The maths, science and reasoning guide
Written guides to mathematics, science and pattern reasoning. Worked examples, the reasoning behind the rules, and where the usual claims are overstated. Written to be read, not skimmed.
Mathematics
- Counting things without listing them
Combinatorics is the mathematics of counting arrangements. Three rules cover most of it, and they turn impossible-looking questions into arithmetic.
- Probability, and why your intuition about it is wrong
Probability is counting, divided. The arithmetic is easy; the hard part is that human intuition about chance is reliably and predictably mistaken.
- Prime numbers: the atoms of arithmetic
Primes are the numbers that cannot be broken down. They are the building blocks of every other number, they never run out, and they keep the internet secure.
- Fibonacci, the golden ratio, and how things grow
The Fibonacci sequence appears in pinecones, sunflowers and puzzle papers. Here is what it actually is, where the golden ratio comes from, and which claims about it are nonsense.
- Clock arithmetic: what happens when numbers wrap around
Modular arithmetic is the mathematics of cycles. You already use it to tell the time, and it underlies calendars, check digits and cryptography.
- Pascal's triangle, and the patterns hidden in it
A triangle built by adding neighbours contains the combinations, the binomial expansions, the triangular numbers and the powers of two all at once.
- Binary, and why ten was never special
Base ten is an accident of anatomy. Understanding that other bases work identically explains computers, and makes place value obvious rather than mysterious.
- What a proof is, and why checking a million cases is not one
Mathematics is the only field where you can be completely certain. That certainty comes from proof, which is a stricter thing than evidence.
- Estimation: getting the right answer to within a factor of ten
Fermi estimation is the skill of producing a roughly right number from almost no information. It is more useful than most exact arithmetic you will ever do.
- Angles, polygons, and why the rules are not arbitrary
A handful of facts about angles explain every regular shape, why only three tile a floor, and where 360 degrees came from.
Science
- How science actually decides what is true
Science is not a collection of facts but a method for finding out when you are wrong. Understanding falsifiability changes how you read every claim.
- Measurement, units, and the errors that destroy spacecraft
A number without a unit is not a measurement. Unit discipline catches more mistakes than careful arithmetic, and its absence has ended missions.
- Forces and motion: three laws that took two thousand years
Newton's laws are short, and every one of them contradicts an intuition people arrive with. That is why they were so hard to find.
- Energy, and why perpetual motion is impossible
Energy cannot be created or destroyed, only moved and converted. That single rule rules out an entire category of invention and explains why nothing is ever 100% efficient.
- Waves: one set of rules for sound, light and everything else
Sound, light, radio and earthquakes all obey the same equations. Learning waves once explains all of them.
- Atoms, elements, and why the periodic table is shaped like that
The periodic table is not a list. Its shape is a direct picture of how electrons arrange themselves, which is why it predicts chemistry.
- Electricity: what is actually moving, and what is not
Current, voltage and resistance are three different things that get used interchangeably in ordinary speech. Separating them makes circuits straightforward.
- Gravity and orbits: falling, and missing the ground
An orbit is not a balance between gravity and some outward force. It is continuous free fall that keeps missing, which is why astronauts float.
- Evolution by natural selection: three conditions and a consequence
Natural selection needs only variation, inheritance and differential survival. Given those three, complexity without a designer is not surprising but inevitable.
Reasoning and pattern logic
- How matrix reasoning puzzles actually work
A matrix reasoning puzzle gives you a grid with one cell missing. Here is how the grids are built, how to find the rule, and why checking the answers first makes you slower.
- Rotation, reflection, and how to tell them apart
Rotation and reflection often produce identical-looking results, and puzzles exploit that. Here is the reliable test for which one you are looking at.
- Reading a sequence: how progressions are built
Sequence puzzles change one or more attributes at a steady rate. Here is how to separate the attributes and find the step.
- Odd one out: reasoning by exclusion
Odd-one-out puzzles ask which figure does not belong. The difficulty is that several groupings are usually valid — here is how to find the one the puzzle means.
- When more than one rule is running at once
Hard puzzles are rarely built from exotic rules. They are two or three ordinary rules applied to different attributes at the same time. Here is how to pull them apart.
- The traps that catch good solvers
Most marks lost on reasoning tests are not lost to hard puzzles. They are lost to a small set of predictable mistakes, all of which are avoidable.
- Getting faster without starting to guess
Speed on reasoning tests is a side effect of recognition, not something you can train by rushing. Here is what actually produces it.
- Analogies: A is to B as C is to what
Visual analogy puzzles ask you to find a relationship in one pair and apply it to another. The skill is describing the relationship precisely enough to transfer.
- Symmetry, and the four kinds worth knowing
Symmetry is one of the most common rules in abstract reasoning and one of the least precisely understood. Here are the four types and how to test for each.
- Why these puzzles feel hard, and what to do about it
The difficulty in reasoning puzzles is usually not the reasoning. It is holding several things in mind at once. Here is how to work around a real limit.
More articles
- Sets, Venn diagrams and the words that trip people up
And, or, not and all are used loosely in English and precisely in mathematics. The gap between the two causes more confusion than any symbol.
- Graphs and networks: the maths of things that are connected
Graph theory started with a walk over seven bridges. It now runs route planning, social networks and the internet.
- Averages, and the three ways they mislead you
Mean, median and mode answer different questions. Choosing the wrong one, or quoting an average with no spread, is how honest-looking numbers deceive.
- Ratio, proportion, and why doubling the size is not doubling the material
Proportional reasoning is the most reused idea in school mathematics, and the square-cube law it leads to explains why giants and giant insects cannot exist.
- Why there are letters in mathematics
Algebra is not arithmetic made harder. It is the point at which you stop answering questions one at a time and start describing the rule that answers all of them.
- Number patterns hiding inside picture puzzles
Many visual puzzles are numeric rules wearing shapes as a disguise. Here is how to spot them and the differencing technique that cracks most of them.
- Nets, folding and three-dimensional rotation
Three-dimensional questions ask you to fold a flat net into a solid or rotate a solid in your head. Both have reliable methods that avoid doing it visually.
- How to practise so that it actually sticks
Most practice on reasoning tests produces very little improvement. The difference between practice that works and practice that does not is small and specific.
- Deduction and induction: the two moves behind every puzzle
Reasoning puzzles use two different kinds of thinking, and knowing which one a question wants tells you how certain your answer can be.
- What these tests actually measure, and what they do not
Abstract reasoning tests measure something real but narrow. Being honest about the limits makes the practice more useful, not less.