A fraud, free-market, antifragility tale

Crypto's only rival is public opinion.

In less than a week, mainstream media, bloggers, and TikTokers turned on FTX's founder.

While some were surprised, almost everyone with a keyboard and a Twitter account predicted the FTX collapse. These financial oracles should have warned the 1.2 million people Sam Bankman-Fried duped.

After happening, unexpected events seem obvious to our brains. It's a bug and a feature because it helps us cope with disasters and makes our reasoning suck.

Nobody predicted the FTX debacle. Bloomberg? Politicians. Non-famous. No cryptologists. Who?

When FTX imploded, taking billions of dollars with it, an outrage bomb went off, and the resulting shockwave threatens the crypto market's existence.

As someone who lost more than $78,000 in a crypto scam in 2020, I can only understand people’s reactions.  When the dust settles and rationality returns, we'll realize this is a natural occurrence in every free market.

What specifically occurred with FTX? (Skip if you are aware.)

FTX is a cryptocurrency exchange where customers can trade with cash. It reached #3 in less than two years as the fastest-growing platform of its kind.

FTX's performance helped make SBF the crypto poster boy. Other reasons include his altruistic public image, his support for the Democrats, and his company Alameda Research.

Alameda Research made a fortune arbitraging Bitcoin.

Arbitrage trading uses small price differences between two markets to make money. Bitcoin costs $20k in Japan and $21k in the US. Alameda Research did that for months, making $1 million per day.

Later, as its capital grew, Alameda expanded its trading activities and began investing in other companies.

Let's now discuss FTX.

SBF's diabolic master plan began when he used FTX-created FTT coins to inflate his trading company's balance sheets. He used inflated Alameda numbers to secure bank loans.

SBF used money he printed himself as collateral to borrow billions for capital. Coindesk exposed him in a report.

One of FTX's early investors tweeted that he planned to sell his FTT coins over the next few months. This would be a minor event if the investor wasn't Binance CEO Changpeng Zhao (CZ).

The crypto space saw a red WARNING sign when CZ cut ties with FTX. Everyone with an FTX account and a brain withdrew money. Two events followed. FTT fell from $20 to $4 in less than 72 hours, and FTX couldn't meet withdrawal requests, spreading panic.

SBF reassured FTX users on Twitter. Good assets.

He lied.

SBF falsely claimed FTX had a liquidity crunch. At the time of his initial claims, FTX owed about $8 billion to its customers. Liquidity shortages are usually minor. To get cash, sell assets. In the case of FTX, the main asset was printed FTT coins.

Sam wouldn't get out of trouble even if he slashed the discount (from $20 to $4) and sold every FTT. He'd flood the crypto market with his homemade coins, causing the price to crash.

SBF was trapped. He approached Binance about a buyout, which seemed good until Binance looked at FTX's books.

Binance's tweet ended SBF, and he had to apologize, resign as CEO, and file for bankruptcy.

Bloomberg estimated Sam's net worth to be zero by the end of that week. 0!

But that's not all. Twitter investigations exposed fraud at FTX and Alameda Research. SBF used customer funds to trade and invest in other companies.

Thanks to the Twitter indie reporters who made the mainstream press look amateurish. Some Twitter detectives didn't sleep for 30 hours to find answers. Others added to existing threads. Memes were hilarious.

One question kept repeating in my bald head as I watched the Blue Bird. Sam, WTF?

Then I understood.

SBF wanted that FTX becomes a bank.

Think about this. FTX seems healthy a few weeks ago. You buy 2 bitcoins using FTX. You'd expect the platform to take your dollars and debit your wallet, right?

No. They give I-Owe-Yous.

FTX records owing you 2 bitcoins in its internal ledger but doesn't credit your account. Given SBF's tricks, I'd bet on nothing.

What happens if they don't credit my account with 2 bitcoins? Your money goes into FTX's capital, where SBF and his friends invest in marketing, political endorsements, and buying other companies.

Over its two-year existence, FTX invested in 130 companies. Once they make a profit on their purchases, they'll pay you and keep the rest.

One detail makes their strategy dumb. If all FTX customers withdraw at once, everything collapses.

Financially savvy people think FTX's collapse resembles a bank run, and they're right. SBF designed FTX to operate like a bank.

You expect your bank to open a drawer with your name and put $1,000 in it when you deposit $1,000. They deposit $100 in your drawer and create an I-Owe-You for $900. What happens to $900?

Let's sum it up: It's boring and headache-inducing.

When you deposit money in a bank, they can keep 10% and lend the rest. Fractional Reserve Banking is a popular method. Fractional reserves operate within and across banks.

Fractional reserve banking generates $10,000 for every $1,000 deposited. People will pay off their debt plus interest.

As long as banks work together and the economy grows, their model works well.

SBF tried to replicate the system but forgot two details. First, traditional banks need verifiable collateral like real estate, jewelry, art, stocks, and bonds, not digital coupons. Traditional banks developed a liquidity buffer. The Federal Reserve (or Central Bank) injects massive cash into troubled banks.

Massive cash injections come from taxpayers. You and I pay for bankers' mistakes and annual bonuses. Yes, you may think banking is rigged. It's rigged, but it's the best financial game in 150 years. We accept its flaws, including bailouts for too-big-to-fail companies.

Anyway.

SBF wanted Binance's bailout. Binance said no, which was good for the crypto market.

Free markets are resilient.

Nassim Nicholas Taleb coined the term antifragility.

“Some things benefit from shocks; they thrive and grow when exposed to volatility, randomness, disorder, and stressors and love adventure, risk, and uncertainty. Yet, in spite of the ubiquity of the phenomenon, there is no word for the exact opposite of fragile. Let us call it antifragile. Antifragility is beyond resilience or robustness. The resilient resists shocks and stays the same; the antifragile gets better.”

The easiest way to understand how antifragile systems behave is to compare them with other types of systems.

• Glass is like a fragile system. It snaps when shocked.

• Similar to rubber, a resilient system. After a stressful episode, it bounces back.

• A system that is antifragile is similar to a muscle. As it is torn in the gym, it gets stronger.

Time-changed things are antifragile. Culture, tech innovation, restaurants, revolutions, book sales, cuisine, economic success, and even muscle shape. These systems benefit from shocks and randomness in different ways, but they all pay a price for antifragility.

Same goes for the free market and financial institutions. Taleb's book uses restaurants as an example and ends with a reference to the 2008 crash.

“Restaurants are fragile. They compete with each other. But the collective of local restaurants is antifragile for that very reason. Had restaurants been individually robust, hence immortal, the overall business would be either stagnant or weak and would deliver nothing better than cafeteria food — and I mean Soviet-style cafeteria food. Further, it [the overall business] would be marred with systemic shortages, with once in a while a complete crisis and government bailout.”

Imagine the same thing with banks.

Independent banks would compete to offer the best services. If one of these banks fails, it will disappear. Customers and investors will suffer, but the market will recover from the dead banks' mistakes.

This idea underpins a free market. Bitcoin and other cryptocurrencies say this when criticizing traditional banking.

The traditional banking system's components never die. When a bank fails, the Federal Reserve steps in with a big taxpayer-funded check. This hinders bank evolution. If you don't let banking cells die and be replaced, your financial system won't be antifragile.

The interdependence of banks (centralization) means that one bank's mistake can sink the entire fleet, which brings us to SBF's ultimate travesty with FTX.

FTX has left the cryptocurrency gene pool.

FTX should be decentralized and independent. The super-star scammer invested in more than 130 crypto companies and linked them, creating a fragile banking-like structure. FTX seemed to say, "We exist because centralized banks are bad." But we'll be good, unlike the centralized banking system.

FTX saved several companies, including BlockFi and Voyager Digital.

FTX wanted to be a crypto bank conglomerate and Federal Reserve. SBF wanted to monopolize crypto markets. FTX wanted to be in bed with as many powerful people as possible, so SBF seduced politicians and celebrities.

Worst? People who saw SBF's plan flaws praised him. Experts, newspapers, and crypto fans praised FTX. When billions pour in, it's hard to realize FTX was acting against its nature.

Then, they act shocked when they realize FTX's fall triggered a domino effect. Some say the damage could wipe out the crypto market, but that's wrong.

Cell death is different from body death.

FTX is out of the game despite its size. Unfit, it fell victim to market natural selection.

Next?

The challengers keep coming. The crypto economy will improve with each failure.

Free markets are antifragile because their fragile parts compete, fostering evolution. With constructive feedback, evolution benefits customers and investors.

FTX shows that customers don't like being scammed, so the crypto market's health depends on them. Charlatans and con artists are eliminated quickly or slowly.

Crypto isn't immune to collapse. Cryptocurrencies can go extinct like biological species. Antifragility isn't immortality. A few more decades of evolution may be enough for humans to figure out how to best handle money, whether it's bitcoin, traditional banking, gold, or something else.

Keep your BS detector on. Start by being skeptical of this article's finance-related claims. Even if you think you understand finance, join the conversation.

We build a better future through dialogue. So listen, ask, and share. When you think you can't find common ground with the opposing view, remember:

Sam Bankman-Fried lied.

Only two of the following three options can be achieved: consistency, availability, and partition tolerance

Someone told me that growing from 30 to 60 is the biggest adjustment for a team or business.

I remember thinking, That's random. Each company is unique. I've seen teams of all types confront the same issues during development periods. With new enterprises starting every year, we should be better at navigating growing difficulties.

As a team grows, its processes and systems break down, requiring reorganization or declining results. Why always? Why isn't there a perfect scaling model? Why hasn't that been found?

The Three Things Productive Organizations Must Have

Any company should be efficient and productive. Three items are needed:

First, it must verify that no two team members have conflicting information about the roadmap, strategy, or any input that could affect execution. Teamwork is required.

Second, it must ensure that everyone can receive the information they need from everyone else quickly, especially as teams become more specialized (an inevitability in a developing organization). It requires everyone's accessibility.

Third, it must ensure that the organization can operate efficiently even if a piece is unavailable. It's partition-tolerant.

From my experience with the many teams I've been on, invested in, or advised, achieving all three is nearly impossible. Why a perfect organization model cannot exist is clear after analysis.

The CAP Theorem: What is it?

Eric Brewer of Berkeley discovered the CAP Theorem, which argues that a distributed data storage should have three benefits. One can only have two at once.

The three benefits are consistency, availability, and partition tolerance, which implies that even if part of the system is offline, the remainder continues to work.

This notion is usually applied to computer science, but I've realized it's also true for human organizations. In a post-COVID world, many organizations are hiring non-co-located staff as they grow. CAP Theorem is more important than ever. Growing teams sometimes think they can develop ways to bypass this law, dooming themselves to a less-than-optimal team dynamic. They should adopt CAP to maximize productivity.

Path 1: Consistency and availability equal no tolerance for partitions

Let's imagine you want your team to always be in sync (i.e., for someone to be the source of truth for the latest information) and to be able to share information with each other. Only division into domains will do.

Numerous developing organizations do this, especially after the early stage (say, 30 people) when everyone may wear many hats and be aware of all the moving elements. After a certain point, it's tougher to keep generalists aligned than to divide them into specialized tasks.

In a specialized, segmented team, leaders optimize consistency and availability (i.e. every function is up-to-speed on the latest strategy, no one is out of sync, and everyone is able to unblock and inform everyone else).

Partition tolerance suffers. If any component of the organization breaks down (someone goes on vacation, quits, underperforms, or Gmail or Slack goes down), productivity stops. There's no way to give the team stability, availability, and smooth operation during a hiccup.

Path 2: Partition Tolerance and Availability = No Consistency

Some businesses avoid relying too heavily on any one person or sub-team by maximizing availability and partition tolerance (the organization continues to function as a whole even if particular components fail). Only redundancy can do that. Instead of specializing each member, the team spreads expertise so people can work in parallel. I switched from Path 1 to Path 2 because I realized too much reliance on one person is risky.

What happens after redundancy? Unreliable. The more people may run independently and in parallel, the less anyone can be the truth. Lack of alignment or updated information can lead to people executing slightly different strategies. So, resources are squandered on the wrong work.

Path 3: Partition and Consistency "Tolerance" equates to "absence"

The third, least-used path stresses partition tolerance and consistency (meaning answers are always correct and up-to-date). In this organizational style, it's most critical to maintain the system operating and keep everyone aligned. No one is allowed to read anything without an assurance that it's up-to-date (i.e. there’s no availability).

Always short-lived. In my experience, a business that prioritizes quality and scalability over speedy information transmission can get bogged down in heavy processes that hinder production. Large-scale, this is unsustainable.

Accepting CAP

When two puzzle pieces fit, the third won't. I've watched developing teams try to tackle these difficulties, only to find, as their ancestors did, that they can never be entirely solved. Idealized solutions fail in reality, causing lost effort, confusion, and lower production.

As teams develop and change, they should embrace CAP, acknowledge there is a limit to productivity in a scaling business, and choose the best two-out-of-three path.

Most crypto is probably too.

The goals of Web 3 and the metaverse are admirable and attractive. Who doesn't want an internet owned by users? Who wouldn't want a digital realm where anything is possible? A better way to collaborate and visit pals.

Companies pursue profits endlessly. Infinite growth and revenue are expected, and if a corporation needs to sacrifice profits to safeguard users, the CEO, board of directors, and any executives will lose to the system of incentives that (1) retains workers with shares and (2) makes a company answerable to all of its shareholders. Only the government can guarantee user protections, but we know how successful that is. This is nothing new, just a problem with modern capitalism and tech platforms that a user-owned internet might remedy. Moxie, the founder of Signal, has a good articulation of some of these current Web 2 tech platform problems (but I forget the timestamp); thoughts on JRE aside, this episode is worth listening to (it’s about a bunch of other stuff too).

Moxie Marlinspike, founder of Signal, on the Joe Rogan Experience podcast.

Source: https://open.spotify.com/episode/2uVHiMqqJxy8iR2YB63aeP?si=4962b5ecb1854288

Web 3 champions are premature. There was so much spectacular growth during Web 2 that the next wave of founders want to make an even bigger impact, while investors old and new want a chance to get a piece of the moonshot action. Worse, crypto enthusiasts believe — and financially need — the fact of its success to be true, whether or not it is.

I’m doubtful that it will play out like current proponents say. Crypto has been the white-hot focus of SV’s best and brightest for a long time yet still struggles to come up any mainstream use case other than ‘buy, HODL, and believe’: a store of value for your financial goals and wishes. Some kind of the metaverse is likely, but will it be decentralized, mostly in VR, or will Meta (previously FB) play a big role? Unlikely.

METAVERSE

The metaverse exists already. Our digital lives span apps, platforms, and games. I can design a 3D house, invite people, use Discord, and hang around in an artificial environment. Millions of gamers do this in Rust, Minecraft, Valheim, and Animal Crossing, among other games. Discord's voice chat and Slack-like servers/channels are the present social anchor, but the interface, integrations, and data portability will improve. Soon you can stream YouTube videos on digital house walls. You can doodle, create art, play Jackbox, and walk through a door to play Apex Legends, Fortnite, etc. Not just gaming. Digital whiteboards and screen sharing enable real-time collaboration. They’ll review code and operate enterprises. Music is played and made. In digital living rooms, they'll watch movies, sports, comedy, and Twitch. They'll tweet, laugh, learn, and shittalk.

The metaverse is the evolution of our digital life at home, the third place. The closest analog would be Discord and the integration of Facebook, Slack, YouTube, etc. into a single, 3D, customizable hangout space.

I'm not certain this experience can be hugely decentralized and smoothly choreographed, managed, and run, or that VR — a luxury, cumbersome, and questionably relevant technology — must be part of it. Eventually, VR will be pragmatic, achievable, and superior to real life in many ways. A total sensory experience like the Matrix or Sword Art Online, where we're physically hooked into the Internet yet in our imaginations we're jumping, flying, and achieving athletic feats we never could in reality; exploring realms far grander than our own (as grand as it is). That VR is different from today's.

https://podcasts.google.com/feed/aHR0cHM6Ly9leHBvbmVudC5mbS9mZWVkLw/episode/aHR0cHM6Ly9leHBvbmVudC5mbS8_cD00MzM?hl=en&ved=2ahUKEwjH5u6r4rv2AhUjc98KHeybAP8QjrkEegQIChAF&ep=6

Ben Thompson released an episode of Exponent after Facebook changed its name to Meta. Ben was suspicious about many metaverse champion claims, but he made a good analogy between Oculus and the PC. The PC was initially far too pricey for the ordinary family to afford. It began as a business tool. It got so powerful and pervasive that it affected our personal life. Price continues to plummet and so much consumer software was produced that it's impossible to envision life without a home computer (or in our pockets). If Facebook shows product market fit with VR in business, through use cases like remote work and collaboration, maybe VR will become practical in our personal lives at home.

Before PCs, we relied on Blockbuster, the Yellow Pages, cabs to get to the airport, handwritten taxes, landline phones to schedule social events, and other archaic methods. It is impossible for me to conceive what VR, in the form of headsets and hand controllers, stands to give both professional and especially personal digital experiences that is an order of magnitude better than what we have today. Is looking around better than using a mouse to examine a 3D landscape? Do the hand controls make x10 or x100 work or gaming more fun or efficient? Will VR replace scalable Web 2 methods and applications like Web 1 and Web 2 did for analog? I don't know.

My guess is that the metaverse will arrive slowly, initially on displays we presently use, with more app interoperability. I doubt that it will be controlled by the people or by Facebook, a corporation that struggles to properly innovate internally, as practically every large digital company does. Large tech organizations are lousy at hiring product-savvy employees, and if they do, they rarely let them explore new things.

These companies act like business schools when they seek founders' results, with bureaucracy and dependency. Which company launched the last popular consumer software product that wasn't a clone or acquisition? Recent examples are scarce.

Web 3

Investors and entrepreneurs of Web 3 firms are declaring victory: 'Web 3 is here!' Web 3 is the future! Many profitable Web 2 enterprises existed when Web 2 was defined. The word was created to explain user behavior shifts, not a personal pipe dream.

Origins of Web 2: http://www.oreilly.com/pub/a/web2/archive/what-is-web-20.html

One of these Web 3 startups may provide the connecting tissue to link all these experiences or become one of the major new digital locations. Even so, successful players will likely use centralized power arrangements, as Web 2 businesses do now. Some Web 2 startups integrated our digital lives. Rockmelt (2010–2013) was a customizable browser with bespoke connectors to every program a user wanted; imagine seeing Facebook, Twitter, Discord, Netflix, YouTube, etc. all in one location. Failure. Who knows what Opera's doing?

Silicon Valley and tech Twitter in general have a history of jumping on dumb bandwagons that go nowhere. Dot-com crash in 2000? The huge deployment of capital into bad ideas and businesses is well-documented. And live video. It was the future until it became a niche sector for gamers. Live audio will play out a similar reality as CEOs with little comprehension of audio and no awareness of lasting new user behavior deceive each other into making more and bigger investments on fool's gold. Twitter trying to buy Clubhouse for $4B, Spotify buying Greenroom, Facebook exploring live audio and 'Tiktok for audio,' and now Amazon developing a live audio platform. This live audio frenzy won't be worth their time or energy. Blind guides blind. Instead of learning from prior failures like Twitter buying Periscope for $100M pre-launch and pre-product market fit, they're betting on unproven and uncompelling experiences.

NFTs

NFTs are also nonsense. Take Loot, a time-limited bag drop of "things" (text on the blockchain) for a game that didn't exist, bought by rich techies too busy to play video games and foolish enough to think they're getting in early on something with a big reward. What gaming studio is incentivized to use these items? Who's encouraged to join? No one cares besides Loot owners who don't have NFTs. Skill, merit, and effort should be rewarded with rare things for gamers. Even if a small minority of gamers can make a living playing, the average game's major appeal has never been to make actual money - that's a profession.

No game stays popular forever, so how is this objective sustainable? Once popularity and usage drop, exclusive crypto or NFTs will fall. And if NFTs are designed to have cross-game appeal, incentives apart, 30 years from now any new game will need millions of pre-existing objects to build around before they start. It doesn’t work.

Many games already feature item economies based on real in-game scarcity, generally for cosmetic things to avoid pay-to-win, which undermines scaled gaming incentives for huge player bases. Counter-Strike, Rust, etc. may be bought and sold on Steam with real money. Since the 1990s, unofficial cross-game marketplaces have sold in-game objects and currencies. NFTs aren't needed. Making a popular, enjoyable, durable game is already difficult.

With NFTs, certain JPEGs on the internet went from useless to selling for $69 million. Why? Crypto, Web 3, early Internet collectibles. NFTs are digital Beanie Babies (unlike NFTs, Beanie Babies were a popular children's toy; their destinies are the same). NFTs are worthless and scarce. They appeal to crypto enthusiasts seeking for a practical use case to support their theory and boost their own fortune. They also attract to SV insiders desperate not to miss the next big thing, not knowing what it will be. NFTs aren't about paying artists and creators who don't get credit for their work.

South Park's Underpants Gnomes

NFTs are a benign, foolish plan to earn money on par with South Park's underpants gnomes. At worst, they're the world of hucksterism and poor performers. Or those with money and enormous followings who, like everyone, don't completely grasp cryptocurrencies but are motivated by greed and status and believe Gary Vee's claim that CryptoPunks are the next Facebook. Gary's watertight logic: if NFT prices dip, they're on the same path as the most successful corporation in human history; buy the dip! NFTs aren't businesses or museum-worthy art. They're bs.

Gary Vee compares NFTs to Amazon.com. vm.tiktok.com/TTPdA9TyH2

We grew up collecting: Magic: The Gathering (MTG) cards printed in the 90s are now worth over $30,000. Imagine buying a digital Magic card with no underlying foundation. No one plays the game because it doesn't exist. An NFT is a contextless image someone conned you into buying a certificate for, but anyone may copy, paste, and use. Replace MTG with Pokemon for younger readers.

When Gary Vee strongarms 30 tech billionaires and YouTube influencers into buying CryptoPunks, they'll talk about it on Twitch, YouTube, podcasts, Twitter, etc. That will convince average folks that the product has value. These guys are smart and/or rich, so I'll get in early like them. Cryptography is similar. No solid, scaled, mainstream use case exists, and no one knows where it's headed, but since the global crypto financial bubble hasn't burst and many people have made insane fortunes, regular people are putting real money into something that is highly speculative and could be nothing because they want a piece of the action. Who doesn’t want free money? Rich techies and influencers won't be affected; normal folks will.

Imagine removing every $1 invested in Bitcoin instantly. What would happen? How far would Bitcoin fall? Over 90%, maybe even 95%, and Bitcoin would be dead. Bitcoin as an investment is the only scalable widespread use case: it's confidence that a better use case will arise and that being early pays handsomely. It's like pouring a trillion dollars into a company with no business strategy or users and a CEO who makes vague future references.

New tech and efforts may provoke a 'get off my lawn' mentality as you approach 40, but I've always prided myself on having a decent bullshit detector, and it's flying off the handle at this foolishness. If we can accomplish a functional, responsible, equitable, and ethical user-owned internet, I'm for it.

Postscript:

I wanted to summarize my opinions because I've been angry about this for a while but just sporadically tweeted about it. A friend handed me a Dan Olson YouTube video just before publication. He's more knowledgeable, articulate, and convincing about crypto. It's worth seeing:

This post is a summary. See the original one here.

Using math to write efficient code in any language

Programmers design, build, test, and maintain software. Employ cases and personal preferences determine the programming languages we use throughout development. Mobile app developers use JavaScript or Dart. Some programmers design performance-first software in C/C++.

A generic source code includes language-specific grammar, pre-implemented function calls, mathematical operators, and control statements. Some mathematical principles assist us enhance our programming and problem-solving skills.

We all use basic mathematical concepts like formulas and relational operators (aka comparison operators) in programming in our daily lives. Beyond these mathematical syntaxes, we'll see discrete math topics. This narrative explains key math topics programmers must know. Master these ideas to produce clean and efficient software code.

Expressions in mathematics and built-in mathematical functions

A source code can only contain a mathematical algorithm or prebuilt API functions. We develop source code between these two ends. If you create code to fetch JSON data from a RESTful service, you'll invoke an HTTP client and won't conduct any math. If you write a function to compute the circle's area, you conduct the math there.

When your source code gets more mathematical, you'll need to use mathematical functions. Every programming language has a math module and syntactical operators. Good programmers always consider code readability, so we should learn to write readable mathematical expressions.

Linux utilizes clear math expressions.

Inbuilt max and min functions can minimize verbose if statements.

How can we compute the number of pages needed to display known data? In such instances, the ceil function is often utilized.

import math as m
results = 102
items_per_page = 10 
pages = m.ceil(results / items_per_page)
print(pages)

Learn to write clear, concise math expressions.

Combinatorics in Algorithm Design

Combinatorics theory counts, selects, and arranges numbers or objects. First, consider these programming-related questions. Four-digit PIN security? what options exist? What if the PIN has a prefix? How to locate all decimal number pairs?

Combinatorics questions. Software engineering jobs often require counting items. Combinatorics counts elements without counting them one by one or through other verbose approaches, therefore it enables us to offer minimum and efficient solutions to real-world situations. Combinatorics helps us make reliable decision tests without missing edge cases. Write a program to see if three inputs form a triangle. This is a question I commonly ask in software engineering interviews.

Graph theory is a subfield of combinatorics. Graph theory is used in computerized road maps and social media apps.

Logarithms and Geometry Understanding

Geometry studies shapes, angles, and sizes. Cartesian geometry involves representing geometric objects in multidimensional planes. Geometry is useful for programming. Cartesian geometry is useful for vector graphics, game development, and low-level computer graphics. We can simply work with 2D and 3D arrays as plane axes.

GetWindowRect is a Windows GUI SDK geometric object.

High-level GUI SDKs and libraries use geometric notions like coordinates, dimensions, and forms, therefore knowing geometry speeds up work with computer graphics APIs.

How does exponentiation's inverse function work? Logarithm is exponentiation's inverse function. Logarithm helps programmers find efficient algorithms and solve calculations. Writing efficient code involves finding algorithms with logarithmic temporal complexity. Programmers prefer binary search (O(log n)) over linear search (O(n)). Git source specifies O(log n):

Logarithms aid with programming math. Metas Watchman uses a logarithmic utility function to find the next power of two.

Employing Mathematical Data Structures

Programmers must know data structures to develop clean, efficient code. Stack, queue, and hashmap are computer science basics. Sets and graphs are discrete arithmetic data structures. Most computer languages include a set structure to hold distinct data entries. In most computer languages, graphs can be represented using neighboring lists or objects.

Using sets as deduped lists is powerful because set implementations allow iterators. Instead of a list (or array), store WebSocket connections in a set.

Most interviewers ask graph theory questions, yet current software engineers don't practice algorithms. Graph theory challenges become obligatory in IT firm interviews.

Recognizing Applications of Recursion

A function in programming isolates input(s) and output(s) (s). Programming functions may have originated from mathematical function theories. Programming and math functions are different but similar. Both function types accept input and return value.

Recursion involves calling the same function inside another function. In its implementation, you'll call the Fibonacci sequence. Recursion solves divide-and-conquer software engineering difficulties and avoids code repetition. I recently built the following recursive Dart code to render a Flutter multi-depth expanding list UI:

Recursion is not the natural linear way to solve problems, hence thinking recursively is difficult. Everything becomes clear when a mathematical function definition includes a base case and recursive call.

Conclusion

Every codebase uses arithmetic operators, relational operators, and expressions. To build mathematical expressions, we typically employ log, ceil, floor, min, max, etc. Combinatorics, geometry, data structures, and recursion help implement algorithms. Unless you operate in a pure mathematical domain, you may not use calculus, limits, and other complex math in daily programming (i.e., a game engine). These principles are fundamental for daily programming activities.

Master the above math fundamentals to build clean, efficient code.