"GTO" (Game Theory Optimal) has been used a lot in poker recently. To clarify its meaning and application, the aim of this article is to define what it is, when to use it when playing, what strategies to apply for how to play GTO poker, for beginner and more advanced players!

Poker GTO

In poker, you can choose between two main winning strategies:

Exploitative play maximizes expected value (EV) by countering opponents' sub-optimal plays and weaker tendencies. Yes, playing this way opens you up to being exploited, but the weaker opponents you're targeting won't change their game to counteract this, allowing you to reap maximum profits over the long run.

GTO (Game-Theory Optimal): You try to play perfect poker, which forces your opponents to make mistakes (which is where almost all of your profit will be derived from). It mixes bluffs or semi-bluffs with value bets, clarifies bet sizes, and more.

GTO vs. Exploitative: Which is Better in Poker?

Before diving into GTO poker strategy, it's important to know which of these two play styles is more profitable for beginners and advanced players. The simple answer is probably both, but usually more exploitable.

Most players don't play GTO poker and can be exploited in their gameplay and strategy, allowing for more profits to be made using an exploitative approach. In fact, it’s only in some of the largest games at the highest stakes that GTO concepts are fully utilized and seen in practice, and even then, exploitative plays are still sometimes used.

Knowing, understanding, and applying GTO poker basics will create a solid foundation for your poker game. It's also important to understand GTO so you can deviate from it to maximize profits.

GTO Poker Strategy

According to Ed Miller's book "Poker's 1%," the most fundamental concept that only elite poker players understand is frequency, which could be in relation to cbets, bluffs, folds, calls, raises, etc.

GTO poker solvers (downloadable online software) give solutions for how to play optimally in any given spot and often recommend using mixed strategies based on select frequencies.

In a river situation, a solver may tell you to call 70% of the time and fold 30%. It may also suggest calling 50% of the time, folding 35% of the time, and raising 15% of the time (with a certain range of hands).

Frequencies are a fundamental and often unrecognized part of poker, but they run through these 5 GTO concepts.

1. Preflop ranges

To compensate for positional disadvantage, out-of-position players must open tighter hand ranges.

Premium starting hands aren't enough, though. Considering GTO poker ranges and principles, you want a good, balanced starting hand range from each position with at least some hands that can make a strong poker hand regardless of the flop texture (low, mid, high, disconnected, etc).

Below is a GTO preflop beginner poker chart for online 6-max play, showing which hand ranges one should open-raise with. Table positions are color-coded (see key below).

NOTE: For GTO play, it's advisable to use a mixed strategy for opening in the small blind, combining open-limps and open-raises for various hands. This cannot be illustrated with the color system used for the chart.

Choosing which hands to play is often a math problem, as discussed below.

Other preflop GTO poker charts include which hands to play after a raise, which to 3bet, etc. Solvers can help you decide which preflop hands to play (call, raise, re-raise, etc.).

2. Pot Odds

Always make +EV decisions that profit you as a poker player. Understanding pot odds (and equity) can help.

Postflop Pot Odds

Let’s say that we have JhTh on a board of 9h8h2s4c (open-ended straight-flush draw).  We have $40 left and $50 in the pot. He has you covered and goes all-in. As calling or folding are our only options, playing GTO involves calculating whether a call is +EV or –EV. (The hand was empty.)

Any remaining heart, Queen, or 7 wins the hand. This means we can improve 15 of 46 unknown cards, or 32.6% of the time.

What if our opponent has a set? The 4h or 2h could give us a flush, but it could also give the villain a boat. If we reduce outs from 15 to 14.5, our equity would be 31.5%.

We must now calculate pot odds.

(bet/(our bet+pot)) = pot odds

= $50 / ($40 + $90)

= $40 / $130

= 30.7%

To make a profitable call, we need at least 30.7% equity. This is a profitable call as we have 31.5% equity (even if villain has a set). Yes, we will lose most of the time, but we will make a small profit in the long run, making a call correct.

Pot odds aren't just for draws, either. If an opponent bets 50% pot, you get 3 to 1 odds on a call, so you must win 25% of the time to be profitable. If your current hand has more than 25% equity against your opponent's perceived range, call.

Preflop Pot Odds

Preflop, you raise to 3bb and the button 3bets to 9bb. You must decide how to act. In situations like these, we can actually use pot odds to assist our decision-making.

This pot is:

(our open+3bet size+small blind+big blind)

(3bb+9bb+0.5bb+1bb)

= 13.5

This means we must call 6bb to win a pot of 13.5bb, which requires 30.7% equity against the 3bettor's range.

Three additional factors must be considered:

Being out of position on our opponent makes it harder to realize our hand's equity, as he can use his position to put us in tough spots. To profitably continue against villain's hand range, we should add 7% to our equity.

Implied Odds / Reverse Implied Odds: The ability to win or lose significantly more post-flop (than pre-flop) based on our remaining stack.

While statistics on 3bet stats can be gained with a large enough sample size (i.e. 8% 3bet stat from button), the numbers don't tell us which 8% of hands villain could be 3betting with. Both polarized and depolarized charts below show 8% of possible hands.

7.4% of hands are depolarized.

Polarized Hand range (7.54%):

Each hand range has different contents. We don't know if he 3bets some hands and calls or folds others.

Using an exploitable strategy can help you play a hand range correctly. The next GTO concept will make things easier.

3. Minimum Defense Frequency:

This concept refers to the % of our range we must continue with (by calling or raising) to avoid being exploited by our opponents. This concept is most often used off-table and is difficult to apply in-game.

These beginner GTO concepts will help your decision-making during a hand, especially against aggressive opponents.

MDF formula:

MDF = POT SIZE/(POT SIZE+BET SIZE)

Here's a poker GTO chart of common bet sizes and minimum defense frequency.

Take the number of hand combos in your starting hand range and use the MDF to determine which hands to continue with. Choose hands with the most playability and equity against your opponent's betting range.

Say you open-raise HJ and BB calls. Qh9h6c flop. Your opponent leads you for a half-pot bet. MDF suggests keeping 67% of our range.

Using the above starting hand chart, we can determine that the HJ opens 254 combos:

We must defend 67% of these hands, or 170 combos, according to MDF. Hands we should keep include:

Flush draws

Open-Ended Straight Draws

Gut-Shot Straight Draws

Overcards

Any Pair or better

So, our flop continuing range could be:

Some highlights:

Fours and fives have little chance of improving on the turn or river.

We only continue with AX hearts (with a flush draw) without a pair or better.

We'll also include 4 AJo combos, all of which have the Ace of hearts, and AcJh, which can block a backdoor nut flush combo.

Let's assume all these hands are called and the turn is blank (2 of spades). Opponent bets full-pot. MDF says we must defend 50% of our flop continuing range, or 85 of 170 combos, to be unexploitable. This strategy includes our best flush draws, straight draws, and made hands.

Here, we keep combining:

Nut flush draws

Pair + flush draws

GS + flush draws

Second Pair, Top Kicker+

One combo of JJ that doesn’t block the flush draw or backdoor flush draw.

On the river, we can fold our missed draws and keep our best made hands. When calling with weaker hands, consider blocker effects and card removal to avoid overcalling and decide which combos to continue.

4. Poker GTO Bet Sizing

To avoid being exploited, balance your bluffs and value bets. Your betting range depends on how much you bet (in relation to the pot). This concept only applies on the river, as draws (bluffs) on the flop and turn still have equity (and are therefore total bluffs).

On the flop, you want a 2:1 bluff-to-value-bet ratio. On the flop, there won't be as many made hands as on the river, and your bluffs will usually contain equity. The turn should have a "bluffing" ratio of 1:1. Use the chart below to determine GTO river bluff frequencies (relative to your bet size):

This chart relates to your opponent's pot odds. If you bet 50% pot, your opponent gets 3:1 odds and must win 25% of the time to call. Poker GTO theory suggests including 25% bluff combinations in your betting range so you're indifferent to your opponent calling or folding.

Best river bluffs don't block hands you want your opponent to have (or not have). For example, betting with missed Ace-high flush draws is often a mistake because you block a missed flush draw you want your opponent to have when bluffing on the river (meaning that it would subsequently be less likely he would have it, if you held two of the flush draw cards). Ace-high usually has some river showdown value.

If you had a 3-flush on the river and wanted to raise, you could bluff raise with AX combos holding the bluff suit Ace. Blocking the nut flush prevents your opponent from using that combo.

5. Bet Sizes and Frequency

GTO beginner strategies aren't just bluffs and value bets. They show how often and how much to bet in certain spots. Top players have benefited greatly from poker solvers, which we'll discuss next.

GTO Poker Software

In recent years, various poker GTO solvers have been released to help beginner, intermediate, and advanced players play balanced/GTO poker in various situations.

PokerSnowie and PioSolver are popular GTO and poker study programs.

While you can't compute players' hand ranges and what hands to bet or check with in real time, studying GTO play strategies with these programs will pay off. It will improve your poker thinking and understanding.

Solvers can help you balance ranges, choose optimal bet sizes, and master cbet frequencies.

GTO Poker Tournament

Late-stage tournaments have shorter stacks than cash games. In order to follow GTO poker guidelines, Nash charts have been created, tweaked, and used for many years (and also when to call, depending on what number of big blinds you have when you find yourself shortstacked).

The charts are for heads-up push/fold. In a multi-player game, the "pusher" chart can only be used if play is folded to you in the small blind. The "caller" chart can only be used if you're in the big blind and assumes a small blind "pusher" (with a much wider range than if a player in another position was open-shoving).

Divide the pusher chart's numbers by 2 to see which hand to use from the Button. Divide the original chart numbers by 4 to find the CO's pushing range. Some of the figures will be impossible to calculate accurately for the CO or positions to the right of the blinds because the chart's highest figure is "20+" big blinds, which is also used for a wide range of hands in the push chart.

Both of the GTO charts below are ideal for heads-up play, but exploitable HU shortstack strategies can lead to more +EV decisions against certain opponents. Following the charts will make your play GTO and unexploitable.

Poker pro Max Silver created the GTO push/fold software SnapShove. (It's accessible online at www.snapshove.com or as iOS or Android apps.)

Players can access GTO shove range examples in the full version. (You can customize the number of big blinds you have, your position, the size of the ante, and many other options.)

In Conclusion

Due to the constantly changing poker landscape, players are always improving their skills. Exploitable strategies often yield higher profit margins than GTO-based approaches, but knowing GTO beginner and advanced concepts can give you an edge for a few reasons.

It creates a solid gameplay base.

Having a baseline makes it easier to exploit certain villains.

You can avoid leveling wars with your opponents by making sound poker decisions based on GTO strategy.

It doesn't require assuming opponents' play styles.

Not results-oriented.

This is just the beginning of GTO and poker theory. Consider investing in the GTO poker solver software listed above to improve your game.

Don't overcomplicate to-do lists. DWTS (Done, Waiting, Top 3, Soon) organizes your to-dos.

How Are You Going to Manage Everything?

Modern America is busy. Work involves meetings. Anytime, Slack communications arrive. Many software solutions offer a @-mention notification capability. Emails.

Work obligations continue. At home, there are friends, family, bills, chores, and fun things.

How are you going to keep track of it all? Enter the todo list. It’s been around forever. It’s likely to stay forever in some way, shape, or form.

Everybody has their own system. You probably modified something from middle school. Post-its? Maybe it’s an app? Maybe both, another system, or none.

I suggest a format that has worked for me in 15 years of professional and personal life.

Try it out and see if it works for you. If not, no worries. You do you! Hopefully though you can learn a thing or two, and I from you too.

It is merely a Google Doc, yes.

It's a giant list. One task per line. Indent subtasks on a new line. Add or move new tasks as needed.

I recommend using Google Docs. It's easy to use and flexible for structuring.

Prioritizing these tasks is key. I organize them using DWTS (Done, Waiting, Top 3, Soon). Chronologically is good because it implicitly provides both a priority (high, medium, low) and an ETA (now, soon, later).

Yes, I recognize the similarities to DWTS (Dancing With The Stars) TV Show. Although I'm not a fan, it's entertaining. The acronym is easy to remember and adds fun to something dull.

What each section contains

Done

All tasks' endpoint. Finish here. Don't worry about it again.

Waiting

You're blocked and can't continue. Blocked tasks usually need someone. Write Person Task so you know who's waiting.

Blocking tasks shouldn't last long. After a while, remind them kindly. If people don't help you out of kindness, they will if you're persistent.

Top 3

Mental focus areas. These can be short- to mid-term goals or recent accomplishments. 2 to 5 is a good number to stay focused.

Top 3 reminds us to prioritize. If they don't fit your Top 3 goals, delay them.

Every 1:1 at work is a project update. Another chance to list your top 3. You should know your Top 3 well and be able to discuss them confidently.

Soon

Here's your short-term to-do list. Rank them from highest to lowest.

I usually subdivide it with empty lines. First is what I have to do today, then week, then month. Subsections can be arranged however you like.

Inventories by Concept

Tasks that aren’t in your short or medium future go into the backlog. 
Eventually you’ll complete these tasks, assign them to someone else, or mark them as “wont’ do” (like done but in another sense).

Backlog tasks don't need to be organized chronologically because their timing and priority may change. Theme-organize them. When planning/strategic, you can choose themes to focus on, so future top 3 topics.

More Tips on Todos

Decide Upon a Morning Goal

Morning routines are universal. Coffee and Wordle. My to-do list is next. Two things:

• As needed, update the to-do list: based on the events of yesterday and any fresh priorities.

• Pick a few jobs to complete today: Pick a few goals that you know you can complete today. Push the remainder below and move them to the top of the Soon section. I typically select a few tasks I am confident I can complete along with one stretch task that might extend into tomorrow.

Finally. By setting and achieving small goals every day, you feel accomplished and make steady progress on medium and long-term goals.

Tech companies call this a daily standup. Everyone shares what they did yesterday, what they're doing today, and any blockers. The name comes from a tradition of holding meetings while standing up to keep them short. Even though it's virtual, everyone still wants a quick meeting.

Your team may or may not need daily standups. Make a daily review a habit with your coffee.

Review Backwards & Forwards on a regular basis

While you're updating your to-do list daily, take time to review it.

Review your Done list. Remember things you're proud of and things that could have gone better. Your Done list can be long. Archive it so your main to-do list isn't overwhelming.

Future-gaze. What you considered important may no longer be. Reorder tasks. Backlog grooming is a workplace term.

Backwards-and-forwards reviews aren't required often. Every 3-6 months is fine. They help you see the forest as often as the trees.

Final Remarks

Keep your list simple. Done, Waiting, Top 3, Soon. These are the necessary sections. If you like, add more subsections; otherwise, keep it simple.

I recommend a morning review. By having clear goals and an action-oriented attitude, you'll be successful.

In this essay, I studied REAL NFT use examples and their potential uses.

Knowledge of the Hype Cycle

Gartner's Hype Cycle.

It proposes 5 phases for disruptive technology.

1. Technology Trigger: the emergence of potentially disruptive technology.

2. Peak of Inflated Expectations: Early publicity creates hype. (Ex: 2021 Bubble)

3. Trough of Disillusionment: Early projects fail to deliver on promises and the public loses interest. I suspect NFTs are somewhere around this trough of disillusionment now.

4. Enlightenment slope: The tech shows successful use cases.

5. Plateau of Productivity: Mainstream adoption has arrived and broader market applications have proven themselves. Here’s a more detailed visual of the Gartner Hype Cycle from Wikipedia.

In the speculative NFT bubble of 2021, @beeple sold Everydays: the First 5000 Days for $69 MILLION in 2021's NFT bubble.

@nbatopshot sold millions in video collectibles.

This is when expectations peaked.

Let's examine NFTs' real-world applications.

Watch this video if you're unfamiliar with NFTs.

Online Art

Most people think NFTs are rich people buying worthless JPEGs and MP4s.

Digital artwork and collectibles are revolutionary for creators and enthusiasts.

NFT Profile Pictures

You might also have seen NFT profile pictures on Twitter.

My profile picture is an NFT I coined with @skogards factoria app, which helps me avoid bogus accounts.

Profile pictures are a good beginning point because they're unique and clearly yours.

NFTs are a way to represent proof-of-ownership. It’s easier to prove ownership of digital assets than physical assets, which is why artwork and pfps are the first use cases.

They can do much more.

NFTs can represent anything with a unique owner and digital ownership certificate. Domains and usernames.

Usernames & Domains

@unstoppableweb, @ensdomains, @rarible sell NFT domains.

NFT domains are transferable, which is a benefit.

Godaddy and other web2 providers have difficult-to-transfer domains. Domains are often leased instead of purchased.

Tickets

NFTs can also represent concert tickets and event passes.

There's a limited number, and entry requires proof.

NFTs can eliminate the problem of forgery and make it easy to verify authenticity and ownership.

NFT tickets can be traded on the secondary market, which allows for:

1. marketplaces that are uniform and offer the seller and buyer security (currently, tickets are traded on inefficient markets like FB & craigslist)

2. unbiased pricing

3. Payment of royalties to the creator

4. Historical ticket ownership data implies performers can airdrop future passes, discounts, etc.

5. NFT passes can be a fandom badge.

The $30B+ online tickets business is increasing fast.

NFT-based ticketing projects:

• @GetProtocol

• @SeatlabNFT

• @YellowHeartNFT

• @GUTStickets

• @WicketEvents

Gaming Assets

NFTs also help in-game assets.

Imagine someone spending five years collecting a rare in-game blade, then outgrowing or quitting the game. Gamers value that collectible.

The gaming industry is expected to make $200 BILLION in revenue this year, a significant portion of which comes from in-game purchases.

Royalties on secondary market trading of gaming assets encourage gaming businesses to develop NFT-based ecosystems.

Digital assets are the start. On-chain NFTs can represent real-world assets effectively.

Real estate has a unique owner and requires ownership confirmation.

Real Estate

Tokenizing property has many benefits.

1. Can be fractionalized to increase access, liquidity

2. Can be collateralized to increase capital efficiency and access to loans backed by an on-chain asset

3. Allows investors to diversify or make bets on specific neighborhoods, towns or cities +++

I've written about this thought exercise before.

I made an animated video explaining this.

We've just explored NFTs for transferable assets. But what about non-transferrable NFTs?

SBTs are Soul-Bound Tokens. Vitalik Buterin (Ethereum co-founder) blogged about this.

NFTs are basically verifiable digital certificates.

Diplomas & Degrees

That fits Degrees & Diplomas. These shouldn't be marketable, thus they can be non-transferable SBTs.

Anyone can verify the legitimacy of on-chain credentials, degrees, abilities, and achievements.

The same goes for other awards.

For example, LinkedIn could give you a verified checkmark for your degree or skills.

Authenticity Protection

NFTs can also safeguard against counterfeiting.

Counterfeiting is the largest criminal enterprise in the world, estimated to be $2 TRILLION a year and growing.

Anti-counterfeit tech is valuable.

This is one of @ORIGYNTech's projects.

Identity

Identity theft/verification is another real-world problem NFTs can handle.

In the US, 15 million+ citizens face identity theft every year, suffering damages of over $50 billion a year.

This isn't surprising considering all you need for US identity theft is a 9-digit number handed around in emails, documents, on the phone, etc.

Identity NFTs can fix this.

• NFTs are one-of-a-kind and unforgeable.

• NFTs offer a universal standard.

• NFTs are simple to verify.

• SBTs, or non-transferrable NFTs, are tied to a particular wallet.

• In the event of wallet loss or theft, NFTs may be revoked.

This could be one of the biggest use cases for NFTs.

Imagine a global identity standard that is standardized across countries, cannot be forged or stolen, is digital, easy to verify, and protects your private details.

Since your identity is more than your government ID, you may have many NFTs.

@0xPolygon and @civickey are developing on-chain identity.

Memberships

NFTs can authenticate digital and physical memberships.

Voting

NFT IDs can verify votes.

If you remember 2020, you'll know why this is an issue.

Online voting's ease can boost turnout.

Informational property

NFTs can protect IP.

This can earn creators royalties.

NFTs have 2 important properties:

• Verifiability IP ownership is unambiguously stated and publicly verified.

• Platforms that enable authors to receive royalties on their IP can enter the market thanks to standardization.

Content Rights

Monetization without copyrighting = more opportunities for everyone.

This works well with the music.

Spotify and Apple Music pay creators very little.

Crowdfunding

Creators can crowdfund with NFTs.

NFTs can represent future royalties for investors.

This is particularly useful for fields where people who are not in the top 1% can’t make money. (Example: Professional sports players)

Mirror.xyz allows blog-based crowdfunding.

Financial NFTs

This introduces Financial NFTs (fNFTs). Unique financial contracts abound.

Examples:

• a person's collection of assets (unique portfolio)

• A loan contract that has been partially repaid with a lender

• temporal tokens (ex: veCRV)

Legal Agreements

Not just financial contracts.

NFT can represent any legal contract or document.

Messages & Emails

What about other agreements? Verbal agreements through emails and messages are likewise unique, but they're easily lost and fabricated.

Health Records

Medical records or prescriptions are another types of documentation that has to be verified but isn't.

Medical NFT examples:

• Immunization records

• Covid test outcomes

• Prescriptions

• health issues that may affect one's identity

• Observations made via health sensors

Existing systems of proof by paper / PDF have photoshop-risk.

I tried to include most use scenarios, but this is just the beginning.

NFTs have many innovative uses.

For example: @ShaanVP minted an NFT called “5 Minutes of Fame” 👇

Here are 2 Twitter threads about NFTs:

1. This piece of gold by @chriscantino

2. This conversation between @punk6529 and @RaoulGMI on @RealVision“The World According to @punk6529”

If you're wondering why NFTs are better than web2 databases for these use scenarios, see this Twitter thread I wrote:

If you liked this, please share it.

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.