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Reasoning: patterns, puzzles and thinking traps

Reasoning in nine levels: number patterns to continue, problems to solve step by step, knights and knaves logic puzzles, probability questions that surprise most people, and the thinking traps and mental models that come up in everyday decisions. Each topic starts with a short lesson.

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The levels

Patterns and traps

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Thinking tools

The lessons

Before its first questions, each of these levels shows a short lesson: an example to try, the method, and why it works.

1.1 Number patterns

Look at the gaps between the numbers. If every gap is the same, keep adding (or taking away) it. If not, try dividing: maybe each number is the one before times the same amount. Some patterns do both: × 2, then + 1.

Why it works: Most simple patterns are one repeated step. Writing the gaps down turns a guess into a check.

1.2 Trickier patterns

When the gaps aren't equal, look at the gaps between the gaps: 3, 5, 7, 9 grow by 2 each time. Also try adding the two numbers before, or splitting the list into two series that take turns.

Why it works: A pattern is a rule that makes every number from the ones before. Only call it found when it fits every number shown.

1.3 Think twice

These questions have a quick answer that feels right and is wrong. Before you type, check your answer against the question: a notebook and pen at 110 together, with the notebook 100 more? Then the pen is 5, not 10.

Why it works: Our fast, automatic thinking jumps to an answer; slow thinking checks it (Kahneman's System 1 and System 2). Checking takes a second and saves the trap.

2.1 Problem-solving strategies

Pick a strategy before you start. Work backwards from the end, undoing each step. Imagine a simpler case: if all the animals were chickens, how many legs? Draw it: 3 gaps need 4 fence posts.

Why it works: George Pólya's How to Solve It: understand the problem, make a plan, carry it out, then look back and check. A strategy turns a puzzle into steps.

2.2 Knights and knaves

Knights always tell the truth and knaves always lie. Try each possibility: if A is a knight, is what A says true? If A is a knave, is it false? Keep only the possibility where everyone's words fit.

Why it works: Testing every case and throwing out the contradictions is how logicians solve these (Raymond Smullyan's puzzles). Exactly one answer survives.

2.3 How likely, really?

Turn percentages into people. Out of 1,000: how many really have it, and how many of those does the test catch? How many of the rest does it wrongly flag? Then compare the two groups of positives.

Why it works: People are much better with counts than with percentages (Gerd Gigerenzer's natural frequencies). When something is rare, false alarms can outnumber real cases.

3.1 Thinking traps

Each story shows a common thinking trap: a mental shortcut or habit that can lead us wrong. Read it, then ask what went wrong in the thinking, and match it to the trap.

Why it works: Knowing the names of these traps (from books like The Art of Thinking Clearly) makes them easier to spot in your own thinking.

3.2 Mental models

A mental model is a tool for thinking: "and then what?", "what would make this fail?", "what am I giving up?". Read each story and pick the tool that's helping.

Why it works: Having a few good thinking tools ready (from books like The Great Mental Models) helps you think a problem through instead of just reacting.

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