from the left number by the same rule.
StandardTry it yourself first
Which number replaces the ? in the figure?
6 → 647 → 1288 → 2569 → ?
Figure: in every pair the right number is made from the left number by the same rule.
AdvancedTry it yourself first
Which number replaces the ? in the figure?
2 → 97 → 648 → 81? → 100
Figure: in every pair the right number is made from the left number by the same rule.
AdvancedTry it yourself first
Which number replaces the ? in the figure?
2

Grids and tables

Row / column rule, constant sum, letter grid

Grids of numbers in which the third number of a row or column is made from the first two, constant-sum grids and grids of letters.

Third number = a rule on the first two, in every row or column

c = f(a, b)

A 3 × 3 (or 4 × 3) grid in which the third number of every row (or column) is made from the first two by the same rule.

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A grid with a constant sum

Every row and column adds up to the same total

When every row, every column (and sometimes the diagonals) has the same total; the blank equals total − the other numbers.

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A grid of letters

One step along rows, another down columns

In a grid of letters each row and each column runs with a steady gap of its own; read the letters as numbers (A = 1 … Z = 26).

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3

Triangles and circles

The middle number from the corners or sectors

The rule that makes the middle number from the three corners of a triangle and from the four sectors of a circle.

Triangles — the middle number from the corners

middle = f(top, left, right)

In several triangles the same rule makes the middle number from the three corners; one number in the last triangle is a ?.

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Circles — the middle number from four sectors

Opposite pairs

The rule making the middle number from the sector numbers of circles split in four: pairs of opposite sectors.

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4

Strategy guide

Where to begin and the common traps

The order of finding the rule in any figure, common traps and ways to check.

The order of checks

Beginning from the look of the figure

What to look at first for each kind of figure and which rules to try first.

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Common traps

The mistakes that keep happening

The commonest ways of losing marks in missing-character questions.

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5

Mixed practice

All types mixed — first recognise the figure

Pairs, grids, triangles, circles and letter grids in three levels.

Basic practice

Add, subtract, multiply rules

Simple grids, pairs, constant sum, triangles and circles.

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Standard practice

Rules with multiples, squares and pairs

Rules like 2a + 3b, square rules, the ? in any place, and pair rules for triangles and circles.

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Advanced practice

Square rules, three variables and solving backwards

Grids with complex rules, pairs of cubes and squares, triangles and circles in which a corner or sector is unknown.

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6

Revision — cards

Flip each card and test yourself

Think of the answer yourself first, then flip.

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