Thinking Better: The Art of the Shortcut in Math and Life

by Marcus du Sautoy

One of  the world’s  great mathematicians  shows why math is the ultimate  timesaver

We are often told that hard work is the key to success. But success isn’t about hard work – it’s about shortcuts. Shortcuts allow us to solve one problem quickly so that we can tackle an even bigger one. They make us capable of doing great things. And according to Marcus du Sautoy, math is the very art of the shortcut.

Thinking Better is a celebration of how math lets us do more with less. Du Sautoy explores how diagramming revolutionized therapy, why calculus is the greatest shortcut ever invented, whether you must really practice for ten thousand hours to become a concert violinist, and why shortcuts give us an advantage over even the most powerful AI. Throughout, we meet artists, scientists, and entrepreneurs who use mathematical shortcuts to change the world.

Delightful, illuminating, and above all practical, Thinking Better is for anyone who has wondered why you should waste time climbing the mountain when you could go around it much faster.

Genres: Nonfiction, Science, Mathematics, Self Help, Psychology, Personal Development, Popular Science

372 pages, Kindle Edition

Published October 19, 2021

About the author

Marcus Peter Francis du Sautoy, OBE is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford.

Reviews

[Colman Glagovich · Rating 4 out of 5]

It seemed to me the title was forced by the publisher, and each chapter tried to bring somewhat loosely related concepts back to the theme of shortcuts. Framing this book more as a fun overview by an excited author on why we should be amazed that even the mathematics we learn before college can solve numerous interesting problems is more useful. The peppering of historical math figures such as gauss, euclid, leibniz, and others was a great way to bring context to the discoveries mentioned.

[Kaethe]

4 April 2023

After a month I can acknowledge that this is just not what I 've been in the mood for

[Diane Hernandez · Rating 2 out of 5]

Thinking Better is a history of how shortcuts have opened our eyes to better, and faster, methodologies. It advocates for outside-the-box thinking. Unfortunately, the book doesn’t contain ways to create these shortcuts for your own problems in either life or math.

I’m disappointed and feel misled by the publisher’s blurb. However, at least you will know what you are getting going in. 2 stars.

Thanks to Basic Books, Perseus Books and NetGalley for a copy in exchange for my honest review.

[Michael David · Rating 2 out of 5]

As someone interested in optimizing my thinking and becoming more self-aware with respect to my biases, I went to the Big Bad Wolf sale with cognitive psychology books in mind. I knew I wouldn't be able to read 15 books in a short span of time, and so tried to be judicious with my purchases.

Because of the lack of a synopsis, I bought this book largely from the availability heuristic. I liked the cover, it dealt with optimizing thinking and mathematical shortcuts, and might offer me insights toward more statistical thinking (which is clearly unusual in human beings). I was also decided to read this book first, because this was the book that I was the most unsure of. I knew that I wasn't going to be disappointed by Ralf Dobelli or Carl Bergstrom, but I believe in eating the frog and so I started reading it right away.

By itself, it's actually a good book. It chronicles the achievements and developments of mathematics and how it's made our lives easier through the development of shortcuts. However, I think it's poorly marketed: the book hasn't contributed towards my thinking better. If this were sold as a text of popular mathematics celebrating the known shortcuts that have paved the way for technological advancements, I wouldn't have bought this book, but would have rated it probably a 4/5.

However, as it was being sold as a book that would optimize thinking, it's rather only okay. Although it offered useful shortcuts as the Gaussian sum, and ended wonderfully, emphasizing that many things in life really have no shortcuts, it didn't contribute to how I would better address problems in my life. (I did like the section featuring Ognjen Amidzic's belief that genius is genetic, and that there are limitations to intelligence that genetics might be detrimental to.)

For those enjoying smart vignettes of mathematical discovery, this book's for you. For those who want to actually think better , it's time to re-read Thinking, Fast and Slow.

[Gonzalo Corvera · Rating 2 out of 5]

First of all, just to be clear, two stars for me means "it was OK".

I don't have a mathematical mind, so I like articles or books about math for the layman which convert mathematical concepts into more visual or simply more understandable terms; this book doesn't really do that. I found many of the subjects interesting, but I finally had to take the author's word for it without really grasping the underlying technicalities. A math student probably would "get it" more easily, but then I suspect it could be too simple. The only chapter where I think he managed to make the complicated simple, is the one on quantum computing.

The main premise is that shortcuts are not lazy, and that is a reasonably well defended idea throughout the book, but I actually think it could be presented more dynamically.

[Mireille · Rating 4 out of 5]

Alles is wiskunde?/!

Een boek over logisch en sneller denken met wiskunde als basis: ga in Beter denken van Marcus du Sautoy op reis door je geheugen, taalbegrip en natuurlijk de wiskunde.
Het ‘beter’ uit de titel slaat op logischer wiskundiger denken. Welke shortcuts/afsnijroutes zijn er om sneller verder te komen in denkprocessen en het berekenen van ingewikkelde vraagstukken?

De auteur beschrijft niet alleen droge stof, maar vertelt bijvoorbeeld ook regelmatig over wiskundigen die hem inspireerden dat vak uit te oefenen. Uitgebreide uitleg over stellingen die patronen blootleggen en andere dagelijkse onderwerpen staat net zo goed in het boek. Waaruit bestaat een jpeg-bestand eigenlijk?

De hoofdstukken zijn ingedeeld volgens een vast stramien. De auteur zet de lezer eerst aan het denken met een vraagstuk waarvoor een snellere oplossing bedacht kan worden en gaat vervolgens in op de context. Aan het eind legt hij de wiskundige shortcut uit. Als niet-wiskundige moet je een enkele keer iets dieper nadenken bij genoemde termen, maar voor het merendeel is Du Sautoy echt goed te begrijpen voor een breed publiek. Dat komt ook doordat hij regelmatig een brug slaat tussen de wiskunde en andere (wetenschaps)domeinen zoals kunst, geheugen en taal.

Neem bijvoorbeeld Du Sautoys motto dat “alles wiskunde is” (en dus overzichtelijk en oplosbaar). Toch erkent ook hij dat je soms een idioom moet veranderen om iets inzichtelijk te krijgen. Hij laat zien hoe wiskunde ook een taal is met een eigen grammatica, hetzij zonder onregelmatige werkwoorden. De algebra blijkt dan een shortcut an sich voor een logisch getallenraadsel in woorden. Waarom krijg je als antwoord altijd 49 wanneer je twee dobbelstenen werpt en de getallen op onder- en bovenkant op verschillende manieren met elkaar vermenigvuldigt en de uitkomsten optelt? Zet het op wiskundige wijze neer en je snapt het: x x y + (7-x) x (7-y) + x x (7-y) + (7-x) x y = 7x7 = 49.

Op de vraag hoeveel cadeaus je hebt aan het eind van het lied ‘12 Days of Christmas’ kun je een meetkundige vergelijking maken, als je tenminste een beetje verstand hebt van driehoeksgetallen en kwadraten.
Hierbij maakt Du Sautoy de kanttekening dat niet elke bètawetenschapper vaardig genoeg is om zijn formulewereld te vatten in woorden en diverse contexten. Anders gezegd laat hij duidelijk blijken dat hij óók taal en kunst echte wetenschapsgebieden vindt.

Is álles oplosbaar met cijfers en formules? Nee of nog niet. Een interessante casus is bijvoorbeeld het beschrijven van hyperruimte, waarin de vierde dimensie een rol heeft. Hoe geef je dat weer op papier? Met het tekenen van een wormgat kom je in het menselijk voorstellingsvermogen het verst.
En het uitzoeken van de kortste route op een festival? Een algoritme heeft al snel miljoenen jaren nodig om dat te berekenen, dus zelf experimenteren gaat toch het snelst.

Het is door alle wetenschappelijke context in Beter denken grappig en boeiend om te merken hoeveel dagelijkse kwesties omgezet kunnen worden in wiskundige taal. Een boek om vaker een stuk in te lezen vanwege alle feitjes die met zoveel enthousiasme zijn opgeschreven. Na het lezen (zonder shortcut) heb je ongetwijfeld nieuwe gedachten in je hoofd!

[Francisco Rodríguez · Rating 2 out of 5]

I am not liking this book. Previous maths books by the author were amazing because the author is great at explaining the essence and beauty of maths itself. There are occasional morsels of that in this book, which are enjoyable, but unfortunately most of the book is taken over by the idea of "the shortcut" applied to real life situations supposedly inspired by the maths. It just doesn't work. The author tries too hard to find "shortcuts" in real-life somehow related to the maths, but the relation is not there... it ends up being shoehorned in, filler material with lots of words but little content, interviewing people trying unsuccessfully to extract "shortcuts" from them, which on several occasions ends up being a bit cringe and child-like. Real-life examples are unsupported by evidence (no indication of serious research, no references) and most of the time they are just obvious trivial boring statements.

An example of what I mean: a really nice mathematical fact is given - to calculate something in the real domain, you can take a shortcut through the complex domain and back (amazing, no need to find real-life shortcuts!). The author then interviews the creator of lastminute.com, shoe-horning the following "analogy" to the shortcut over complex numbers "Hoberman’s success was that he took advantage of the amazing shortcut that the internet provided in those early days of the dot-com boom. Time and again it allowed one to cut out the middle men. In the case of lastminute.com it was the travel agent.".... ok. It's at best loosely related, and at worst a very boring and trivial comment. I'd rather read only about the mathy bits!!

Another example: talking about the maths of map-making (projecting a sphere into a planar map) suddenly the discussion turns to mind maps "Called mind maps, their purpose is to tease out interesting connections between different ideas that you might be exploring. Mind maps have for years been the staple diet of students trying to cram for exams because they help to create an integrated story of a subject that in words can feel too difficult to navigate."... ok.

Another example: In a chapter about probability and chance, the conversation goes into markets, interviewing a historian, "if you were reading this pitstop in the hope that I have some cunning shortcuts to how to invest your savings, my advice would be to combine the skills of the mathematicians together with exploiting the deep knowledge that someone like Helen has gleaned through her training as a historian to be able to guess the next episode in the soap opera that we call the markets."

Some things I liked, though: the scaling advantage of big cities (doubling size gives extra 15% more of everything, on top of doubling), the counting-to-60-in-2-hands system using fingerbones, the trick to win Nim, the removing-beans game, converting each pile to binary and adding them, the trinary system with negative digits (-1,0,1) (never knew you could do that!!), going over 1/e of a full population is enough to almost guarantee getting the "best" if you keep the next highest.

[Adelyne · Rating 5 out of 5]

5 stars.

This was brilliant! I was attracted to it in a bookstore from its subtitle "The Art of the Shortcut", but it turned out to be so much more. Rooted in finding shortcuts in everyday life, I loved how each chapter started off with a little puzzle to get the reader thinking, and then the rest of the chapter is devoted to explaining a mathematical concept/discovery that is related to finding a shortcut for solving the puzzle posed at the start. These discussions went into just enough detail to challenge readers who are more well-versed with mathematical concepts (I consider myself "intermediate"), yet would have been equally enjoyable if just glossed over if a reader is less familiar or less interested in the nitty-gritty of the methods (I had to do this for two of the chapters where the explanations went over my head).

The puzzles at the start are also a good conversation topic for think-y friends, and thinking about why the logic works the way it does was quite fun for me. As mentioned in the previous paragraph though, I didn't quite get all of the concepts so this is a rare one that goes on the to-reread pile for me.

[Alice Bell · Rating 5 out of 5]

“It is not knowledge, but the act of learning, not possession but the act of getting there, which grants the greatest enjoyment ….A shortcut is not about a fast way to finish your journey but rather a stepping stone to beginning a new one”.

So many interesting uses of maths and interesting shortcuts. A bit of history, a bit of problem solving, some applications and some quirky results. A very enjoyable read.

[Enoch · Rating 5 out of 5]

I enjoyed this book a lot. The way it was written and the way ideas were presented resonated with me. The book got me excited about math and the joy of identifying a shortcut. I am going going to apply the concepts at my day job. There are likely ways to do things well and faster if you can think from a new angle.

[jaroiva · Rating 5 out of 5]

Jsem nekritická, protože Marcus du Sautoy je můj oblíbený autor.
Myslela jsem si původně, že to bude víc o logice, ale je to zase spíš "jen" samá matematika. To mně nevadí, ale někoho by to mohlo možná zklamat. Tak s tím počítejte.

(česky právě vychází pod názvem Umění zkratky)

[Rusty · Rating 4 out of 5]

Review to come ...

[Hannah de Bruin · Rating 5 out of 5]

This book definitely made the math nerd in me very happy and I thoroughly enjoyed it! It gave me some good fun facts that I like to tell my students even if they think it’s silly math stuff.

[Marc Antony Haddad · Rating 4 out of 5]

Good tickle for the brain !

[Hanie Noor · Rating 5 out of 5]

Exploring mathematical shortcuts through the minds of prominent figures in the field apart from Du Sautoy's personal experiences. An enjoyable book on thought process and another great read on mental models. Du Sautoy unravels ways in finding alternatives to problem solving while minimising gut response.
 
There are 10 (ten) different mathematical shortcuts in thinking better introduced in this book i.e., patterns, geometric, data, probable shortcut, etc. I do find it really eye-opening and such a fun read however, the book is more on the topic of shortcuts than how it would help us 'think better' (in terms of explanations of real life applications). Nonetheless, still a good peek into mathematician’s thinking which we could form our own mental models based on the approaches. Discusses few good topics and I was reading it simultaneously with Thinking in Numbers by Daniel Tammet (biographical so, more approachable) that revolves around similar topic.
 
One part that I love about the book is how it views shortcuts--it's not necessarily the simplest way but sometimes the most complex; how shortcuts aren't all in straight lines; sometimes all it takes is just one simple number to get a pure signal in working around our solutions, and etc. Maths isn't just about finding solutions but understanding the whole process and the world around us. It's good to remember that not everything have shortcuts, every problems has solutions but not necessarily shortcuts, therefore we have to do the work and put in the effort. On a side note, it might not deliver the premise as intended to but it's good to understand the maths we used to learn in schools are more meaningful than we think.
 
One of my 5🌟 reads for 2022. Thanks to Times Reads for sending me a copy.

[Conner Castagno]

Did not finish! I got about 70 pages into this book and all that it talked about was math shortcuts. I was wanting more real life application but based upon some other reiews I don't think that's what it's going to be so I'm not interested in reading it anymore.

[Musab · Rating 1 out of 5]

A thoroughly unremarkable book to read. A scene of scattered ideas loosely tethered to the misleading title and subtitle of the book. Thinking better should be renamed to somewhat cool mathematical shortcuts. The central question posed in each chapter was fiercely hidden and forgotten until the very end with a small passerby explanation. A serious lack of cohesion both in chapters and across them made reading rather dull and particularly unexciting. The writer has failed in his goal to invoke better thinking. Perhaps the only redeeming qualities are some neat facts littered about and some rarely captivating personal stories.

[Serge · Rating 2 out of 5]

Mediocre. Viewing mathematics under the light of "shortcuts" is somewhat interesting ... but deeply reductive. There is not much "thinking" in this book, as it often veers into a collection of factoids and name dropping. It progressively turns into your typical poorly edited "non-fiction" book that get so easily published by mainstream publishers. Skim or avoid ...

[Anas Alrowaili · Rating 3 out of 5]

I think it's a great idea I couldn't comprehend it due to my lack of understanding of math so I would say the fault is on me I need to learn about math to be able to appreciate the book more.

Now the book in general speaks about shortcuts and how did they come into existence highlighting many big names that contributed to the building of civilization and science.
Each chapter will start with a brain teaser honestly couldn't crack many of them 😅

I loved the fact the book included a tribute to Muslim scholars and how much their understanding of the previous civilizations' knowledge has enabled them to build and come up with original ideas to contribute to the world.

The book dwel onto geometry and psychology and many more subjects just to show how math shortcuts affect everything in our lives.

And I learned that Indians was the first people to create the number sefer aka zero which lead to nothingness

Then to the modern age with computers and machine capabilities and how it's best to work with them not against them and how our math led to programming which led to these incredible machines.

He taps into the brain and how we are born to do something and can't be trained to become masters in it as its DNA is encoded.

Hope I was able to give a fair review with my limited understanding of the subject

Can't give a proper verdict but I'll say if you love math go for it.

[Peter · Rating 1 out of 5]

I was very disappointed by this book. I've seen Marcus du Sautoy on the TV and listened to him on the radio many times and I like what he has to say. This book however has a very weak point and that is he doesn't explain the maths clearly. He sort of half explains it and leaves you to work it out. When he's talking about turning triangular numbers into square numbers using geometry we don't even get a diagram to show precisely what he means. Instead we have to guess it and read it over and over again. I'm not sure that he really expects us to learn a Maths method by any of his examples. It's as if he suggesting it can be done and you might nearly understand how to do it. This may not be true all the way through the book but I've just got fed up with trying to figure out what he doesn't explain having to look in other texts and and resources to understand. I think this kind of book is aimed at non mathematicians who are really interested in the topic but lack the training and as such I think it fails.
It's a good job I have an uncontrollable urge to discover Maths or I'd have list interest in the subject.

[Arthur Ryman · Rating 4 out of 5]

In general, I like whatever du Sautoy produces and I applaud him for his efforts to make mathematics more accessible to a broader audience. However, I was somewhat dissappointed by two aspects of this book. First, his premise for the book is that math is primarily about finding shortcuts. Certainly math can produce shortcuts, but there is much more to math than that. I found the author's attempts to justify the shortcut paradigm to be a little tortured. Does everything that contributes to the solution of a problem qualify as a shortcut? Second, the author recycled many well-known math stories including Gauss adding up the first 100 integers and Euler solving the bridges of Konigsberg problem. Any reader of popular math books will have seen these many times. Some fresher material would have been appreciated. I listened to the audiobook, which was not the best experience when it came to equations. The narrator simply spelled out the symbols. However, there weren't too many of those. It would have been a better experience if the producer of the audiobook had provided an Accompanying PDF of all the equations and diagrams (if any).

[Ian Cummings · Rating 4 out of 5]

Very interesting book about the uses of mathematics presented in a very unique way. It alerted me of some very cool things like DNA computing and desire paths. It also reminded me of Dijkstra’s algorithm for finding the shortest path through a graph which I then tried to apply to a Project Euler problem I’ve been stuck on for a while. Upon further review, the problem was looking for the longest path through a graph, which is a much harder problem, but can be found on a DAG w/ Dijkstra’s algorithm(though I’m still working on it). One criticism I have is the author’s though mention that all the biggest hedge funds are run by mathematicians is true, those same funds have used mathematical models to send us into global recessions. Obviously a good knowledge of math can help you make money, but it can also lose you (and everyone else) a lot as well. So even knowing a lot of the math concepts it was still valuable to me, and I would recommend it to someone who doesn’t have a strong background in math to learn a little more in a fun way.

[Arturo Herrero · Rating 2 out of 5]

Sautoy en caída libre. Mezcla de historia de las matemáticas, con divulgación inconexa y cultura popular.

“El propósito de un atajo es ofrecer la oportunidad de emprender un nuevo viaje. Gauss resumió sus opiniones sobre la búsqueda del conocimiento en una carta enviada a su amigo Farkas Bolyai el 2 de septiembre de 1808, en la que afirma lo siguiente:

«No es el conocimiento, sino el acto de aprender, y no es la posesión, sino el acto de llegar a ella, lo que proporciona la máxima satisfacción. Cuando he esclarecido y agotado un asunto, me desentiendo de él para sumirme de nuevo en la oscuridad. El hombre que nunca se da por satisfecho es muy extraño: si ha completado una estructura, no es con el fin de habitar pacíficamente en ella, sino para comenzar a construir otra. Imagino que un conquistador del mundo debe sentir esto mismo: apenas conquistado un reino, ya extiende sus brazos para abarcar otros.»”

[Tyler · Rating 3 out of 5]

This interesting book felt like having an inspiring high school math teacher giving a beginning of the term pump-up speech for the relevance of the coming content. Again, the book was interesting. Many of the historical anecdotes were entertaining. I appreciated how his discussion of “short-cuts” in psychotherapy balanced the aspirations of CBT with the realism of depth oriented approaches.

The author left one with a greater sense of the mysteries and helpfulness imbedded in mathematics. However, this book is not practical. I left with no new ideas for how I could make new shortcuts in my life. People with mathematics and engineering degrees may be able to get some new ideas for problem-solving out of this text. However, I left with the thought of “it must be nice (at times) to have your specialization.”

[Alan · Rating 3 out of 5]

Thinking Better is an interesting tour through some of the major ideas in mathematics and computer science that have saved humanity from the tedious method of brute force. du Sautoy strikes a good balance between technical detail and high-level overviews. While the stories themselves were entertaining and held my interest, I think the attempt at extrapolating the general ideas into useful advice for life fell somewhat short. I think the book would have been better served by either eschewing the "shortcut to the shortcut" sections altogether and expanding on some of the existing stories, or recasting it as a way to provide some level of inspiration for readers to find shortcuts in their everyday lives. While I personally felt the book as a whole was somewhat 'low calorie', I can still recommend it as an entertaining read.

[Rudi · Rating 4 out of 5]

Mss is het boek voor mij als wiskunde leek iets te lang om 5 sterren , ( en ik vond geen shortcut)
Het enthousiasme en de leuke weetjes zorgen dat het boek gemakkelijk leesbaar blijft , ( behalve de enkele blz die wat dieper ingaan op de wiskunde en die mijn begrip te boven gingen, of mij teveel uren gingen kosten om het te kunnen volgen )
Een shortcut lijkt mij wat als een sprong in de tijd ( met de wijzers van de klok mee ) en het boek beschrijft ook stukjes van de levens van sommige ontdekkingen ( tegen de wijzers in )
Het enthousiasme van de schrijver kan lezers meenemen en er zal waarschijnlijk niemand boos zijn als men de moeilijkere stukjes overslaat ( tenzij men boos wordt op zichzelf maar dat kan men mee overslaan: )
4 op 5 voor deze lezer een boek over tijds (winst?) in de wiskunde , de korte of de lange weg , weg is weg ,

[Brian Beckner · Rating 2 out of 5]

A Few Gems Lots of Fluff

The author took what is a really important and significant idea that underlies innovation, life and mathematics and packaged it with filler material and incorrect assessments that were outside his circle of competence and prone to bias. For example, he misrepresents the early geometers in the introduction as people building civilization which is entirely inconsistent with the theme of the book of math as a language and means to see beauty. He later criticizes a pioneer of truth for his mathematical reasoning to discover truth for what this pioneer believes to be true. Math is a powerful tool but exists entirely abstract and independent of reasoning in the natural sciences. I will use the truths the author does communicate to undermine the falsities which he perpetuates.

[Martin Richardson · Rating 1 out of 5]

I must be thick and lacking in patience. This book lost me right near the beginning when the author assumes you have a degree level understanding of mathematics and just ploughs into explaining things using X or N as a number.

And this was prior to page 30!

Without a basic understanding if these formulas and how they work you're frankly screwed and whilst I wanted to carry on reading in the hope it might all slot into place I needed up getting bored as I just didn't understand the calculus involved.

Feels like it's written by someone obsessed with math and wants to tell the world math is the answer to everything.

My shortcut to reading this was to stop early on and read the reviews.

[Nadia Kanan · Rating 3 out of 5]

DNF
Bookclub I wanted to attend chose this book. Math problems at the beginning of each chapter were nice brain teasers, definitely had fun trying to crack them, but the whole book was just not for me. It is showing the power of math, how math created all the shortcuts to solve complicated problems in no time, how we used all of the shortcuts to reach all the technological advancement we are living in now. A nice read if you are in the mood to learn about Gause, the history of many experiments and equations, and where is each "shortcut" applicable today, from city planning, to sending rockets to the moon.

[K Das · Rating 2 out of 5]

2/5

Picked up a math based fact book that I thought held some promise. Naval Ravikant in his almanac had suggested people should read more math and physics books to also get acquainted with the ways of the world. Unfortunately this book had much promise but very little effect.

It starts of interestingly with some good real world examples and fun ways one can use math. That’s just the first 3 chapters. The remaining 80% of the book neither reads like a fun facts book nor a full fledged math tricks book. It has some random information on equations in there and non compelling story telling. Nothing memorable in here for me to even hang on to as key learnings.