Skip to content
Mintnib™

Mintnib › PAT › Method library

The PAT Method Library

Fifty habits rather than fifty topics: what to do when a physics problem does not tell you what to do. Each one says when to reach for it. 50 methods in 5 groups.

Schools and organisations we have worked with

Shown with their permission. Not endorsements. See all

The PAT no longer runs, and these methods still matter

Oxford retired the Physics Aptitude Test after the October 2025 sitting. For Physics, for Physics and Philosophy and for Engineering Science, Oxford now uses the ESAT. Nobody reading this page is about to sit the PAT, and nothing here should be read as advice about a test you are preparing for.

We publish the methods anyway, because they are the part that transfers. Every habit below is what an Oxford physics interview rewards and what the ESAT rewards: estimating a quantity nobody has measured, letting the units tell you the shape of an answer, staying in symbols until the last line, deciding what is small enough to neglect and saying so out loud, and testing a result against a world whose size you already know. None of that retired with the paper.

If you have a test to sit this cycle, start at our ESAT pages. That is the test these courses now use, and our ESAT library goes much deeper. Come back here for the habits.

This page says nothing about how long any paper was, what you were allowed to take into it, or how it was marked. Those things changed between cycles and are not a claim a page like this should make. What is described here is how to think about a physics problem.

How this PAT method material was checked

The habits themselves are standard; the wording and the choice of fifty are ours. The library was first written as an index into our PAT practice papers, which are withdrawn, so no entry links to a question and none is missed.

How to use this library

Read the method name first, then the sentence under it, which says when to reach for it. These are not topics, and the difference is the whole point. A topic index tells you what a question is about, which you already know by the time you are stuck. This one tells you what to do next, which is the thing that was actually being tested.

The five groups are five habits: estimate when you cannot calculate, let the units do work, stay in symbols and substitute last, decide which effect matters most, and check the answer against physical reality. Run them in that order against an unfamiliar problem and you will almost always have something to say inside a minute, which in an interview is most of what is being watched.

When a problem defeats you, find the method it needed here and try that method on other problems. Seeing one habit in four different settings is what makes it transfer, and transfer is the reason we keep this page for a test that no longer runs.

Contents

Order-of-magnitude estimation

Some questions cannot be answered exactly and are not meant to be. The skill is to turn a quantity nobody has measured into a product of quantities you can each defend, to keep the arithmetic in powers of ten, and to stop at the precision your worst input supports. An interviewer asking how many piano tuners work in a city is not testing arithmetic: they are watching whether you decompose the problem out loud, commit to numbers, and stay honest about how uncertain the result is. These methods cover the decomposition itself, the choice of inputs, and the discipline of reporting an estimate rather than a false answer.

Break one impossible estimate into factors you can each guess

Reach for this when the quantity is impossible to look up but can be written as a product of two or three factors you can each guess to within a factor of a few.

Choose a round number you can defend out loud

Reach for this when a messy input would let the arithmetic crowd out the thinking, and a round value you can justify keeps the estimate both quick and honest.

Bracket the answer between a bound too big and a bound too small

Reach for this when you cannot name the answer but can name a value it is certainly below and a value it is certainly above.

Estimate by comparison with something whose size you already know

Reach for this when you have no formula but do know the size of something comparable, so the estimate becomes a ratio rather than a calculation.

One significant figure is often the honest answer

Reach for this whenever the inputs are guesses, because carrying more digits than the weakest input supports invents a precision that is not there.

Do the powers of ten first, the digits afterwards

Reach for this when the numbers span many decades, so the exponents can be added in your head and the leading digits tidied up at the end.

Count by dividing the big thing by the small thing

Reach for this when the question is really how many of one thing fit inside another, whether in length, area, volume or time.

A rate multiplied by a time, and be honest about the time

Reach for this when a quantity accumulates, so the estimate is a rate times a duration and the duration is usually the part worth arguing about.

Find the one factor your estimate is most at the mercy of

Reach for this when an estimate is challenged, because the factor worth defending or refining is the one whose plausible range moves the answer most, once the power it is raised to is counted.

Between two bounds, take the geometric mean, not the average

Reach for this when your two bounds are a factor of ten or more apart, where an arithmetic mean would sit far too close to the upper bound.

Dimensional analysis

Every physical equation is a statement about quantities that carry units, and that constraint is strong enough to do real work. Dimensions can hand you the form of a relationship you have never derived, rule out a route before you follow it, rebuild a formula you have forgotten, and catch in seconds a slip that would otherwise survive to the final line. What they cannot supply is the dimensionless factor out front, or how the answer depends on any dimensionless group the variables can form, and knowing where that boundary sits is part of the method. These methods cover deriving, checking and recovering with dimensions, together with the bookkeeping habits that keep units usable.

Assume a product of powers, then let the dimensions fix the exponents

Reach for this when asked how a quantity depends on the others and no derivation is available, since a product of powers plus dimensional consistency usually pins the exponents.

Check the dimensions before you check the arithmetic

Reach for this at the end of every symbolic answer, because a dimensional slip is far cheaper to find here than inside the arithmetic.

A dimensionally impossible route is wrong before it is finished

Reach for this when several routes look plausible, since a route that would add unlike quantities can be abandoned before any effort is spent on it.

Collect the variables into dimensionless groups

Reach for this when a problem has too many variables, because the behaviour usually depends on a few dimensionless combinations rather than on each variable separately.

Recover a forgotten formula from the units it must have

Reach for this when you cannot remember a formula but do remember the units of everything that goes into it.

Read a constant's dimensions off the equation that defines it

Reach for this whenever a constant appears with no stated units, because the equation that introduces it always fixes them.

Whatever sits inside a sine, a log or an exponential carries no units

Reach for this whenever an angle, an exponent or a logarithm appears, since anything carrying units inside one of them signals an earlier mistake.

Convert every prefix once, before any algebra

Reach for this the moment a problem mixes prefixes or unit systems, so the algebra never has to carry a stray factor of a thousand.

Dimensions give you the form, never the factor of 2π

Reach for this when reporting a dimensionally derived result, so you claim the scaling you have earned and not a numerical factor you have not.

Per what? Name the denominator of every rate and every density

Reach for this whenever the words per, density, intensity or specific appear, because the denominator decides what the quantity actually means.

Working in symbols, substituting last

Substituting numbers early throws away everything except the number. Carrying symbols to the last line keeps the answer checkable by dimensions and by limits, makes cancellations visible, and turns a single result into a general one. It also makes the most common kind of physics question, the one that asks what happens to B when A doubles, answerable by inspection rather than by two full calculations. These methods cover working symbolically, scaling and proportionality arguments, and the small algebraic habits that keep a long derivation readable.

Carry the symbols to the last line, then substitute once

Reach for this by default, since a symbolic final line can be checked dimensionally, checked in limits and reused, while a decimal can be checked only for plausibility.

If it cancels, ask why you were never given it

Reach for this when a quantity you expected to need never appears in the answer, because understanding why it cancels is usually the point of the question.

Measure each variable in units of something the problem already supplies

Reach for this when the problem supplies a natural length, time or energy, so the algebra reduces to a statement about pure numbers.

Keep the intermediates exact and round only the final line

Reach for this in any multi-step calculation, where early rounding both loses accuracy and hides the cancellations that reveal the structure.

If this doubles, what happens to that?

Reach for this when the question asks about a change rather than a value, so a ratio answers it without ever evaluating the constant.

Divide one case by the other and the common factors walk out

Reach for this when the same physics is applied twice, since dividing one statement by the other removes every factor common to both.

Write the proportionality first, worry about the constant later

Reach for this when the relationship matters more than the number, so the physics is settled before the bookkeeping starts.

A straight line on log-log axes is a power law, and the gradient is its exponent

Reach for this when data or a graph is claimed to follow a power law, because on log-log axes the exponent is simply the gradient.

Give the recurring cluster one symbol

Reach for this when the same combination of symbols keeps reappearing, so the algebra stays readable and the structure stays visible.

Solve the general case once, then read the special cases off it

Reach for this when a question has several parts differing only in their numbers, since one general result answers all of them at once.

Identifying the dominant effect

A real problem contains more effects than any solution can carry, so the work is deciding which ones matter and saying so out loud. That means comparing terms by size rather than by appearance, using small-quantity approximations only where they are genuinely valid, identifying the two effects that balance, and separating processes that run on very different timescales. It also means recognising the case where a neglected term is the entire answer, because the large contributions cancelled. These methods cover approximating deliberately, and being able to defend the approximation afterwards.

Name what you are neglecting, and why it is small

Reach for this in any modelling answer, because naming the neglected effect turns a silent assumption into a decision a reader can follow and challenge.

Small angle: sine, tangent and the angle itself agree

Reach for this when angles stay below about ten degrees (0.17 radians), which is what turns a pendulum or a lens problem into a linear one.

Keep the first correction and drop the rest

Reach for this when a quantity is raised to a power and one part of it is small even after multiplying by the power, so the exact expression becomes a value plus a small correction.

Push a parameter towards 0 or ∞ and see what survives

Reach for this when a formula is hard to interpret, since its behaviour at the extremes usually says plainly what it means.

Size the terms against each other before you keep them all

Reach for this before simplifying, because whether a term can be dropped is a question about numbers, not about how the term looks.

Separate the fast process from the slow one, and freeze whichever you can

Reach for this when two processes run at very different rates, so one can be treated as instantaneous and the other as frozen.

Two effects balance, and the third is a correction

Reach for this when an equation has three or more competing terms, since the physics normally lives in whichever two balance each other.

When the big terms cancel, the small one is the answer

Reach for this when the leading contributions cancel, which is exactly when the term you were about to neglect becomes the whole answer.

Pick the crudest model that still contains the effect you are asked about

Reach for this at the start of an open problem, choosing the crudest idealisation that still contains the effect you have been asked about.

Close to equilibrium, almost every restoring force looks like Hooke's law

Reach for this when a system sits near a stable point, since almost any restoring force is linear there and the motion is simple harmonic.

Sanity-checking against physical reality

An answer is not finished when the algebra stops. Every result can be tested against a case you already know, against a symmetry, against a conservation law, against the direction it ought to move in, and against whether a quantity that size could exist in the world at all. These checks cost seconds and catch the errors that re-reading the working will not, because they interrogate the result rather than the process that produced it. These methods cover the checks worth running on every answer, in roughly the order that finds mistakes fastest.

Test the result on a case whose answer you already know

Reach for this the moment you have a general result, by setting a parameter to a value whose answer you already know.

Check the sign before you admire the magnitude

Reach for this whenever direction carries meaning, since a sign error changes what the answer says while leaving it looking perfectly respectable.

Ask whether that number could be true of the world you live in

Reach for this before writing any number down, by asking whether a quantity of that size would contradict something you already know about the world.

If nothing distinguishes the two, the answer cannot either

Reach for this when a configuration is unchanged by a swap, a reflection or a rotation, because the answer must respect the same invariance.

Audit the answer against what has to be conserved

Reach for this after any collision, circuit or energy-transfer answer, checking that the conserved totals really are unchanged.

Does it grow where it obviously should grow?

Reach for this when a formula contains several variables, testing that each one moves the answer in the direction physical intuition demands.

Nothing may exceed a bound that nature enforces

Reach for this whenever an answer approaches a hard limit such as the speed of light, absolute zero, unit probability or perfect efficiency.

Sketch it: the shape catches what the algebra hides

Reach for this when a problem is described in words or an answer is hard to picture, since a sketch exposes errors the algebra can conceal.

Predict the answer first, then explain any disagreement

Reach for this before a long calculation, so you have an independent expectation to compare the finished result against.

Count the unknowns against the equations before you start solving

Reach for this when you are unsure whether a problem is solvable as stated, before spending effort on algebra that cannot close.

Keep going