This blog post is about the book “Math with Bad Drawings” written by Ben Orlin

Ultimate Tic-Tac-Toe

The author introduces the book with a normal tic-tac-toe game that we all are familiar with. He then goes about describing a slightly different version of tic-tac-toe involving 9 mini boards of tic-tac-toe. In these mini boards version, the rules of the game, i.e. constraints of the game is what makes it interesting. This 9 mini boards that the author calls “Ultimate tic-tac-toe” is a cue to understand real math. Math as most of us know it is the boring 3 by 3 tic-tac-toe but ultimate tic-tac-toe is how Math ought to have to be. It is a fantastic way to introduce a book that looks like a graphic novel but probably touches upon many of the concepts that one comes across in a undergrad math course.

What does Math Look like to Students ?

In this brief chapter, the author talks about the common situation for most of the students who study math at school level. For them, math contains some numbers and symbols that dance across a page. They memorize certain formulas to be spit out in various examinations. They come across symbols in equalities and inequalities but they can’t really understand them deeply. They try solving word problems that have no connection to real life and worse of all they get graded on these problems and the marks or scores are used as a yardstick to decide who is good and who is bad at the subject. The author offers a deep apology to his own students and also to the subject of mathematics itself for his own transgressions from his teaching career

What does Math Look like to Mathematicians ?

Mathematics like any other subject is a language,a language for defining, understanding various mathematical objects. Well, it is definitely not as easy like any other humanities subject where you can be reasonably good at it, without spending too much effort at it. However for an outsider, expressing something using mathematics looks difficult as it looks like the subject is nothing more than a set of numbers and symbols.

From the perspective of a mathematician though, math is a wonderful way to express thought. This chapter gives a few glimpses of how mathematicians think

• they form mental images of various mathematical expressions
• they work with quick estimates and approximations before formalizing
• they employ tools to turn the static in to dynamic

How Science and Math see each other ?

I found this chapter an awesome exploration of how mathematicians and scientist(physicists, biologists, chemists,…) see each other. In the very beginning, math and science were the same. If you look at the works of stalwarts such as Einstein, they never bothered to demarcate math and science. It was all one and same.

However in the current age, we see that math and science are separate disciplines. We have separate classes, class teachers, graduate programs etc. They are now separate disciplines in their own regard. From the perspective of science, math is a tool that is used to answer specific problems in the investigation of real life theories. In a sense, science views math as a faithful servant always present to be able to take care of the details or give specific and concrete solutions to the problem at hand.

Whereas science sees itself as the protagonist of an action movie, mathematics sees itself as the auteur director of an experimental art project

Math cares about not things but ideas. Math lives not in the material universe of science but in the conceptual universe of logic.

The chapter brings out the paradox of mathematics

Math sees itself as a dreamy poet. Science sees it as a supplier of specialized technical equipment. And herein we find one of the great paradoxes of human inquiry: These two views, both valid, are hard to reconcile. If math is an equipment supplier, why is its equipment so strangely poetic? And if math is a poet, then why is its poetry so unexpectedly useful?

The author brings out several examples where odd mathematical curiosities that lead many mathematicians to spend decades were ultimately found pivotal in understanding scientific real life problems. At the outset an abstract mathematician spending time on a problem that has no relation to reality might appear a waste of time, but history has several examples where unknown mathematical theories were crucial in the development of scientific advances.

The below visual summarized the symbiotic relationship

Perhaps, we ought to see them as a symbiotic pair of very different creatures, like an insect-eating bird perched on the back of a rhino. The rhino gets its itchy problems solved. The bird gets nourished. Both emerge happy.

Good Mathematician vs. Great Mathematician

The author tries to dispel some of the myths surround mathematicians and gives examples to distinguish between a good mathematician and a great mathematician

Good Mathematician	Great Mathematician
A good mathematician can think quickly	A great mathematician can think slowly
A good mathematician has the patience to reach complicated answers	A great mathematician has the patience to reach simple answers
A good mathematician can remember all the details	A great mathematician can forget all the details
A good mathematician tackles the problem head-on	A great mathematician circles around it
A good mathematician selects the most powerful tool for the job	A great mathematician selects the least powerful tool for the job
A good mathematician can achieve understanding	A great mathematician can pass it along
A good mathematician wants to be the best	A great mathematician wants to learn from the best

We Built This City on Triangles

The prelude to this section talks about “geometry” and the undeniable fact that is comes with constraints. These constraints actually give birth to creativity and there are five chapters in the book that deal with this subject.

This chapter is about triangle, the humble planar geometry object that most of us would have come across at some point or the other. The author elevates triangle to the status of shining star of geometry by highlighting some of the unique properties and practical applications of triangle and its properties in real life objects like bridges, railroads, buildings etc.

I know there is something gorgeous in the narrative of the triangle. Its three-sided-ness makes it unique; its uniqueness makes it strong; and its strength makes it essential to modern architecture. Perhaps it’s a stretch to claim that the triangle “saved the world” but if you ask me, it did one better. The triangle allowed the world to become what it is.

Irrational Paper

Well, I had never given much thought to the dimensions of various paper sizes such as A2,A3, A4, etc. The author being a curious mathematician explore the reason for the somewhat arbitrary looking dimensions of A4 size paper, which is 210mm by 297mm. If you pause for a second and consider those numbers, you will be start wondering too, like the author, why would anyone come up with such a size. Author being more curious realizes that the ratio of their dimensions is $\sqrt{2}$. The chapter then goes on to show how all the various types of papers have their sizes tied to this ratio. For someone who has never thought about this before or noticed before, this chapter will provide an aha moment

The Square-Cube Fables

The author shows the way in which a linear increase in dimension gives rise to squared increase in area and cubed increase in volume. This logic is used to explain why giants human beings cannot exist. When some one grows proportionally, there is an isometric scaling. However in order to survive and withstand the increase in mass, we need to have allmoetric scaling.

When a shape’s length grows, its surface area grows faster and its volume grows faster still. This means that big shapes are interior-heavy. We have lots of inner volume per unit of surface area. Small shapes like ants are the opposite; surface-heavy. Being surface-heavy means you never need to fear heights. When one falls, their mass speeds up the fall and surface areas slows it down. That’s why bricks plummet and paper flutters, and why penguins can’t fly and eagles can

The same logic is used to explain why certain type of animals survive in certain climates

Bigger animals, being more interior-heavy, will have an easy time keeping warm. Smaller ones, being surface-heavy, will struggle. That’s why you’re most vulnerable to cold in your surface-heavy extremities: fingers, toes, and ears. This also explains why cold climates support only big mammals: polar bears, seals, yaks, moose, mooses, meeses, and (depending on your zoology professor) the Sasquatch. A surface-heavy mouse wouldn’t stand a chance in the Arctic. Even at moderate latitudes, mice cope with heat loss by eating a quarter of their body weight in food each day.

The last section of this chapter uses the same square-cube logic to show the reason why our night sky is blinding, scorching white. Using a simple equation

Total Brightness = Number of stars x Minimum Brightness

the author asks the reader to ponder over the question of the effect of infinite sky. The answer to the above question has nothing to size, but age. Whether or not the universe is infinitely big, we know for certain that it is finitely old, born less than 14 billion years ago. The light hasn’t had time to reach us.

Game of Dice

I found this chapter a fascinating exploration of geometry, human biases and culture that has shaped the modern dice that we all get to see. The author takes the reader through a thought experiment involving designing something that can generate random outcomes in a game of chance. The author proceeds by first exploring congruency as a prerequisite. If the faces are congruent, can one design a fair dice ? Not really. The next prerequisite is symmetry. Shapes like dipyramid, trapezohedron are explored. This category of dice are even sided dice such as 8 sided or 14 sided or 26 sided. However were these two properties of congruency and symmetry enough to produce dice that was widely accepted. Not really. Since the even sided dice doesn’t tumble but gets rocked a little back and forth between the two clusters of faces, it was no longer a recipe for fun which is integral in playing any game of chance. The best dice need more than symmetric faces. They need symmetric everything. Here come the platonic solids. The author explores the potential usage of platonic solids as a way to create a fair dice and points to one single issue - they can only randomize outcomes among 4, 6, 8, 12 or 20 corresponding to tetrahedron, cube, octahedron, dodecahedron(dodeka - 12, hedra - face), isosahedron(eikosi-20, hedra - face).

The next step in the pursuit of a fair dice is to use the platonic solids and make it a long dice. The only issue with long dice is that two of the sides never occur and it takes a long table length to see its outcome. So a compromise between the length and face length is the cube that we see today. Why not use coin flips to simulate outcomes in a game ? Well there are issues associated with tabulating the outcome of the multiple flips. Also it is not fun any more

I loved this chapter where the author pulls apart various aspects of math in discussing the evolution of a fair dice. Here is a list of loosely governed rules that shaped the way things are

1. Good Dice play fair
2. Good Dice look fair
3. Good Dice work well anywhere
4. Good Dice are simple to roll
5. Good Dice are hard to control
6. Good Dice feel nice

Along the way, there are interesting remarks that made me stop reading this chapter and go around the net and figure why such things survived; why does a fair die have (1,6),(2,5)(3,4) on the opposite sides. It lead to me a rabbit hole where there sees to be no clear answer. Some of have explained mathematically saying the design comes close to average of single die outcome - 3.5. Some of have said that structurally the design gives uniform distribution. Some say it is just a cultural thing that has stuck. Strange how simple stuff like designing a dice for a game of chance has so many interesting aspects to it.

Oral History of the death star

This chapter is an interesting chapter in which the author weaves a fictional story relating to the design of death star. I haven’t grown up watching Starwars, nor am I a big scifi movie fan. Hence I did not have much background about death star. Did some reading around it and then dived in to this chapter. The author talks about various practical and geometric considerations that come up in hypothetical conversations in a team comprising architects, physicists and a few other personas who are all trying to build a gigantic structure to scare everyone in the galaxy. May be scifi fans would love this chapter more than a casual reader would.

The 10 People You Meet in Line for the Lottery

The author gives a sample of 10 different personas to drive home the point of expected value of a lottery and why they exist in the first place. If some one ever struggled with the concept of expected value, this section will bring out the concept in a wonderful way that makes it easy to understand. Along the way the author touches on behavioral economics, game theory and a few other related math concepts, all while making fun of 10 personas

Children of the Coin

The author tries to explain binomial distribution using a few visuals about coin tosses. While not mentioning CLT, the author makes it clear through interesting visuals that extremes vanish when you toss multiple coins and tabulate the various combinations of heads and tails that are obtained.

This simple coin tossing experiment can be used to explain genetics and that’s what the author does. For some reason, I had forgotten many aspects of how genetics works and I reread an old book on genetics to understand the cartoons. Indeed, a simple cartoon shows that there could be 2^46 distinct offspring ? what is the sanctity of the number ?

Here is a quick way to break down that number

Chromosome Pairs & Independent Assortment: Humans have 23 pairs of chromosomes (one set from each parent). During meiosis, chromosomes are shuffled and randomly assigned to reproductive cells (sperm/egg). The independent assortment of chromosomes alone gives ≈8.4 million possible combinations.

Recombination (Crossing Over): During meiosis, genetic material is exchanged between chromosome pairs, creating new combinations beyond just independent assortment. This process significantly increases genetic diversity.

Since both sperm and egg undergo these random shuffling processes, the total number of unique offspring a couple can theoretically produce is roughly 70 trillion unique individuals(8.4 million squared combinations). This accounts for all possible unique genetic combinations that could arise from a single couple due to meiosis.

Using a few visuals, the author manages to breakdown complex genetic concepts such as chromosomes, independent assortment, Crossover. Ultimately the number of possible unique individuals is 70 trillion squared ~ 350 duodecillion

I found this single chapter is worth the time and effort buying and reading this book. Thoroughly enjoyed the fact that one can explain genetics using simple pictures. Wow!

What does Probability Mean in your Profession ?

The author takes a sample of professions and comes up with funny events relating to each profession that can have span the lower end to higher end of probability spectrum

Weird Insurance

The chapter starts with a story of how merchants sent goods via waterways by splitting their merchandise across several boats so that one specific boat would not do damage to their goods. This form of social insurance has lead to the modern day for-profit enterprises where we trade in a monthly or yearly payment to insurance companies and insure against several kinds of events. There are bunch of funny insurance scenarios mentioned in the book Never knew that there are services that offer change of heart wedding insurance and there have been attempts to game the service and hence the insurers had to impose stricter conditions so that they don’t get ripped off

• multiple-birth insurance
• alien-abduction insurance
• failed-exam insurance
• change of heart wedding insurance
• insurance for college football players incase they get worse

How to Break the Economy with a Single Pair of Dice

The author starts off explaining the difference between dependent events and independent events in the probabilistic world.

Independence irons out extremes, Dependence amplifies them

A nice explanation of Wall street mortgage crisis is illustrated using this simple principle

Are houses like thousands of separate dice, each roll independent of the others? Or are they like a single die, multiplies thousands of times by a hall of mirrors ?

Never knew this trivia about the origin of Gaussian Copula that was heavily used in modeling CDOs and CDS

Copula originated in life insurance, where it helped to adjust the probability of one spouse’s survival after the other spouse’s death. Replace “spouse” with “house” and “death” with “default” and you have a model for calculating the dependencies with mortgages

CDOs and CDS are explained using plain simple English and diagrams that any person can easily understand in a few pages, the crux of mortgage crisis. If all the defaults in the pool happened in a dependent fashion, then the investors might easily lose a ton of money and that’s what happened.

CDS was created as a parallel insurance structure where each individual investor can act as a insurance company.During the financial crisis, this was massively appealing to a set of investors who wanted to sell insurance to the CDO issuers and Wall street ended up selling dozens of insurance policies on the same house. By 2008, this represented 60T dollars. It had to collapse and it did so in a spectacular fashion

Why Not to Trust Statistics

The author a nice set of visuals to walk the reader through some of the commonly used summary statistics such as mean, median, mode, percentile, percent change, range, standard deviation, correlation coefficient. Along the way, he gives situations when they are to be used and what they hide. The message from the chapter is that any summary statistic by definition hides information or chucks information. In that sense, it is always better not to trust only one summary statistic. It is better to look at a combination of stats to get a better understanding of what’s going on

The Last 0.400 hitter

Frankly until I read this chapter, I had never paid attention to the rules of baseball. I had seen the some visuals about baseball, saw “Moneyball” and read the book too. But for some reason, I had never taken time to understand the rules of the game. But this chapter made me look at the rules of the game and understand them atleast from a 10,000 ft view. Sometimes I wonder how could I have read Moneyball and understood it, without having a clue about how baseball is played. Ignorance can be hidden for many decades until you realize that you knew nothing all along

The chapter starts off by telling the story of Henry Chadwick who created the Batting Average metric in baseball. The metric became a scorecard metric and everyone was using it to rank players. The fact that players could be ranked and the fact that clubs could poach other players resulted in to a massive baseball economy that we see it today. Baseball was then introduced with terms such as OBP(on-base percentage) and SL(slugging percentage) that turned out to be better metrics. Well they were better metrics but they were not used until the story mentioned in Moneyball started. All along the way, the author uses plain English and funny cartoons to bring out the main ideas behind the measures. Really loved it. I found the following to be very insightful - Had never thought of verbalizing it this way, until I read this book

Statistics, like probability, bridges two worlds. First, there’s the messy day-to-day reality of bad hops and lucky bounces. Second there’s the long-run paradise of smooth averages and stable tendencies. Probability begins with the long-run world, and imagines how things turn out on a single day. Statistics does the opposite: it begins with the day-to-day mess, and strives to infer the invisible long-run distribution from which that data emerged.

Put another way: a probabilist starts with a deck of cards and describes what hands might come up. A statistician looks at the hands on the table, and tries to infer the nature of the deck.

Love the way the author verbalizes the difference between probability and statistics

Barbarians at the Gate

This was a fun chapter to read. p-hacking is something I had come across several times in several books but the way the author explains using pictures is brilliant. He first explains (True/False) positives and (True/False) negatives using a visual that I guess should come to your mind if you want to explain it a kid. Wonderful of showing something visually that sticks

I love the way the author summarizes p-value as a yard stick to measure between Ordinary coincidence and Crazy coincidence. The choice of words are so useful in communicating key ideas about the key factors behind p-values

• Size of of the difference between alternate and null
• Size of the dataset under study
• Variance within each group

p-value was first introduced by Fischer as a way to calibrate p-value and since then the value of 0.05 has taken a cult level status in the world of statistical testing. The author goes on to show how over reliance on this calibrated p-value has created a replication crisis in the academic community. Overall a breeze to read this chapter and I had many aha moments along the way

The Scoreboard Wars

This chapter is all about things we measure and do they really measure what we want to measure. It starts off with one of the popular measures to rank colleges in US, called the challenge index. Instead of looking at the average AP Calculus score, Jay Mathew a bureau chief devised challenge index that took in to consideration attempts rather than pass rate. Based on the challenge index, many news papers and magazines routinely create rankings and there is always a massive debate on whether the measure is the right measure or not. If you look around, you will find measures every where - in your work environment, in social environment etc. where some sort of metric is cooked up and things are sorted according to the metric. Overtime the rankings become so important that no one bothers to address the limitations of the metric.

Fantastic way to look at the effectiveness of the metric is by thinking in terms of whether it is a window or whether it is a scoreboard

A “window” is a number that offers a glimpse of reality. It does not feed into any incentive scheme. It cannot earn plaudits or incur punishments. It is a rough, partial, imperfect thing - yet still useful to the curious observer. Think of a psychologist asking a subject to rate his happiness on a scale from 1 to 10. This figure is just a crude simplification only the most hopeless “1” would believe that the number is happiness.

The second kind off metric is a “scoreboard”. It reports a definite, final outcome. It is not a detached observation, but a summary judgment, an incentive scheme, carrying consequences.

Any metric can be a window or scoreboard based on who is looking at it. Test scores are a window for the teacher but a scoreboard for the student. The author shows several conversation drawings that illustrate this aspect of how a window can turn in to scoreboard quickly and then the focus sometimes is always on the scoreboard feature of the metric

Think about your own earnings at work. One might look at it as a scoreboard, wanting to increase it quarter on quarter or year on year. However if you look at it from the company’s shareholders perspective, wage is a window to how well the company is managing its labor costs. In the world of hedge fund pods, if you are PM, the yearly performance is your scorecard and the sum of performance is also a score card for the company. Hence sometimes the scorecard vs. window thinking might help you understand the reason a specific measure, given the context you are looking at

Never thought about metrics from this perspective of window vs scoreboard until I read this book.

Book Shredders

This chapter is about the ways statisticians are mining literary texts for insights about the language. It starts off with the author talking about Ben Blatt’s book who tears apart literary books and analyzes great authors use of adverbs. Based on his statistical analysis, he concludes that most of the great authors use adverbs sparingly. Even amongst the collection written by a single author, his or her best works almost always contain lesser usage of adverbs.

The chapter mentions about the study of 1-grams of over 4 percent of the book ever printed and reports the findings of the study: Less than half of the 1-grams from the year 1990 tuned out to be actual words, while more than 2/3 of the 1-grams from the year 2000 were words.

There is a section on statistical analysis of texts for gender prediction where the author talks about various services online that can predict the gender of the text based on the writing. One such online project is called Apply Magic Sauce that can do a pyscho-demographic profile from digital identities. I did a quick check my own writing and found the tool did not work with only text. Out of curiosity I tried Gender Guesser and it seemed to get it right. Somehow the fact the one can look based on word frequencies and things like unigrams, bigrams, an algo can predict the gender is very interesting.

Any mention of statistical analysis of text would be remiss if the stats work around Federalist papers are not mentioned. Unsurprisingly, the author gives the meat of the story - How “Upon” was one of the key words that was used by statisticians to assign authorship to 12 Federalist papers, whose author was a mystery.

The Final Speck of Diamond Dust

The author starts off by posing a simple question on why water, the life saving liquid for humans should cost so less compared to diamonds, which seem to just shiny rocks. Until 1870s the most prominent for of theory was “labor theory of value” where every good was priced based on the labor needed to produce it.

Post 1870, marginalism crept the economics world and every concept was explained from the perspective of “marginality”. The main idea behind marginality is

The economy’s first cup of water is worth far more than its first shard of diamond. But prices don’t depend on the first increment or even the average increment. They depend on the last increment, on the final speck of diamond dust.

The essence is that marginality will drive supply and demand equation to a point where market establishes equilibrium price and that’s what we pay for various goods today and that as per marginality theory is why diamonds cost more than water.

The highlight of the chapter like the other ones is to drive home the concept of pervasiveness of marginality concept in economics.

Bracketology

The chapter starts by giving a crash course of how the modern taxation has evolved over the world. It started off with just one tax rate for individuals earning beyond a certain slab and then slowly graduated toward having more tax rates at progressively higher income levels. Once you start looking at marginal tax rates a bunch of step functions, there are nice extensions or thought experiments that one can indulge in. And of course the author does exactly that making this chapter a fun treatise on current and possible state of marginal tax rate curve.

A quick online search on various countries and their taxation rules yields some interesting facts

The country with the maximum number of tax brackets for personal income tax varies over time, but historically, Belgium and Germany have had some of the most complex tax structures. However, Japan holds one of the highest numbers of tax brackets.

Countries with the Most Tax Brackets:

• Japan – 12 tax brackets (5% to 45%)
• Luxembourg – 23 tax brackets (8% to 42%)
• France – Historically had many brackets, now simplified to 5
• Germany – Uses a progressive formula rather than fixed brackets but effectively has continuous taxation

Luxembourg technically has the highest number of fixed brackets (23), but Germany and Japan have complex progressive taxation methods.

Never knew that Germany has continuous tax rate regime until now, all thanks to this chapter that made me curious to get a check on various tax rates across the world

One State, Two State, Red State, Blue State

I had never particularly paid attention to the way in which presidents in US are chosen. Well every four years there is a Red vs Blue geocoded map that is splashed around in the media and my knowledge was always restricted to just that visual. The author gives a fantastic tour of the process behind president’s election and along the way shows the way various states differ in terms of their voting power and method. It was slightly strange to read about Winner-take-all seats that is followed in most of the US states. The reasoning becomes clear once the author explains the rationale from a game theory perspective.

Systems lead to incentives. Winner-take-all system gives rise to a tendency where there are some states which have always voted to Democrats or Republicans only. In a sense, these states are actually never that important in shaping the policies for a presidential candidate. The excess focus on swing states brings out the limitations of the current electoral college framework. There is also a clear rationale mentioned on why proportional allocation of electoral votes based on voting in various regions in a state will not be put in place - It goes against the incentives of state politicians.

The Chaos of History

This is an interesting chapter and talks about a few of the concepts mentioned in the book Four ways of thinking book.

The author tries to distill the interactive and random nature of some of the world’s phenomena using visuals. If you look at simple pendulum, it is the most vivid example for most of us, when we hear periodic. But you tweak the pendulum by a bit by introducing another pendulum to the first pendulum - a double pendulum - and now it enters a state of randomness and chaos. A small change in initial conditions of a double pendulum gives rise to completely different transition points. It is no longer a periodic system. It is no longer a random system. It is a chaotic system

The section on “game of life” and author’s interpretation is probably THE MOST appealing section to me. Well, it starts off by showing a simple game of life simulation and draws out a deep philosophical and scientific opinion

Simplicity on a small scale gives way to strange emergent behavior on a large one

Amos Tversky

The big choices we make in life are practically random. The small choices probably tell us more about who we are. Which field we go into may depend on which high school teacher we happen to meet. Who we marry may depend on who happens to be around at the right time of life. On the other hand, the small decisions are very systematic. That I became a psychologist is probably not very revealing. What kind of psychologist I am may reflect deep traits

The author’s take on the above statement

Small choices obey predictable causes. But large scale events are the products of a devlishly complicated and interconnected system, in which every motion depends on the context.

A person is predictable; people are not; Their delicate interrelationships amplify some patterns and extinguish others, without any clear rhyme or reason.

The above sections has tremendous implications on how one perceives life as. Wherever you are, it could be a result of many conscious internal action driven outcomes and many external forces that are beyond our control. In that sense, whatever we get to do in our lives in any given hour, given day, it might be helpful to have this attitude that we really can’t say where we land up in the future. It is only the tiny steps that we have control and that’s all our brains can comprehend and take corrective course if needed. In that sense it is better to have a compass in life than a map. Having a compass might give you a general direction to go and then you start taking tiny conscious steps and iterate along the way. If some one asks you how have you achieved or managed to get the current state, I think it is better to have the mindset mentioned in Fluke and attribute to any of the innumerable factors that are out of your control.

It is lovely to see this analogy in many philosophical texts in various forms. We can’t attribute or analyze how we ended up at a certain place in life. All we can do is to control the tiny steps we take each day and that we shouldn’t fritter away by thinking that there was something wrong in the past and worse, there is some inherent limitations in oneself.

Takeway

I loved this book. I wonder why I did not read this book for so long, despite buying it more than four years ago. It has been lying on my book shelf for so many years. The book talks about so many mathematical concepts that will motivate any young or adult reader to explore those concepts(if one is not already familiar with them), visualize the concepts (using authors drawings). Whatever be one’s level of mathematical education, this book has so much to offer and is a pleasure to read. Worth every minute of my time spent reading the book. Authors like Ben Orlin who communicate visually and use vivid examples to explain math concepts are rare in today’s world. I plan to read his other books in the coming days.