Reasoning Puzzles
No grid to fill and nothing to shade. These five give you a run of terms or a page of clues and ask you to work out the rule behind them.
Inferring A Rule Is A Different Skill From Applying One
Everywhere else on this site the rule is handed to you and the work is carrying it out accurately. Here it is the opposite: the rule is what you are looking for, and once you have it the answer takes seconds.
That changes the method. Guessing and checking is genuinely the right approach — try the gaps, then the ratios, then the differences of the differences — because the space of plausible rules is small and testing one is cheap.
Start With The Gaps, Not The Numbers
For any run of numbers, write down the differences between consecutive terms before you do anything else. A constant difference is an arithmetic run; a difference that itself grows steadily is a quadratic; a constant ratio is geometric. Three lines of arithmetic settle most series without a flash of insight.
Letter series works the same way once you replace each letter with its position in the alphabet, and the pages here say so in the first step of every explanation.
Why One Wrong Term Is Harder Than A Missing One
A missing term at the end has the whole run to constrain it. A wrong term in the middle means the run you are looking at is partly a lie, and the first job is deciding which part to trust. That is why the broken term is never the first or last: a wrong first term cannot be proved wrong, because nothing precedes it and the series would simply start elsewhere.
| Type | You are given | You supply |
|---|---|---|
| Number series | A run of numbers with the last missing | The next term |
| Wrong number | A complete run, one term altered | Which term breaks the rule |
| Letter series | A run of letters | The next letter |
| Clock angles | A time | The angle between the hands |
| Zebra puzzle | A page of clues | The whole arrangement |
Printing A Reasoning Puzzle
These are the most portable puzzles on the site: a number series is one line of text, and a zebra puzzle is a page of clues and a grid. Both print cleanly with the interface stripped out, and the zebra grid in particular is easier to work on paper, where you can cross squares off faster than you can click them.
Why Guessing Is The Correct Method Here
Everywhere else on this site guessing is a failure — the grids are built so that no guess is ever needed. Reasoning puzzles are the deliberate exception. You are hunting for a rule, the space of plausible rules is small, and testing one costs seconds. Trying the gaps, then the ratios, then the second differences is the method, not a shortcut around it.
That is worth saying because people arrive at series puzzles expecting a flash of insight and feel they have cheated when they find the answer by trial. They have not. Systematically testing a short list of candidate rules is exactly what an experienced solver does.
| Try | It is this if | Example |
|---|---|---|
| Differences between terms | The difference is constant | 3, 7, 11, 15 — add 4 |
| Differences of the differences | The second difference is constant | 2, 5, 10, 17 — squares plus one |
| Ratios | Each term divides evenly into the next | 3, 6, 12, 24 — double |
| Sum of the previous two | Each term equals the two before it | 1, 2, 3, 5, 8 |
| Two series woven together | Odd and even positions behave differently | 1, 10, 2, 20, 3, 30 |
| Position in the alphabet | You are looking at letters | C, F, I, L — add three |
Reasoning Puzzles Frequently Asked Questions
What is a reasoning puzzle?
A puzzle where you infer a hidden rule from examples rather than apply a rule you were given. A run of numbers, a run of letters, or a set of clues about an arrangement — in each case the rule is the thing you are hunting, and the answer follows once you have it.
How do I solve a number series?
Write the differences between consecutive terms first. A constant difference means each step adds the same amount; a difference that grows steadily means a squared relationship; a constant ratio means multiplication. If none of those fit, check whether two series are woven together, or whether each term is the sum of the two before it.
Why is the wrong term never at the start or the end?
Because a wrong first term cannot be proved wrong. Nothing precedes it, so any value is consistent with the rest and the series simply starts somewhere else. The same is true at the end in reverse. Only a term with neighbours on both sides can contradict anything.
Do clock angle puzzles need the hour hand to move?
Yes, and that is the whole point of the type. The hour hand moves half a degree every minute, so at twenty past three it is not on the three — it has drifted ten degrees past it. Every wrong answer offered here is a named version of forgetting that.
What is a zebra puzzle?
A logic grid: several categories, a set of clues relating them, and exactly one arrangement that satisfies every clue. It is also called a logic grid puzzle or an Einstein puzzle. Every clue in the puzzles here is needed — remove any one and the answer stops being deducible.
Related Practice
Figure series is this same rule-hunting in shapes rather than numbers. Number logic puzzles hand you the rule instead, and word puzzles ask the same question of letters.