The 2008 Bitcoin white paper solves the classic computer science consensus problem.

Issue Statement

The Byzantine Generals Problem (BGP) is called after an allegory in which several generals must collaborate and attack a city at the same time to win (figure 1-left). Any general who retreats at the last minute loses the fight (figure 1-right). Thus, precise messengers and no rogue generals are essential. This is difficult without a trusted central authority.

In their 1982 publication, Leslie Lamport, Robert Shostak, and Marshall Please termed this topic the Byzantine Generals Problem to simplify distributed computer systems.

Consensus in a distributed computer network is the issue. Reaching a consensus on which systems work (and stay in the network) and which don't makes maintaining a network tough (i.e., needs to be removed from network). Challenges include unreliable communication routes between systems and mis-reporting systems.

Solving BGP can let us construct machine learning solutions without single points of failure or trusted central entities. One server hosts model parameters while numerous workers train the model. This study describes fault-tolerant Distributed Byzantine Machine Learning.

Bitcoin invented a mechanism for a distributed network of nodes to agree on which transactions should go into the distributed ledger (blockchain) without a trusted central body. It solved BGP implementation. Satoshi Nakamoto, the pseudonymous bitcoin creator, solved the challenge by cleverly combining cryptography and consensus mechanisms.

Disclaimer

This is not financial advice. It discusses a unique computer science solution.

Bitcoin

Bitcoin's white paper begins:

“A purely peer-to-peer version of electronic cash would allow online payments to be sent directly from one party to another without going through a financial institution.” Source: https://www.ussc.gov/sites/default/files/pdf/training/annual-national-training-seminar/2018/Emerging_Tech_Bitcoin_Crypto.pdf

Bitcoin's main parts:

1. The open-source and versioned bitcoin software that governs how nodes, miners, and the bitcoin token operate.

2. The native kind of token, known as a bitcoin token, may be created by mining (up to 21 million can be created), and it can be transferred between wallet addresses in the bitcoin network.

3. Distributed Ledger, which contains exact copies of the database (or "blockchain") containing each transaction since the first one in January 2009.

4. distributed network of nodes (computers) running the distributed ledger replica together with the bitcoin software. They broadcast the transactions to other peer nodes after validating and accepting them.

5. Proof of work (PoW) is a cryptographic requirement that must be met in order for a miner to be granted permission to add a new block of transactions to the blockchain of the cryptocurrency bitcoin. It takes the form of a valid hash digest. In order to produce new blocks on average every 10 minutes, Bitcoin features a built-in difficulty adjustment function that modifies the valid hash requirement (length of nonce). PoW requires a lot of energy since it must continually generate new hashes at random until it satisfies the criteria.

6. The competing parties known as miners carry out continuous computing processing to address recurrent cryptography issues. Transaction fees and some freshly minted (mined) bitcoin are the rewards they receive. The amount of hashes produced each second—or hash rate—is a measure of mining capacity.

Cryptography, decentralization, and the proof-of-work consensus method are Bitcoin's most unique features.

Bitcoin uses encryption

Bitcoin employs this established cryptography.

1. Hashing

2. digital signatures based on asymmetric encryption

Hashing (SHA-256) (SHA-256)

Hashing converts unique plaintext data into a digest. Creating the plaintext from the digest is impossible. Bitcoin miners generate new hashes using SHA-256 to win block rewards.

A new hash is created from the current block header and a variable value called nonce. To achieve the required hash, mining involves altering the nonce and re-hashing.

The block header contains the previous block hash and a Merkle root, which contains hashes of all transactions in the block. Thus, a chain of blocks with increasing hashes links back to the first block. Hashing protects new transactions and makes the bitcoin blockchain immutable. After a transaction block is mined, it becomes hard to fabricate even a little entry.

Asymmetric Cryptography Digital Signatures

Asymmetric cryptography (public-key encryption) requires each side to have a secret and public key. Public keys (wallet addresses) can be shared with the transaction party, but private keys should not. A message (e.g., bitcoin payment record) can only be signed by the owner (sender) with the private key, but any node or anybody with access to the public key (visible in the blockchain) can verify it. Alex will submit a digitally signed transaction with a desired amount of bitcoin addressed to Bob's wallet to a node to send bitcoin to Bob. Alex alone has the secret keys to authorize that amount. Alex's blockchain public key allows anyone to verify the transaction.

Solution

Now, apply bitcoin to BGP. BGP generals resemble bitcoin nodes. The generals' consensus is like bitcoin nodes' blockchain block selection. Bitcoin software on all nodes can:

Check transactions (i.e., validate digital signatures)

2. Accept and propagate just the first miner to receive the valid hash and verify it accomplished the task. The only way to guess the proper hash is to brute force it by repeatedly producing one with the fixed/current block header and a fresh nonce value.

Thus, PoW and a dispersed network of nodes that accept blocks from miners that solve the unfalsifiable cryptographic challenge solve consensus.

Suppose:

1. Unreliable nodes

2. Unreliable miners

Bitcoin accepts the longest chain if rogue nodes cause divergence in accepted blocks. Thus, rogue nodes must outnumber honest nodes in accepting/forming the longer chain for invalid transactions to reach the blockchain. As of November 2022, 7000 coordinated rogue nodes are needed to takeover the bitcoin network.

Dishonest miners could also try to insert blocks with falsified transactions (double spend, reverse, censor, etc.) into the chain. This requires over 50% (51% attack) of miners (total computational power) to outguess the hash and attack the network. Mining hash rate exceeds 200 million (source). Rewards and transaction fees encourage miners to cooperate rather than attack. Quantum computers may become a threat.

Visit my Quantum Computing post.

Quantum computers—what are they? Quantum computers will have a big influence. towardsdatascience.com

Nodes have more power than miners since they can validate transactions and reject fake blocks. Thus, the network is secure if honest nodes are the majority.

Summary

Table 1 compares three Byzantine Generals Problem implementations.

Bitcoin white paper and implementation solved the consensus challenge of distributed systems without central governance. It solved the illusive Byzantine Generals Problem.

Resources

Resources

1. https://en.wikipedia.org/wiki/Byzantine_fault

2. Source-code for Bitcoin Core Software — https://github.com/bitcoin/bitcoin

3. Bitcoin white paper — https://bitcoin.org/bitcoin.pdf

4. https://en.wikipedia.org/wiki/Bitcoin

5. https://www.microsoft.com/en-us/research/publication/byzantine-generals-problem/

6. https://www.microsoft.com/en-us/research/uploads/prod/2016/12/The-Byzantine-Generals-Problem.pdf

7. https://en.wikipedia.org/wiki/Hash_function

8. https://en.wikipedia.org/wiki/Merkle_tree

9. https://en.wikipedia.org/wiki/SHA-2

10. https://en.wikipedia.org/wiki/Public-key_cryptography

11. https://en.wikipedia.org/wiki/Digital_signature

12. https://en.wikipedia.org/wiki/Proof_of_work

13. https://en.wikipedia.org/wiki/Quantum_cryptography

14. https://dci.mit.edu/bitcoin-security-initiative

15. https://dci.mit.edu/51-attacks

16. Genuinely Distributed Byzantine Machine Learning, El-Mahdi El-Mhamdi et al., 2020. ACM, New York, NY, https://doi.org/10.1145/3382734.3405695

Microsoft co-founder Bill Gates assesses digital assets while the bull is caged.

Bill Gates is well-respected.

Reasonably. He co-founded and led Microsoft during its 1980s and 1990s revolution.

After leaving Microsoft, Bill Gates pursued other interests. He and his wife founded one of the world's largest philanthropic organizations, Bill & Melinda Gates Foundation. He also supports immunizations, population control, and other global health programs.

When Gates criticized Bitcoin, cryptocurrencies, and NFTs, it made news.

Bill Gates said at the 58th Munich Security Conference...

“You have an asset class that’s 100% based on some sort of greater fool theory that somebody’s going to pay more for it than I do.”

Gates means digital assets. Like many bitcoin critics, he says digital coins and tokens are speculative.

And he's not alone. Financial experts have dubbed Bitcoin and other digital assets a "bubble" for a decade.

Gates also made fun of Bored Ape Yacht Club and NFTs, saying, "Obviously pricey digital photographs of monkeys will help the world."

Why does Bill Gates dislike digital assets?

According to Gates' latest comments, Bitcoin, cryptos, and NFTs aren't good ways to hold value.

Bill Gates is a better investor than Elon Musk.

“I’m used to asset classes, like a farm where they have output, or like a company where they make products,” Gates said.

The Guardian claimed in April 2021 that Bill and Melinda Gates owned the most U.S. farms. Over 242,000 acres of farmland.

The Gates couple has enough farmland to cover Hong Kong.

Bill Gates is a classic investor. He wants companies with an excellent track record, strong fundamentals, and good management. Or tangible assets like land and property.

Gates prefers the "old economy" over the "new economy"

Gates' criticism of Bitcoin and cryptocurrency ventures isn't surprising. These digital assets lack all of Gates's investing criteria.

Volatile digital assets include Bitcoin. Their costs might change dramatically in a day. Volatility scares risk-averse investors like Gates.

Gates has a stake in the old financial system. As Microsoft's co-founder, Gates helped develop a dominant tech company.

Because of his business, he's one of the world's richest men.

Bill Gates is invested in protecting the current paradigm.

He won't invest in anything that could destroy the global economy.

When Gates criticizes Bitcoin, cryptocurrencies, and NFTs, he's suggesting they're a hoax. These soapbox speeches are one way he protects his interests.

Digital assets aren't a bad investment, though. Many think they're the future.

Changpeng Zhao and Brian Armstrong are two digital asset billionaires. Two crypto exchange CEOs. Binance/Coinbase.

Digital asset revolution won't end soon.

If you disagree with Bill Gates and plan to invest in Bitcoin, cryptocurrencies, or NFTs, do your own research and understand the risks.

But don’t take Bill Gates’ word for it.

He’s just an old rich guy with a lot of farmland.

He has a lot to lose if Bitcoin and other digital assets gain global popularity.

This post is a summary. Read the full article here.

Introduction

Markets rely on knowledge.

The internet provided practically endless knowledge and wisdom. Humanity has never seen such leverage. Technology's progress drives us to adapt to a changing world, changing our routines and behaviors.

In a digital age, people may struggle to live in the analogue world of their upbringing. Can those who can't adapt change their lives? I won't answer. We should teach those who are willing to learn, nevertheless. Unravel the modern world's riddles and give them wisdom.

Adapt or die . Accept the future or remain behind.

This essay will help you comprehend Bitcoin better than most market participants and the general public. Let's dig into Bitcoin.

Join me.

Ascension

Bitcoin.org was registered in August 2008. Bitcoin whitepaper was published on 31 October 2008. The document intrigued and motivated people around the world, including technical engineers and sovereignty seekers. Since then, Bitcoin's whitepaper has been read and researched to comprehend its essential concept.

I recommend reading the whitepaper yourself. You'll be able to say you read the Bitcoin whitepaper instead of simply Googling "what is Bitcoin" and reading the fundamental definition without knowing the revolution's scope. The article links to Bitcoin's whitepaper. To avoid being overwhelmed by the whitepaper, read the following article first.

Bitcoin isn't the first peer-to-peer digital currency. Hashcash or Bit Gold were once popular cryptocurrencies. These two Bitcoin precursors failed to gain traction and produce the network effect needed for general adoption. After many struggles, Bitcoin emerged as the most successful cryptocurrency, leading the way for others.

Satoshi Nakamoto, an active bitcointalk.org user, created Bitcoin. Satoshi's identity remains unknown. Satoshi's last bitcointalk.org login was 12 December 2010. Since then, he's officially disappeared. Thus, conspiracies and riddles surround Bitcoin's creators. I've heard many various theories, some insane and others well-thought-out.

It's not about who created it; it's about knowing its potential. Since its start, Satoshi's legacy has changed the world and will continue to.

Block-by-block blockchain

Bitcoin is a distributed ledger. What's the meaning?

Everyone can view all blockchain transactions, but no one can undo or delete them.

Imagine you and your friends routinely eat out, but only one pays. You're careful with money and what others owe you. How can everyone access the info without it being changed?

You'll keep a notebook of your evening's transactions. Everyone will take a page home. If one of you changed the page's data, the group would notice and reject it. The majority will establish consensus and offer official facts.

Miners add a new Bitcoin block to the main blockchain every 10 minutes. The appended block contains miner-verified transactions. Now that the next block has been added, the network will receive the next set of user transactions.

Bitcoin Proof of Work—prove you earned it

Any firm needs hardworking personnel to expand and serve clients. Bitcoin isn't that different.

Bitcoin's Proof of Work consensus system needs individuals to validate and create new blocks and check for malicious actors. I'll discuss Bitcoin's blockchain consensus method.

Proof of Work helps Bitcoin reach network consensus. The network is checked and safeguarded by CPU, GPU, or ASIC Bitcoin-mining machines (Application-Specific Integrated Circuit).

Every 10 minutes, miners are rewarded in Bitcoin for securing and verifying the network. It's unlikely you'll finish the block. Miners build pools to increase their chances of winning by combining their processing power.

In the early days of Bitcoin, individual mining systems were more popular due to high maintenance costs and larger earnings prospects. Over time, people created larger and larger Bitcoin mining facilities that required a lot of space and sophisticated cooling systems to keep machines from overheating.

Proof of Work is a vital part of the Bitcoin network, as network security requires the processing power of devices purchased with fiat currency. Miners must invest in mining facilities, which creates a new business branch, mining facilities ownership. Bitcoin mining is a topic for a future article.

More mining, less reward

Bitcoin is usually scarce.

Why is it rare? It all comes down to 21,000,000 Bitcoins.

Were all Bitcoins mined? Nope. Bitcoin's supply grows until it hits 21 million coins. Initially, 50BTC each block was mined, and each block took 10 minutes. Around 2140, the last Bitcoin will be mined.

But 50BTC every 10 minutes does not give me the year 2140. Indeed careful reader. So important is Bitcoin's halving process.

What is halving?

The block reward is halved every 210,000 blocks, which takes around 4 years. The initial payout was 50BTC per block and has been decreased to 25BTC after 210,000 blocks. First halving occurred on November 28, 2012, when 10,500,000 BTC (50%) had been mined. As of April 2022, the block reward is 6.25BTC and will be lowered to 3.125BTC by 19 March 2024.

The halving method is tied to Bitcoin's hashrate. Here's what "hashrate" means.

What if we increased the number of miners and hashrate they provide to produce a block every 10 minutes? Wouldn't we manufacture blocks faster?

Every 10 minutes, blocks are generated with little asymmetry. Due to the built-in adaptive difficulty algorithm, the overall hashrate does not affect block production time. With increased hashrate, it's harder to construct a block. We can estimate when the next halving will occur because 10 minutes per block is fixed.

Building with nodes and blocks

For someone new to crypto, the unusual terms and words may be overwhelming. You'll also find everyday words that are easy to guess or have a vague idea of what they mean, how they work, and what they do. Consider blockchain technology.

Nodes and blocks: Think about that for a moment. What is your first idea?

The blockchain is a chain of validated blocks added to the main chain. What's a "block"? What's inside?

The block is another page in the blockchain book that has been filled with transaction information and accepted by the majority.

We won't go into detail about what each block includes and how it's built, as long as you understand its purpose.

What about nodes?

Nodes, along with miners, verify the blockchain's state independently. But why?

To create a full blockchain node, you must download the whole Bitcoin blockchain and check every transaction against Bitcoin's consensus criteria.

What's Bitcoin's size? 

In April 2022, the Bitcoin blockchain was 389.72GB.

Bitcoin's blockchain has miners and node runners.

Let's revisit the US gold rush. Miners mine gold with their own power (physical and monetary resources) and are rewarded with gold (Bitcoin). All become richer with more gold, and so does the country.

Nodes are like sheriffs, ensuring everything is done according to consensus rules and that there are no rogue miners or network users.

Lost and held bitcoin

Does the Bitcoin exchange price match each coin's price? How many coins remain after 21,000,000? 21 million or less?

Common reason suggests a 21 million-coin supply.

What if I lost 1BTC from a cold wallet?

What if I saved 1000BTC on paper in 2010 and it was damaged?

What if I mined Bitcoin in 2010 and lost the keys?

Satoshi Nakamoto's coins? Since then, those coins haven't moved.

How many BTC are truly in circulation?

Many people are trying to answer this question, and you may discover a variety of studies and individual research on the topic. Be cautious of the findings because they can't be evaluated and the statistics are hazy guesses.

On the other hand, we have long-term investors who won't sell their Bitcoin or will sell little amounts to cover mining or living needs.

The price of Bitcoin is determined by supply and demand on exchanges using liquid BTC. How many BTC are left after subtracting lost and non-custodial BTC? 

We have significantly less Bitcoin in circulation than you think, thus the price may not reflect demand if we knew the exact quantity of coins available.

True HODLers and diamond-hand investors won't sell you their coins, no matter the market.

What's UTXO?

Unspent (U) Transaction (TX) Output (O)

Imagine taking a $100 bill to a store. After choosing a drink and munchies, you walk to the checkout to pay. The cashier takes your $100 bill and gives you $25.50 in change. It's in your wallet.

Is it simply 100$? No way.

The $25.50 in your wallet is unrelated to the $100 bill you used. Your wallet's $25.50 is just bills and coins. Your wallet may contain these coins and bills:

2x 10$ 1x 10$

1x 5$ or 3x 5$

1x 0.50$ 2x 0.25$

Any combination of coins and bills can equal $25.50. You don't care, and I'd wager you've never ever considered it.

That is UTXO. Now, I'll detail the Bitcoin blockchain and how UTXO works, as it's crucial to know what coins you have in your (hopefully) cold wallet.

You purchased 1BTC. Is it all? No. UTXOs equal 1BTC. Then send BTC to a cold wallet. Say you pay 0.001BTC and send 0.999BTC to your cold wallet. Is it the 1BTC you got before? Well, yes and no. The UTXOs are the same or comparable as before, but the blockchain address has changed. It's like if you handed someone a wallet, they removed the coins needed for a network charge, then returned the rest of the coins and notes.

UTXO is a simple concept, but it's crucial to grasp how it works to comprehend dangers like dust attacks and how coins may be tracked.

Lightning Network: fast cash

You've probably heard of "Layer 2 blockchain" projects.

What does it mean?

Layer 2 on a blockchain is an additional layer that increases the speed and quantity of transactions per minute and reduces transaction fees.

Imagine going to an obsolete bank to transfer money to another account and having to pay a charge and wait. You can transfer funds via your bank account or a mobile app without paying a fee, or the fee is low, and the cash appear nearly quickly. Layer 1 and 2 payment systems are different.

Layer 1 is not obsolete; it merely has more essential things to focus on, including providing the blockchain with new, validated blocks, whereas Layer 2 solutions strive to offer Layer 1 with previously processed and verified transactions. The primary blockchain, Bitcoin, will only receive the wallets' final state. All channel transactions until shutting and balancing are irrelevant to the main chain.

Layer 2 and the Lightning Network's goal are now clear. Most Layer 2 solutions on multiple blockchains are created as blockchains, however Lightning Network is not. Remember the following remark, as it best describes Lightning.

Lightning Network connects public and private Bitcoin wallets.

Opening a private channel with another wallet notifies just two parties. The creation and opening of a public channel tells the network that anyone can use it.

Why create a public Lightning Network channel?

Every transaction through your channel generates fees.

Money, if you don't know.

See who benefits when in doubt.

Anonymity, huh?

Bitcoin anonymity? Bitcoin's anonymity was utilized to launder money.

Well… You've heard similar stories. When you ask why or how it permits people to remain anonymous, the conversation ends as if it were just a story someone heard.

Bitcoin isn't private. Pseudonymous.

What if someone tracks your transactions and discovers your wallet address? Where is your anonymity then?

Bitcoin is like bulletproof glass storage; you can't take or change the money. If you dig and analyze the data, you can see what's inside.

Every online action leaves a trace, and traces may be tracked. People often forget this guideline.

A tool like that can help you observe what the major players, or whales, are doing with their coins when the market is uncertain. Many people spend time analyzing on-chain data. Worth it?

Ask yourself a question. What are the big players' options?  Do you think they're letting you see their wallets for a small on-chain data fee?

Instead of short-term behaviors, focus on long-term trends.

More wallet transactions leave traces. Having nothing to conceal isn't a defect. Can it lead to regulating Bitcoin so every transaction is tracked like in banks today?

But wait. How can criminals pay out Bitcoin? They're doing it, aren't they?

Mixers can anonymize your coins, letting you to utilize them freely. This is not a guide on how to make your coins anonymous; it could do more harm than good if you don't know what you're doing.

Remember, being anonymous attracts greater attention.

Bitcoin isn't the only cryptocurrency we can use to buy things. Using cryptocurrency appropriately can provide usability and anonymity. Monero (XMR), Zcash (ZEC), and Litecoin (LTC) following the Mimblewimble upgrade are examples.

Summary

Congratulations! You've reached the conclusion of the article and learned about Bitcoin and cryptocurrency. You've entered the future.

You know what Bitcoin is, how its blockchain works, and why it's not anonymous. I bet you can explain Lightning Network and UTXO to your buddies.

Markets rely on knowledge. Prepare yourself for success before taking the first step. Let your expertise be your edge.

This article is a summary of this one.

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.

I recently wrote about getting VCs excited about Marketplace start-ups. SaaS founders became envious!

Understanding how people wire tens of millions is the only Series A hack I recommend.

Few people understand the intellectual process behind investing.

VC is risk management.

Series A-focused VCs must cover two risks.

1. Market risk

You need a large market to cross a threshold beyond which you can build defensibilities. Series A VCs underwrite market risk.

They must see you have reached product-market fit (PMF) in a large total addressable market (TAM).

2. Execution risk

When evaluating your growth engine's blitzscaling ability, execution risk arises.

When investors remove operational uncertainty, they profit.

Series A VCs like businesses with derisked revenue streams. Don't raise unless you have a predictable model, pipeline, and growth.

Please beat these 3 metrics before Series A:

Achieve $1.5m ARR in 12-24 months (Market risk)

Above 100% Net Dollar Retention. (Market danger)

Lead Velocity Rate supporting $10m ARR in 2–4 years (Execution risk)

Hit the 3 and you'll raise $10M in 4 months. Discussing 2/3 may take 6–7 months.

If none, don't bother raising and focus on becoming a capital-efficient business (Topics for other posts).

Let's examine these 3 metrics for the brave ones.

1. Lead Velocity Rate supporting €$10m ARR in 2 to 4 years

Last because it's the least discussed. LVR is the most reliable data when evaluating a growth engine, in my opinion.

SaaS allows you to see the future.

Monthly Sales and Sales Pipelines, two predictive KPIs, have poor data quality. Both are lagging indicators, and minor changes can cause huge modeling differences.

Analysts and Associates will trash your forecasts if they're based only on Monthly Sales and Sales Pipeline.

LVR, defined as month-over-month growth in qualified leads, is rock-solid. There's no lag. You can See The Future if you use Qualified Leads and a consistent formula and process to qualify them.

With this metric in your hand, scaling your company turns into an execution play on which VCs are able to perform calculations risk.

2. Above-100% Net Dollar Retention.

Net Dollar Retention is a better-known SaaS health metric than LVR.

Net Dollar Retention measures a SaaS company's ability to retain and upsell customers. Ask what $1 of net new customer spend will be worth in years n+1, n+2, etc.

Depending on the business model, SaaS businesses can increase their share of customers' wallets by increasing users, selling them more products in SaaS-enabled marketplaces, other add-ons, and renewing them at higher price tiers.

If a SaaS company's annualized Net Dollar Retention is less than 75%, there's a problem with the business.

Slack's ARR chart (below) shows how powerful Net Retention is. Layer chart shows how existing customer revenue grows. Slack's S1 shows 171% Net Dollar Retention for 2017–2019.

Slack S-1

3. $1.5m ARR in the last 12-24 months.

According to Point 9, $0.5m-4m in ARR is needed to raise a $5–12m Series A round.

Target at least what you raised in Pre-Seed/Seed. If you've raised $1.5m since launch, don't raise before $1.5m ARR.

Capital efficiency has returned since Covid19. After raising $2m since inception, it's harder to raise $1m in ARR.

P9's 2016-2021 SaaS Funding Napkin

In summary, less than 1% of companies VCs meet get funded. These metrics can help you win.

If there’s demand for it, I’ll do one on direct-to-consumer.

Cheers!

It's cheaper, faster charging, longer lasting, safer, and better for the environment.

Batteries are the future. Next-gen and planet-saving technology, including solar power and EVs, require batteries. As these smart technologies become more popular, we find that our batteries can't keep up. Lithium-ion batteries are expensive, slow to charge, big, fast to decay, flammable, and not environmentally friendly. MIT just created a new battery that eliminates all of these problems.  So, is this the battery of the future? Or is there a catch?

When I say entirely new, I mean it. This battery employs no currently available materials. Its electrodes are constructed of aluminium and pure sulfur instead of lithium-complicated ion's metals and graphite. Its electrolyte is formed of molten chloro-aluminate salts, not an organic solution with lithium salts like lithium-ion batteries.

How does this change in materials help?

Aluminum, sulfur, and chloro-aluminate salts are abundant, easy to acquire, and cheap. This battery might be six times cheaper than a lithium-ion battery and use less hazardous mining. The world and our wallets will benefit.

But don’t go thinking this means it lacks performance.

This battery charged in under a minute in tests. At 25 degrees Celsius, the battery will charge 25 times slower than at 110 degrees Celsius. This is because the salt, which has a very low melting point, is in an ideal state at 110 degrees and can carry a charge incredibly quickly. Unlike lithium-ion, this battery self-heats when charging and discharging, therefore no external heating is needed.

Anyone who's seen a lithium-ion battery burst might be surprised. Unlike lithium-ion batteries, none of the components in this new battery can catch fire. Thus, high-temperature charging and discharging speeds pose no concern.

These batteries are long-lasting. Lithium-ion batteries don't last long, as any iPhone owner can attest. During charging, metal forms a dendrite on the electrode. This metal spike will keep growing until it reaches the other end of the battery, short-circuiting it. This is why phone batteries only last a few years and why electric car range decreases over time. This new battery's molten salt slows deposition, extending its life. This helps the environment and our wallets.

These batteries are also energy dense. Some lithium-ion batteries have 270 Wh/kg energy density (volume and mass). Aluminum-sulfur batteries could have 1392 Wh/kg, according to calculations. They'd be 5x more energy dense. Tesla's Model 3 battery would weigh 96 kg instead of 480 kg if this battery were used. This would improve the car's efficiency and handling.

These calculations were for batteries without molten salt electrolyte. Because they don't reflect the exact battery chemistry, they aren't a surefire prediction.

This battery seems great. It will take years, maybe decades, before it reaches the market and makes a difference. Right?

Nope. The project's scientists founded Avanti to develop and market this technology.

So we'll soon be driving cheap, durable, eco-friendly, lightweight, and ultra-safe EVs? Nope.

This battery must be kept hot to keep the salt molten; otherwise, it won't work and will expand and contract, causing damage. This issue could be solved by packs that can rapidly pre-heat, but that project is far off.

Rapid and constant charge-discharge cycles make these batteries ideal for solar farms, homes, and EV charging stations. The battery is constantly being charged or discharged, allowing it to self-heat and maintain an ideal temperature.

These batteries aren't as sexy as those making EVs faster, more efficient, and cheaper. Grid batteries are crucial to our net-zero transition because they allow us to use more low-carbon energy. As we move away from fossil fuels, we'll need millions of these batteries, so the fact that they're cheap, safe, long-lasting, and environmentally friendly will be huge. Who knows, maybe EVs will use this technology one day. MIT has created another world-changing technology.