Thinking From the Beginning and Other Ways of Working Things Out

The price of a loaf

A woman sells bread in a market. She has always charged the same as the stall beside her, because that is what bread costs. One evening she sits down with a pencil and works it out from the beginning. Flour. Water. Salt. A little yeast. Firewood for the oven. An hour of her morning. She adds it up and finds that a loaf costs her less than half of what she charges, and less than she had ever guessed.

Now she knows something her neighbour does not. She can lower her price and still eat. She can keep her price and buy shoes. She can see which part of the loaf is expensive, and it turns out to be the firewood, so next month she looks into a better oven.

None of that came from asking around. It came from taking one thing apart into pieces small enough to weigh.

This is the oldest useful trick there is, and it has a plain name: work it out from the beginning. Break the question into parts. Keep breaking until you reach something you can weigh, count, taste, or look up. Then build the answer back up from those pieces, and see whether it matches what everyone told you.

It is a wonderful trick. It is also slow, and most days you should not use it. That second half is the part people forget, and it is what this piece is about.

Two ways to get an answer

There are only two ways to know what a loaf costs.

You can copy. Ask the stall next door, or your mother, or the person who has sold bread for thirty years. This is fast, it is free, and it is usually right, because a price that has survived thirty years has survived a lot of bad years too.

You can build. Sit down with a pencil for an evening and work it out yourself. This is slow, it is tiring, and it is the only way to find out that everybody has been wrong.

Neither one wins. They trade against each other, and the trade is always the same. Copying gives you an answer you cannot examine. Building gives you an answer you can examine, and charges you an evening for it.

So the question is never whether working things out from the beginning is admirable. It is whether this particular question is worth an evening. A loaf you will sell every morning for ten years, yes. A loaf you will bake once, no.

Take it apart until you reach something you can check

The method is easier to do than to describe.

Write down what you want to know as a number with a name attached to it. Not "is this a good idea" but "how much does one loaf cost me". Not "is the roof bad" but "how many litres come through in an hour of rain".

Then split that number into the pieces it is made of. Cost of a loaf splits into ingredients, fuel, and time. Split again. Ingredients split into flour, water, salt, yeast. Keep splitting.

Stop when you reach a piece you can check without asking anyone's opinion. A weight on a scale. A price on a sack. A number of minutes on a clock. Those are the bottom. Everything above them is arithmetic.

Then add it up and compare it with what you were told. When the two numbers are close, the received wisdom is sound and you have learned that too. When they are far apart, you have found something, and the gap tells you where to look.

The whole method is one habit: refuse to stop at a piece you cannot check.

A neat argument that nobody tested

Here is the way this goes wrong, and it is worth knowing before you start, because it has caught cleverer people than any of us.

For a very long time almost everyone believed that a heavy stone falls faster than a light one. It sounds obvious. You can build a tidy argument for it, and people did, and the argument was passed down and taught for centuries.

Any child could have settled it in an afternoon with two stones and a wall. For a very long time, nobody bothered.

That is the failure to guard against. A chain of reasoning can be perfectly tidy and still be about nothing, because tidiness is a property of the argument and not of the world. Working something out from the beginning gives you a chain. It cannot tell you whether the first link was ever true.

So the method only earns its keep when something outside your own head can contradict you. A scale. A price. A clock. A rain gauge. A friend with a reason to disagree with you. Whatever it is, its job is to tell you that you are wrong while it is still cheap to be wrong.

A thousand years ago a scholar in Basra spent years on the question of how we see. He wrote that anyone who wants the truth should treat everything he reads as an enemy until he has tested it himself. He was not the first to say so and not the last. Similar insistence turns up in old Chinese writing on how to judge an argument, and in old Indian writing on how anyone comes to know anything at all. People in every part of the world arrived at the same warning, which suggests the mistake is a human one rather than a local one.

Guessing well

Most of the time you cannot check every piece. You have to guess. There is a way of guessing that is far better than it has any right to be.

Say you want to know how many barbers there are in your town, and there is nobody to ask.

Guess the pieces instead of the answer. The town has about sixty thousand people. A person gets their hair cut maybe six times a year. That is around three hundred and sixty thousand haircuts a year. A barber might do twelve in a day, six days a week, fifty weeks a year, which is about three thousand six hundred haircuts. Three hundred and sixty thousand haircuts divided among barbers who each do three thousand six hundred comes to about a hundred barbers.

The real number might be sixty. It might be a hundred and fifty. It will almost never be five, and it will almost never be five thousand, and that is the point. You now know the size of the thing.

The reason this works is quietly beautiful. Every one of those guesses is wrong. But some of them are too high and some are too low, and when you multiply them together the errors pull against each other and partly cancel out. A single bold guess at the whole answer has nothing pulling against it. A chain of small guesses polices itself.

There is one way to spoil it. If every guess leans the same way, nothing cancels. A person who wants the answer to be large will guess large at every step, and five hopeful guesses in a row do not cancel, they compound. So mark each guess as you make it: measured, looked up, or hoped for. If most of them are hoped for, and they all lean the same way, you have not estimated anything. You have wished.

Start at the bad ending

Here is a different trick, and it is the cheapest one in this whole piece.

Instead of asking how to make the thing go well, say out loud that it has already gone badly, and then explain how.

A family planning a long journey can spend an hour asking how to make the journey smooth, and will produce vague, comfortable answers. Or they can say: we missed the train. Now explain why. Within two minutes somebody says the bags were not packed the night before, somebody else says the bus into town only runs twice an hour, and a child points out that nobody has looked at the ticket since it was bought.

The second conversation is better, and the reason is a fact about how minds work. Explaining something that has already happened is easy and we are good at it. Predicting something that has not happened is hard and we are bad at it. Saying "it failed" turns a prediction into an explanation, and you get the fluency for free.

Do this before you commit to anything you cannot undo. Announce the disaster, collect the reasons, and then go and fix the two or three that were embarrassingly obvious.

Keep asking why

There is a wet patch on the ceiling. Most people paint over it. Some people fix the roof tile. A few ask why five times.

Why is the ceiling wet? Water is coming through. Why is water coming through? A tile is cracked. Why is the tile cracked? A branch scrapes it whenever the wind blows. Why does a branch scrape it? The tree has not been trimmed in four years. Why has the tree not been trimmed? Nobody in the house thinks the tree is theirs.

The repair for the first answer is paint. The repair for the last answer is a person who owns the tree. Only one of those two stops the patch coming back.

Five is not a magic number. Sometimes you get there in three. Sometimes you need eight. The number is only there to push you past the first answer, because the first answer is nearly always a symptom wearing the clothes of a cause.

This has one limitation worth knowing. It follows a single thread. If the ceiling is wet because of a cracked tile and a blocked gutter and a badly sloped roof, all at once, then following one thread five steps down gives you a confident answer that fixes a third of the problem. When you suspect more than one cause, draw the whole thing on paper instead of following a line.

The guess that could go wrong

Play a game. I have a rule in my head. Three numbers fit it: two, four, six. You may test as many sets of three numbers as you like, and each time I will tell you yes or no. When you are ready, tell me the rule.

Almost everyone tries eight, ten, twelve. I say yes. They try twenty, twenty-two, twenty-four. I say yes again. They announce, with justified confidence, that the rule is even numbers going up.

The rule was any three numbers going up.

They never lost. They also never learned anything, because every test they ran was one they expected to pass. The two tests that would have taught them the rule in a single move were one, two, three, and six, four, two. Both of those risked a no.

Take the lesson out of the game and it becomes the single most useful question in this whole essay. Before you commit: what could I see, cheaply, that would tell me I am wrong?

If the honest answer is nothing, you do not have a plan. You have a hope with dates on it.

Ten in a thousand

Now the hardest thing here, and the most valuable, and there is no arithmetic in it beyond counting.

Imagine an illness that ten people in every thousand have. There is a test for it. The test is good: it finds eight of those ten. The test is also imperfect in the other direction, and it worries about ninety-five people out of the nine hundred and ninety who are perfectly well.

Your test comes back worrying. How frightened should you be?

Count the people. In that thousand, the number of people whose test is worrying is eight who are ill plus ninety-five who are not. That is about a hundred worried people. Eight of them are ill.

So a worrying result on a good test, for something rare, still leaves you far more likely to be well than ill. Not certainly well. You go and get the second test. But the terror that arrives with the letter is out of proportion to what the letter says, and now you know why.

Rareness is what does the work here. A rare thing stays rare after a hint of it appears, because the enormous crowd of people who do not have it produces more false hints than the tiny crowd who do.

Two things follow, and they are worth more than any framework.

First, always ask how common the thing is before you ask how good the sign is. A frightening sign for a rare thing is usually a false alarm. A mild sign for a common thing deserves more attention than it gets. Most bad reasoning about health, about crime, about a stranger's character, about which of your customers is a thief, is this one mistake wearing different clothes.

Second, notice that the counting was easy and the same question in percentages is nearly impossible. When this problem is put to doctors in percentages, most of them get it wrong, and badly wrong, by a factor of ten. When it is put to the same doctors as a thousand people standing in a room, most of them get it right. Nothing changed except how it was written down.

That is a general lesson about thinking, not about illness. When a question feels slippery, stop reasoning and rewrite it as a thousand people, or ten sacks of flour, or a hundred days. Half of thinking well is finding the way of writing a thing down that makes the answer visible.

Borrowing a shape from somewhere else

Very little hard thinking starts from nothing. It starts from something you have seen before, in a form so different that you almost miss it.

A doctor has a beam that can destroy a growth deep inside a patient. At the strength required it burns everything on the way in. What can be done?

Most people find that impossible. Then somebody tells them about a general whose army must reach a fort at the centre of a country, along roads that will collapse under the weight of the whole army at once. He divides the army, sends each part along a different road, and they arrive together.

Once you have heard the second story, the first one is easy. Send many weak beams from many directions so that they meet at one point.

Here is the part worth remembering. When people are given that story and then the problem, most still do not see it, until somebody says "the story might help". The shape was in their heads, unused, one sentence away. They could apply it perfectly well. They just never thought to look.

So the difficult part is remembering that you already know something similar. Cleverness was never the obstacle. Which gives reading and travelling and listening to people whose work is nothing like yours a very practical value. A shape you have never met cannot be borrowed, and a shape you met once and never thought about again might as well not be there. When you solve something hard, spend five minutes afterwards asking what kind of problem it was. That five minutes is what files it away where you can find it.

Things that push back

Some problems are made of parts that sit still, and taking them apart works. Others are made of parts that push back at each other, and taking them apart tells you almost nothing.

A shop lowers its prices for a week to sell more. It works. It does it again the next month, and again. Within a year the customers have learned to wait for the sale, and the shop is selling the same amount at a lower price, and lowering the price further no longer helps.

A road is widened to relieve the traffic. It works for a season. Then the people who had avoided that road come back to it, and some who used to take the train now drive, and within two years the road is as slow as before, with more cars on it.

Nothing was miscalculated in either case. The mistake was to treat a thing that reacts as though it would hold still. In a problem like this, adjusting the numbers is the weakest thing you can do, and it is where nearly all the effort goes. The strong moves are further back: change who decides, change what information reaches whom, change what the thing is trying to achieve. The shop's problem is not the size of the discount. It is that the discount taught the customer something.

You can tell you are in one of these problems by a simple sign. Every fix works at first and then stops working. When you see that pattern, put the calculator down and draw the loop.

The gate in the middle of the field

You are walking through open country and you come across a gate. It stands across the path with a fence running off in both directions, and there is nothing to see on either side but grass. The gate is a nuisance. It is old, nobody is watching, and you could lift it off its hinges in a minute.

Before you do, consider that somebody carried the timber out here. That took a day of work and a reason. The reason is not written on the gate, because reasons never are. What survives is the gate. What is lost is the winter the animals got out, or the argument between two families, or the child who wandered onto the road.

Old rules are answers to accidents you did not see. They come to us with the answer kept and the reason thrown away, which makes them look arbitrary exactly when they are load-bearing.

This is the honest limit of working things out from the beginning. If you tear down every rule you cannot immediately justify, you have chosen a policy that guarantees you will rediscover old disasters at full price, one at a time, personally.

The remedy is a small toll, paid before you remove anything. Say out loud what the rule might have been for. If you can give a decent account of why it exists and still think it should go, take it down. If you cannot give any account at all, that is not evidence the rule is stupid. It is evidence that you do not yet know what you are looking at.

When not to work it out

A person running to catch a high ball does not calculate anything. If they did they would need the speed, the angle, the wind, and the spin, and they would need them in the first half second, and they would drop the ball.

What they do instead is this: fix your eyes on the ball, and run so that it stays at the same height in your view. Do that and your feet take you to where it lands. The rule ignores nearly everything, and it works better than calculation would, because every number you do not have to estimate is a number you cannot get wrong.

Good cooks work this way. So do good nurses, good farmers, good drivers, good market traders. They are using rules shaped by thousands of repetitions, in exactly the situations those rules were shaped for. There is nothing lazy in it.

There is also a larger version of this. Nearly everything you depend on today rests on knowledge that no single person could write down. The person who grew your rice, the person who wired your house, the person who set the price of the metal in your bicycle: none of them could explain the whole system, and the system works anyway. You could not have derived any of it in an evening, and you do not need to.

So: work it out from the beginning when the stakes are high, when the pieces can be checked, when the received answer is young or unexamined, and when an evening of your life is cheap next to the decision. Take the borrowed answer when the rule is old and has survived hard times, when the matter is one of taste or bodies or people rather than weights and prices, and when the cost of being wrong is small or quickly repaired.

The answer that comes too fast

One more thing, and it is about you rather than about the world.

A bottle and its cork cost twelve coins together. The bottle costs ten coins more than the cork. What does the cork cost?

An answer arrived. For most people it was two, and it is wrong. Two for the cork and ten more for the bottle makes fourteen. The cork costs one and the bottle costs eleven, and now that you see it you cannot unsee it.

Notice what happened. The wrong answer did not feel like a guess. It felt like knowing. That is the ordinary condition of a mind at work: an answer arrives quickly, dressed in confidence, and the checking of it is a separate act that somebody has to decide to perform.

Nobody performs it by accident. Every method in this piece is a way of forcing the check to happen: the scale, the disaster announced in advance, the test that could say no, the thousand people counted out, the friend who disagrees.

Which way to use, and when

An hour, start to finish

If you only ever do one thing from this piece, do this. It takes an hour and it handles most of what an evening of worrying would not.

Write the question as a number with a name. Ten minutes, and it is the step most often skipped.

Break the number into the pieces it is made of, and keep breaking until each piece can be weighed, counted, or looked up. Ten minutes.

Put a number on each piece, and a range around it, and mark whether you measured it, looked it up, or hoped. Ten minutes.

Add it up. Compare it with what you were told. Note the size of the gap. Five minutes.

Now say the answer is wrong by a lot, and write the three most likely reasons. Ten minutes.

Pick the single cheapest thing you could do this week that would tell you which of the three it is, and put it in your diary. Five minutes.

Last, list every established practice your answer proposes to abandon, and next to each one write what it might have been for. Anything you cannot explain stays where it is, for now. Ten minutes.

The last two steps are the first ones to be dropped when you are in a hurry, and they are the two that separate working something out from making something up.

The person hardest to argue with

Everything above is a set of devices for one purpose, and it is worth naming plainly.

The person hardest to argue with is yourself, because you already agree with everything you say. Your reasons sound good to you by construction. You will not catch your own mistake by being sincere, or careful, or clever, and the more clever you are the longer and tidier the wrong chain you can build.

So the useful part of every method here is the part that lets the world talk back. The scale that says the loaf is heavier than you thought. The thousand people who show that the frightening letter is mostly false alarms. The test that comes back no. The old gate that turns out to have been holding something in. The friend who wants you to be wrong and can say why.

Get one of those into every decision that matters, early, while being wrong is still cheap. That is the whole of it. The rest is practice.

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A note on where this comes from. Nothing here is my invention. Working a problem down to pieces you can check has been taught, in one form or another, wherever people have written about thinking: in old Greek and Arabic writing on how to know anything, in old Chinese writing on how to judge an argument, in old Indian writing on the ways knowledge reaches us. The scholar in Basra who told his readers to treat every book as an enemy until they had tested it lived around a thousand years ago and worked mainly on light. Asking why five times in a row is a practice from Japanese car factories, where it was used to stop machines breaking twice. The illness figures in the counting example are close to those of real screening tests, and the finding that doctors answer the same question far better when it is put as a thousand people rather than as percentages comes from studies of practising doctors. The number-guessing game, the bottle and the cork, the beam and the fort, and the ball-catching rule are all taken from experiments on ordinary people, and the results were the same each time: the fault was rarely in anyone's cleverness, and nearly always in a check that nobody thought to run.

Last updated: August 30, 2026