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Amelia Winger-Bearskin

Amelia Winger-Bearskin

3 years ago

Hate NFTs? I must break some awful news to you...

More on NFTs & Art

Jake Prins

Jake Prins

3 years ago

What are NFTs 2.0 and what issues are they meant to address?

New standards help NFTs reach their full potential.

NFTs 2.0

NFTs lack interoperability and functionality. They have great potential but are mostly speculative. To maximize NFTs, we need flexible smart contracts.

Current requirements are too restrictive.

Most NFTs are based on ERC-721, which makes exchanging them easy. CryptoKitties, a popular online game, used the 2017 standard to demonstrate NFTs' potential.

This simple standard includes a base URI and incremental IDs for tokens. Add the tokenID to the base URI to get the token's metadata.

This let creators collect NFTs. Many NFT projects store metadata on IPFS, a distributed storage network, but others use Google Drive. NFT buyers often don't realize that if the creators delete or move the files, their NFT is just a pointer.

This isn't the standard's biggest issue. There's no way to validate NFT projects.

Creators are one of the most important aspects of art, but nothing is stored on-chain.

ERC-721 contracts only have a name and symbol.

Most of the data on OpenSea's collection pages isn't from the NFT's smart contract. It was added through a platform input field, so it's in the marketplace's database. Other websites may have different NFT information.

In five years, your NFT will be just a name, symbol, and ID.

Your NFT doesn't mention its creators. Although the smart contract has a public key, it doesn't reveal who created it.

The NFT's creators and their reputation are crucial to its value. Think digital fashion and big brands working with well-known designers when more professionals use NFTs. Don't you want them in your NFT?

Would paintings be as valuable if their artists were unknown? Would you believe it's real?

Buying directly from an on-chain artist would reduce scams. Current standards don't allow this data.

Most creator profiles live on centralized marketplaces and could disappear. Current platforms have outpaced underlying standards. The industry's standards are lagging.

For NFTs to grow beyond pointers to a monkey picture file, we may need to use new Web3-based standards.

Introducing NFTs 2.0

Fabian Vogelsteller, creator of ERC-20, developed new web3 standards. He proposed LSP7 Digital Asset and LSP8 Identifiable Digital Asset, also called NFT 2.0.

NFT and token metadata inputs are extendable. Changes to on-chain metadata inputs allow NFTs to evolve. Instead of public keys, the contract can have Universal Profile addresses attached. These profiles show creators' faces and reputations. NFTs can notify asset receivers, automating smart contracts.

LSP7 and LSP8 use ERC725Y. Using a generic data key-value store gives contracts much-needed features:

  • The asset can be customized and made to stand out more by allowing for unlimited data attachment.

  • Recognizing changes to the metadata

  • using a hash reference for metadata rather than a URL reference

This base will allow more metadata customization and upgradeability. These guidelines are:

  • Genuine and Verifiable Now, the creation of an NFT by a specific Universal Profile can be confirmed by smart contracts.

  • Dynamic NFTs can update Flexible & Updatable Metadata, allowing certain things to evolve over time.

  • Protected metadata Now, secure metadata that is readable by smart contracts can be added indefinitely.

  • Better NFTS prevent the locking of NFTs by only being sent to Universal Profiles or a smart contract that can interact with them.

Summary

NFTS standards lack standardization and powering features, limiting the industry.

ERC-721 is the most popular NFT standard, but it only represents incremental tokenIDs without metadata or asset representation. No standard sender-receiver interaction or security measures ensure safe asset transfers.

NFT 2.0 refers to the new LSP7-DigitalAsset and LSP8-IdentifiableDigitalAsset standards.

They have new standards for flexible metadata, secure transfers, asset representation, and interactive transfer.

With NFTs 2.0 and Universal Profiles, creators could build on-chain reputations.

NFTs 2.0 could bring the industry's needed innovation if it wants to move beyond trading profile pictures for speculation.

Stephen Moore

Stephen Moore

3 years ago

Trading Volume on OpenSea Drops by 99% as the NFT Boom Comes to an End

Wasn't that a get-rich-quick scheme?

Bored Ape, edited by author

OpenSea processed $2.7 billion in NFT transactions in May 2021.

Fueled by a crypto bull run, rumors of unfathomable riches, and FOMO, Bored Apes, Crypto Punks, and other JPEG-format trash projects flew off the virtual shelves, snatched up by retail investors and celebrities alike.

Over a year later, those shelves are overflowing and warehouses are backlogged. Since March, I've been writing less. In May and June, the bubble was close to bursting.

Apparently, the boom has finally peaked.

This bubble has punctured, and deflation has begun. On Aug. 28, OpenSea processed $9.34 million.

From that euphoric high of $2.7 billion, $9.34 million represents a spectacular decline of 99%.

OpenSea contradicts the data. A trading platform spokeswoman stated the comparison is unfair because it compares the site's highest and lowest trading days. They're the perfect two data points to assess the drop. OpenSea chooses to use ETH volume measures, which ignore crypto's shifting price. Since January 2022, monthly ETH volume has dropped 140%, according to Dune.

Unconvincing counterargument.

Further OpenSea indicators point to declining NFT demand:

  • Since January 2022, daily user visits have decreased by 50%.

  • Daily transactions have decreased by 50% since the beginning of the year in the same manner.

Off-platform, the floor price of Bored Apes has dropped from 145 ETH to 77 ETH. (At $4,800, a reduction from $700,000 to $370,000). Google search data shows waning popular interest.

Data: Google Trends

It is a trend that will soon vanish, just like laser eyes.

NFTs haven't moved since the new year. Eminem and Snoop Dogg can utilize their apes in music videos or as 3D visuals to perform at the VMAs, but the reality is that NFTs have lost their public appeal and the market is trying to regain its footing.

They've lost popularity because?

Breaking records. The technology still lacks genuine use cases a year and a half after being popular.

They're pricey prestige symbols that have made a few people rich through cunning timing or less-than-savory scams or rug pulling. Over $10.5 billion has been taken through frauds, most of which are NFT enterprises promising to be the next Bored Apes, according to Web3 is going wonderfully. As the market falls, many ordinary investors realize they purchased into a self-fulfilling ecosystem that's halted. Many NFTs are sold between owner-held accounts to boost their price, data suggests. Most projects rely on social media excitement to debut with a high price before the first owners sell and chuckle to the bank. When they don't, the initiative fails, leaving investors high and dry.

NFTs are fading like laser eyes. Most people pushing the technology don't believe in it or the future it may bring. No, they just need a Kool-Aid-drunk buyer.

Everybody wins. When your JPEGs are worth 99% less than when you bought them, you've lost.

When demand reaches zero, many will lose.

Protos

Protos

3 years ago

Plagiarism on OpenSea: humans and computers

OpenSea, a non-fungible token (NFT) marketplace, is fighting plagiarism. A new “two-pronged” approach will aim to root out and remove copies of authentic NFTs and changes to its blue tick verified badge system will seek to enhance customer confidence.

According to a blog post, the anti-plagiarism system will use algorithmic detection of “copymints” with human reviewers to keep it in check.

Last year, NFT collectors were duped into buying flipped images of the popular BAYC collection, according to The Verge. The largest NFT marketplace had to remove its delay pay minting service due to an influx of copymints.

80% of NFTs removed by the platform were minted using its lazy minting service, which kept the digital asset off-chain until the first purchase.

NFTs copied from popular collections are opportunistic money-grabs. Right-click, save, and mint the jacked JPEGs that are then flogged as an authentic NFT.

The anti-plagiarism system will scour OpenSea's collections for flipped and rotated images, as well as other undescribed permutations. The lack of detail here may be a deterrent to scammers, or it may reflect the new system's current rudimentary nature.

Thus, human detectors will be needed to verify images flagged by the detection system and help train it to work independently.

“Our long-term goal with this system is two-fold: first, to eliminate all existing copymints on OpenSea, and second, to help prevent new copymints from appearing,” it said.

“We've already started delisting identified copymint collections, and we'll continue to do so over the coming weeks.”

It works for Twitter, why not OpenSea

OpenSea is also changing account verification. Early adopters will be invited to apply for verification if their NFT stack is worth $100 or more. OpenSea plans to give the blue checkmark to people who are active on Twitter and Discord.

This is just the beginning. We are committed to a future where authentic creators can be verified, keeping scammers out.

Also, collections with a lot of hype and sales will get a blue checkmark. For example, a new NFT collection sold by the verified BAYC account will have a blue badge to verify its legitimacy.

New requests will be responded to within seven days, according to OpenSea.

These programs and products help protect creators and collectors while ensuring our community can confidently navigate the world of NFTs.

By elevating authentic content and removing plagiarism, these changes improve trust in the NFT ecosystem, according to OpenSea.

OpenSea is indeed catching up with the digital art economy. Last August, DevianArt upgraded its AI image recognition system to find stolen tokenized art on marketplaces like OpenSea.

It scans all uploaded art and compares it to “public blockchain events” like Ethereum NFTs to detect stolen art.

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Matthew Cluff

Matthew Cluff

3 years ago

GTO Poker 101

"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.

Sam Hickmann

Sam Hickmann

3 years ago

Nomad.xyz got exploited for $190M

Key Takeaways:

Another hack. This time was different. This is a doozy.

Why? Nomad got exploited for $190m. It was crypto's 5th-biggest hack. Ouch.

It wasn't hackers, but random folks. What happened:

A Nomad smart contract flaw was discovered. They couldn't drain the funds at once, so they tried numerous transactions. Rookie!

People noticed and copied the attack.

They just needed to discover a working transaction, substitute the other person's address with theirs, and run it.


Nomad.xyz got exploited for $190M

In a two-and-a-half-hour attack, $190M was siphoned from Nomad Bridge.

Nomad is a novel approach to blockchain interoperability that leverages an optimistic mechanism to increase the security of cross-chain communication.  — nomad.xyz

This hack was permissionless, therefore anyone could participate.

After the fatal blow, people fought over the scraps.

Cross-chain bridges remain a DeFi weakness and exploit target. When they collapse, it's typically total.

$190M...gobbled.

Unbacked assets are hurting Nomad-dependent chains. Moonbeam, EVMOS, and Milkomeda's TVLs dropped.

This incident is every-man-for-himself, although numerous whitehats exploited the issue... 

But what triggered the feeding frenzy?

How did so many pick the bones?

After a normal upgrade in June, the bridge's Replica contract was initialized with a severe security issue. The  0x00 address was a trusted root, therefore all messages were valid by default.

After a botched first attempt (costing $350k in gas), the original attacker's exploit tx called process() without first 'proving' its validity.

The process() function executes all cross-chain messages and checks the merkle root of all messages (line 185).

The upgrade caused transactions with a'messages' value of 0 (invalid, according to old logic) to be read by default as 0x00, a trusted root, passing validation as 'proven'

Any process() calls were valid. In reality, a more sophisticated exploiter may have designed a contract to drain the whole bridge.

Copycat attackers simply copied/pasted the same process() function call using Etherscan, substituting their address.

The incident was a wild combination of crowdhacking, whitehat activities, and MEV-bot (Maximal Extractable Value) mayhem.

For example, 🍉🍉🍉. eth stole $4M from the bridge, but claims to be whitehat.

Others stood out for the wrong reasons. Repeat criminal Rari Capital (Artibrum) exploited over $3M in stablecoins, which moved to Tornado Cash.

The top three exploiters (with 95M between them) are:

$47M: 0x56D8B635A7C88Fd1104D23d632AF40c1C3Aac4e3

$40M: 0xBF293D5138a2a1BA407B43672643434C43827179

$8M: 0xB5C55f76f90Cc528B2609109Ca14d8d84593590E

Here's a list of all the exploiters:

The project conducted a Quantstamp audit in June; QSP-19 foreshadowed a similar problem.

The auditor's comments that "We feel the Nomad team misinterpreted the issue" speak to a troubling attitude towards security that the project's "Long-Term Security" plan appears to confirm:

Concerns were raised about the team's response time to a live, public exploit; the team's official acknowledgement came three hours later.

"Removing the Replica contract as owner" stopped the exploit, but it was too late to preserve the cash.

Closed blockchain systems are only as strong as their weakest link.

The Harmony network is in turmoil after its bridge was attacked and lost $100M in late June.

What's next for Nomad's ecosystems?

Moonbeam's TVL is now $135M, EVMOS's is $3M, and Milkomeda's is $20M.

Loss of confidence may do more damage than $190M.

Cross-chain infrastructure is difficult to secure in a new, experimental sector. Bridge attacks can pollute an entire ecosystem or more.

Nomadic liquidity has no permanent home, so consumers will always migrate in pursuit of the "next big thing" and get stung when attentiveness wanes.

DeFi still has easy prey...

Sources: rekt.news & The Milk Road.

Shalitha Suranga

Shalitha Suranga

3 years ago

The Top 5 Mathematical Concepts Every Programmer Needs to Know

Using math to write efficient code in any language

Photo by Emile Perron on Unsplash, edited with Canva

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.

A mathematical expression/formula in the Linux codebase, a screenshot by the author

Inbuilt max and min functions can minimize verbose if statements.

Reducing a verbose nested-if with the min function in Neutralinojs, a screenshot by the author

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.

GetWindowRect outputs an LPRECT geometric object, a screenshot by the author

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):

The Git codebase defines a function with logarithmic time complexity, a screenshot by the author

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

A utility function that uses ceil, a screenshot by the author

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.