Last reviewed 22 Sept 2026 · 6 min read
What the question gives
A "machine" is given an input — a line of words and numbers — and rearranges it a little in each step until it can go no further.
Input: tree 41 garden 18 school 63 river 25
Step I: 18 tree 41 garden school 63 river 25
Step II: 18 25 tree 41 garden school 63 river
You are asked for a particular step of a new input, for the last step, or for how many steps that input needs.
Finding the rule
Compare the input with Step I, then Step I with Step II. Ask:
- What moved? Usually one item — sometimes one from each end.
- Where did it go? To the extreme left, the extreme right, or one place along.
- Why that item? The smallest number, the largest number, the word that comes first alphabetically, the longest word, or the word beginning with a vowel.
- What happened to the rest? They usually shift to fill the gap, keeping their order.
In the example: 18 is the smallest number and moved to the left; then 25, the next smallest, joined it. The rule is smallest number to the left, one per step.
The arranged part of the line grows by exactly one item per step. Everything else keeps its relative order, so you only ever have to decide which item is next — never re-shuffle the whole line.
Common rules
| Rule | Signature |
|---|---|
| Smallest number to the left, one per step | numbers gather on the left in increasing order |
| Largest number to the left | numbers gather on the left in decreasing order |
| Alphabetically first word to the left | words gather on the left in dictionary order |
| Alphabetically last word to the right | words gather on the right |
| One from each end per step | two items placed per step; the line halves quickly |
| Alternate: a word then a number | the arranged block alternates in type |
How many steps
If one item is arranged per step and there are items in all, the machine needs steps: the last item is already in place once the others are. If the rule arranges one word and one number per step, the count is about half that. The paper's own example is the best guide — count its steps and match.
Working backwards
A question may give a step and ask for the input. This is only possible when the rule is reversible, so exam keys usually restrict such questions to "which step is this" rather than "what was the input". If asked, undo the last placement first and work back one item at a time, and state any step that cannot be recovered.