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Input-Output

Reasoning · Logical & Analytical Reasoning

7 Part1 worked examples6 practice questions

Imagine a machine in front of you. You feed it a line — say “pen 47 boat 23 house”. The machine changes it a little at a time: the first step gives one line, the second the next, and so on step by step up to the final line. You are not told the machine’s rule; you must find it yourself from the steps. This is input-output.

The question gives an example (an input and its steps) and then asks: what a step of a new input will be, how many steps are needed, where an item will be in a step, or what the input was from a given step. This chapter teaches all machines in three families, and every line of every step is computed.

How to use this chapter

  1. Read — the method, word rules and three families.
  2. Look at the table — after each machine’s example table come questions on that machine.
  3. Try it yourself — first find the rule (the solution shows it), then work out the later steps.
  4. Know the level — Basic, Standard, Advanced.
NoteThe questions you solve are remembered in this browser; see your progress on the right (the “Contents” button on a phone).
1

Foundation — machine, step and rule

The method of finding the rule

Vocabulary, three families, four steps to find the rule and a reminder of alphabetical and number order.

Vocabulary and the three families

Input, step, rule · arranging, shifting, arithmetic

The words of a machine question and the three kinds of machine.

TermMeaning
InputThe starting line fed into the machine
StepOne change by the machine; Step I, Step II …
RuleWhat the machine does at every step — you have to find it
Last stepWhen the machine has nothing more to do (the line is fully arranged)
FamilyWhat happens at each stepExample rule
1. ArrangingOne item reaches its place (an end); the rest stay as they werethe smallest number goes to the left end
2. ShiftingMany items change place by a fixed patternthe first item goes to the end; neighbouring pairs swap
3. ArithmeticThe values of the numbers changeadd 4 to every number; then ascending order

Below is a complete example of an arranging machine. See that at each step only one item reaches a better place and the order of the rest is unchanged — this is the mark of family 1.

A machine rearranges a line of words and numbers step by step by a fixed rule. An example (the input and all its steps) is below. Find the rule yourself.

StepLine
Input25 gate farm wheel basket jungle 98
Step Ibasket 25 gate farm wheel jungle 98
Step IIbasket farm 25 gate wheel jungle 98
Step IIIbasket farm gate 25 wheel jungle 98
Step IVbasket farm gate jungle 25 wheel 98
Step Vbasket farm gate jungle wheel 25 98
Example 1 · Basic
By the rule of the machine in the table above — Input: 80 gate 11 ship door water home Question: What will Step IV be?
Solution, step by step
  1. 1The rule (in turn): the alphabetically first word goes to the left end.
  2. 2Input: 80 gate 11 ship door water home
  3. 3Step I: door 80 gate 11 ship water home
  4. 4Step II: door gate 80 11 ship water home
  5. 5Step III: door gate home 80 11 ship water
  6. 6Step IV: door gate home ship 80 11 water
Answer door gate home ship 80 11 water
Find the rule and then work out the later steps: each step places one item at an end.
BasicTry it yourself first
By the rule of the machine in the table above — Input: task 62 bake heart drive 15 kernel Question: How many steps are needed to complete the rearrangement?
BasicTry it yourself first
By the rule of the machine in the table above — Input: 13 rope 14 lemon time milk nurse Question: What will the last step be?

Four steps to find the rule

Compare, recognise, test, apply

Comparing two consecutive steps, recognising the family, confirming with the last step, and a reminder of word and number order.

Four steps to find the rule
1. Compare the input and Step I: what changed?2. Compare Step I and II: is the same change repeated?3. Test the rule on every step shown4. Confirm with the last step
You seeLikely family
One item left its place for an end; the rest keep their orderArranging
All items changed place but the set of items is the sameShifting
The values of the numbers changedArithmetic

A reminder of alphabetical and number order

The alphabetical order of words is like a dictionary: look at the first letter (a to z); if it is the same, the second; then the third. For example gate, gift, game: all start with g; second letters a, i, a — so gift comes last; in gate and game the third letters are t and m — game first. Ascending order of numbers is small to large, descending large to small.

TrickThe last item reaches its place by itself in the final step (nothing else is left for it), so the total number of steps is often (number of items − 1) when all items are placed at the ends in turn. But when only some items move (such as only the numbers), the steps are as many as the items that move.

A machine rearranges a line of words and numbers step by step by a fixed rule. An example (the input and all its steps) is below. Find the rule yourself.

StepLine
Input54 44 37 30 hill 50 34
Step I44 37 30 hill 50 34 54
Step II30 44 37 hill 50 34 54
Step III30 44 37 hill 34 50 54
Step IV30 34 44 37 hill 50 54
Step V30 34 37 hill 44 50 54
BasicTry it yourself first
By the rule of the machine in the table above — Input: 56 24 36 47 11 honest 17 Question: What will Step V be?
BasicTry it yourself first
By the rule of the machine in the table above — Input: 70 53 49 25 23 wave 16 Question: How many steps are needed to complete the rearrangement?
BasicTry it yourself first
By the rule of the machine in the table above — Input: 48 71 34 64 85 water 20 Question: In Step II which item is second from the left?
BasicTry it yourself first
By the rule of the machine in the table above — Input: 46 95 13 40 25 89 hole Question: What will the last step be?
2

Arranging machines

One item takes its place at every step

Machines that arrange numbers, words and both in turn.

Machines that bring numbers to an end

The smallest or largest number to the left or right end

The rule of placing numbers one by one at an end; the words keep their places.

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Machines that bring words to an end

Alphabetical order and the length of the word

The alphabetically first or last, or shortest or longest word, one by one at an end.

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The mixed rule for words and numbers

One number and one word in turn

Two rules in turn — one on numbers and one on words — and counting their steps.

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3

Shifting machines

Places change, the items stay the same

Rotating, swapping pairs, reversing, swapping the ends and step-dependent rules; when the line comes back.

Simple shifting rules

Rotating, pairs, reversing, ends, halves

Machines that repeat one position-change at every step and how to recognise them.

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Rules that change with the step

Steps that depend on the step number or alternate

The machine that swaps the next pair at each step and machines doing two operations in turn.

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When the line comes back

Counting the length of the cycle

The number of steps until the line becomes the input again when the machine is run on and on.

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4

Machines that work on numbers

Adding and subtracting, reordering, digit sums

Number machines that repeat two operations in turn and their reverse questions.

Adding and subtracting machines

Adding or subtracting for every number, odd and even places

Operations that change the values of the numbers and two operations in turn.

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Machines that reorder and sum digits

Ascending or descending, reversing, digit sums

Operations that change order and form: digit sum, ascending, descending, reversing, swapping the ends.

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Finding the input from a step

Run the operations backwards

Finding the input from a step through operations that can be undone and checking options by running them forward.

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5

Strategy guide

One order for every machine and the traps to avoid

Spotting the family, the order of working out steps and common mistakes.

The table of recognition and the order

First the family, then the rule

A table for spotting the family from the first two steps and what to check in each.

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Common traps and time-savers

The mistakes that keep happening

Reasons for losing marks in machine questions.

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6

Mixed practice

All three families mixed

Arranging, shifting and arithmetic machines, in three levels.

Basic practice

One rule, direct steps

Three machines with simple rules.

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Standard practice

Word order, alternating operations, reverse questions

A word machine, a step-dependent rule and a number machine that reorders.

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Advanced practice

The mixed word-number rule, cycles, reverse questions

Questions with long steps, the length of a cycle and finding the input from a step.

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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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