2.4 Input-Output (Machine Input) Arrangements
Key Takeaways
- Machine input-output sets show an Input line and several Steps, and the rule is found by comparing the Input with Step I and Step I with Step II.
- In sorting-type machines, elements that have not yet moved keep their original relative order, so the unarranged part must never be re-sorted.
- When a machine places one word-number pair per step, the number of steps equals the number of pairs unless a pair is already in its final place.
- For the input '57 bank 32 loan 91 fund 18 cash 64 note', placing words alphabetically with numbers in descending order from the left ends at Step V: 'bank 91 cash 64 fund 57 loan 32 note 18'.
- Operation-based machines keep positions fixed and change the values by a rule such as adding digits or squaring.
2.4 Input-Output (Machine Input) Arrangements
Exam Focus: Machine input-output sets present a line of words and numbers that a "machine" rearranges one step at a time. They have appeared in bank clerk Mains reasoning papers as multi-question sets, and they reward candidates who find the rule from the Input and the first two steps instead of tracing blindly.
How an Input-Output Set Works
A set gives:
- An Input line, typically 6 to 10 elements (words, numbers, or both).
- Several Steps showing the rearrangement in progress.
- A statement that the last step is reached when the rule can no longer change the line.
Questions then ask for a particular step, the position of an element in a step, the number of steps needed, or what a new input looks like after a given step.
The Four Common Machine Types
| Type | What changes at each step | Signal in Step I |
|---|---|---|
| Sorting | One element moves to an end (left or right) in alphabetical or numerical order; the rest keep their relative order | The first word alphabetically, or the largest/smallest number, is now at an end |
| Pairing | A word and a number move together, often word first and number second | A pair such as "bank 91" appears at one end |
| Shifting (cyclic) | Every element moves a fixed number of places | The same elements appear in rotated order |
| Operation-based | Numbers change by an arithmetic rule while positions stay fixed | Values change, order does not |
Sorting and pairing machines are the most common, so master them first.
The Three-Line Rule-Finding Method
- Compare the Input with Step I: Which elements moved, and to which end?
- Compare Step I with Step II: Confirm that the same logic repeats (the next word alphabetically, the next largest number).
- Check the last step: It shows the finished order and confirms the direction (ascending or descending) and the end being filled.
[!IMPORTANT] In sorting and pairing machines, the elements that have not yet moved keep their original relative order. Candidates lose marks by quietly re-sorting the unarranged part.
Worked Example 1: Word-Number Pairs From the Left
Input: 57 bank 32 loan 91 fund 18 cash 64 note
Rule found from the steps: At each step, the word that comes first alphabetically among the remaining words moves to the left end of the unarranged part, immediately followed by the largest remaining number.
| Step | Arrangement |
|---|---|
| Input | 57 bank 32 loan 91 fund 18 cash 64 note |
| Step I | bank 91 57 32 loan fund 18 cash 64 note |
| Step II | bank 91 cash 64 57 32 loan fund 18 note |
| Step III | bank 91 cash 64 fund 57 32 loan 18 note |
| Step IV | bank 91 cash 64 fund 57 loan 32 18 note |
| Step V | bank 91 cash 64 fund 57 loan 32 note 18 |
Step V is the last step because every word-number pair is now in place. Notice how, in Step I, the unmoved elements 57 32 loan fund 18 cash 64 note stay in their original order.
Typical Questions on This Set
- "How many steps are needed to complete the arrangement?" Five, one for each of the 5 word-number pairs.
- "In Step II, which element is 4th from the right end?" Step II reads
bank 91 cash 64 57 32 loan fund 18 note. Counting from the right: note (1), 18 (2), fund (3), loan (4). - "In which step do 'fund' and '57' first appear next to each other?" Step III.
Counting Steps Without Tracing Everything
- A machine that places one element per step needs at most as many steps as elements to be placed.
- A machine that places a pair per step needs as many steps as there are pairs.
- If an element or pair is already in its final position at the start, that step is skipped and the count falls.
Worked Example 2: An Operation-Based Machine
Input: 24 37 15 42 59
Step I: 6 10 6 6 14 — each number is replaced by the sum of its digits (2 + 4 = 6, 3 + 7 = 10, 1 + 5 = 6, 4 + 2 = 6, 5 + 9 = 14).
Step II: 36 100 36 36 196 — each Step I value is squared.
Once the two operations are known, any question about a new input is arithmetic: for input 81, Step I gives 9 and Step II gives 81.
Common Traps
| Trap | How to avoid it |
|---|---|
| Re-sorting the unarranged part | Copy unmoved elements in their original order |
| Assuming an earlier line can be rebuilt from a later step | In sorting machines, many different inputs can produce the same Step III, so "cannot be determined" is often the right answer |
| Miscounting positions "from the right" | Start counting at 1 from the right end |
| Assuming words and numbers follow the same order | Check separately whether words ascend and numbers descend, or the reverse |
| Forgetting which end is being filled | Confirm from Step I whether the arranged block grows from the left or the right |
A machine rearranges words and numbers. At each step, the word that comes first alphabetically among the unarranged words and the smallest unarranged number move to the right end of the unarranged part, in the order word then number; unmoved elements keep their order. For the input '46 rate 83 bond 25 yield 70 debt 12 gold', which element is third from the left in Step III?
Using the same machine and the same input ('46 rate 83 bond 25 yield 70 debt 12 gold'), how many elements are there between '83' and 'debt' in Step II?
Using the same machine and input, which option gives the last step of the arrangement?