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Nick Nolan

Nick Nolan

3 years ago

In five years, starting a business won't be hip.

More on Entrepreneurship/Creators

Navdeep Yadav

Navdeep Yadav

3 years ago

31 startup company models (with examples)

Many people find the internet's various business models bewildering.

This article summarizes 31 startup e-books.

Types of Startup

1. Using the freemium business model (free plus premium),

The freemium business model offers basic software, games, or services for free and charges for enhancements.

Examples include Slack, iCloud, and Google Drive

Provide a rudimentary, free version of your product or service to users.

Graphic Credit: Business Model toolbox

Google Drive and Dropbox offer 15GB and 2GB of free space but charge for more.

Freemium business model details (Click here)

2. The Business Model of Subscription

Subscription business models sell a product or service for recurring monthly or yearly revenue.

Graphic Credit: Business Model toolbox

Examples: Tinder, Netflix, Shopify, etc

It's the next step to Freemium if a customer wants to pay monthly for premium features.

Types of Subscription Business Models

Subscription Business Model (Click here)

3. A market-based business strategy

It's an e-commerce site or app where third-party sellers sell products or services.

Examples are Amazon and Fiverr.

Marketplace Business Model
  • On Amazon's marketplace, a third-party vendor sells a product.

  • Freelancers on Fiverr offer specialized skills like graphic design.

Marketplace's business concept is explained.

4. Business plans using aggregates

In the aggregator business model, the service is branded.

Uber, Airbnb, and other examples

Airbnb Aggregator Business Model

Marketplace and Aggregator business models differ.

Aggregators Vs Market Place

Amazon and Fiverr link merchants and customers and take a 10-20% revenue split.

Uber and Airbnb-style aggregator Join these businesses and provide their products.

5. The pay-as-you-go concept of business

This is a consumption-based pricing system. Cloud companies use it.

Example: Amazon Web Service and Google Cloud Platform (GCP) (AWS)

Pay-as-you-go pricing in AWS

AWS, an Amazon subsidiary, offers over 200 pay-as-you-go cloud services.

“In short, the more you use the more you pay”

Types of Pay-as-you-plan

When it's difficult to divide clients into pricing levels, pay-as-you is employed.

6. The business model known as fee-for-service (FFS)

FFS charges fixed and variable fees for each successful payment.

For instance, PayU, Paypal, and Stripe

Stripe charges 2.9% + 30 per payment.

Fee-for-service (FFS) business model

These firms offer a payment gateway to take consumer payments and deposit them to a business account.

Fintech business model

7. EdTech business strategy

In edtech, you generate money by selling material or teaching as a service.

Most popular revenue model in EdTech

edtech business models

Freemium When course content is free but certification isn't, e.g. Coursera

FREE TRIAL SkillShare offers free trials followed by monthly or annual subscriptions.

Self-serving marketplace approach where you pick what to learn.

Ad-revenue model The company makes money by showing adverts to its huge user base.

Lock-in business strategy

Lock in prevents customers from switching to a competitor's brand or offering.

It uses switching costs or effort to transmit (soft lock-in), improved brand experience, or incentives.

Apple, SAP, and other examples

Graphic Credit: Business Model toolbox

Apple offers an iPhone and then locks you in with extra hardware (Watch, Airpod) and platform services (Apple Store, Apple Music, cloud, etc.).

9. Business Model for API Licensing

APIs let third-party apps communicate with your service.

How do APIs work?

Uber and Airbnb use Google Maps APIs for app navigation.

Examples are Google Map APIs (Map), Sendgrid (Email), and Twilio (SMS).

Types of APIs business model

Business models for APIs

  1. Free: The simplest API-driven business model that enables unrestricted API access for app developers. Google Translate and Facebook are two examples.

  2. Developer Pays: Under this arrangement, service providers such as AWS, Twilio, Github, Stripe, and others must be paid by application developers.

  3. The developer receives payment: These are the compensated content producers or developers who distribute the APIs utilizing their work. For example, Amazon affiliate programs

10. Open-source enterprise

Open-source software can be inspected, modified, and improved by anybody.

For instance, use Firefox, Java, or Android.

Product with Open source business model

Google paid Mozilla $435,702 million to be their primary search engine in 2018.

Open-source software profits in six ways.

  1. Paid assistance The Project Manager can charge for customization because he is quite knowledgeable about the codebase.

  2. A full database solution is available as a Software as a Service (MongoDB Atlas), but there is a fee for the monitoring tool.

  3. Open-core design R studio is a better GUI substitute for open-source applications.

  4. sponsors of GitHub Sponsorships benefit the developers in full.

  5. demands for paid features Earn Money By Developing Open Source Add-Ons for Current Products

Open-source business model

11. The business model for data

If the software or algorithm collects client data to improve or monetize the system.

Open AI GPT3 gets smarter with use.

Graphic Credit: Business Model toolbox

Foursquare allows users to exchange check-in locations.

Later, they compiled large datasets to enable retailers like Starbucks launch new outlets.

12. Business Model Using Blockchain

Blockchain is a distributed ledger technology that allows firms to deploy smart contracts without a central authority.

Examples include Alchemy, Solana, and Ethereum.

blockchain business model

Business models using blockchain

  1. Economy of tokens or utility When a business uses a token business model, it issues some kind of token as one of the ways to compensate token holders or miners. For instance, Solana and Ethereum

  2. Bitcoin Cash P2P Business Model Peer-to-peer (P2P) blockchain technology permits direct communication between end users. as in IPFS

  3. Enterprise Blockchain as a Service (Baas) BaaS focuses on offering ecosystem services similar to those offered by Amazon (AWS) and Microsoft (Azure) in the web 3 sector. Example: Ethereum Blockchain as a Service with Bitcoin (EBaaS).

  4. Blockchain-Based Aggregators With AWS for blockchain, you can use that service by making an API call to your preferred blockchain. As an illustration, Alchemy offers nodes for many blockchains.

13. The free-enterprise model

In the freeterprise business model, free professional accounts are led into the funnel by the free product and later become B2B/enterprise accounts.

For instance, Slack and Zoom

Freeterprise business model

Freeterprise companies flourish through collaboration.

Loom wants you to join your workspace for an enterprise account.

Start with a free professional account to build an enterprise.

14. Business plan for razor blades

It's employed in hardware where one piece is sold at a loss and profits are made through refills or add-ons.

Gillet razor & blades, coffee machine & beans, HP printer & cartridge, etc.

Razor blade/Bait and hook business model

Sony sells the Playstation console at a loss but makes up for it by selling games and charging for online services.

Advantages of the Razor-Razorblade Method

  1. lowers the risk a customer will try a product. enables buyers to test the goods and services without having to pay a high initial investment.

  2. The product's ongoing revenue stream has the potential to generate sales that much outweigh the original investments.

Razor blade business model

15. The business model of direct-to-consumer (D2C)

In D2C, the company sells directly to the end consumer through its website using a third-party logistic partner.

Examples include GymShark and Kylie Cosmetics.

Direct-to-consumer business Model

D2C brands can only expand via websites, marketplaces (Amazon, eBay), etc.

Traditional Retailer vs D2C business model

D2C benefits

  • Lower reliance on middlemen = greater profitability

  • You now have access to more precise demographic and geographic customer data.

  • Additional space for product testing

  • Increased customisation throughout your entire product line-Inventory Less

16. Business model: White Label vs. Private Label

Private label/White label products are made by a contract or third-party manufacturer.

Most amazon electronics are made in china and white-labeled.

Amazon supplements and electronics.

White-label business model

Contract manufacturers handle everything after brands select product quantities on design labels.

17. The franchise model

The franchisee uses the franchisor's trademark, branding, and business strategy (company).

For instance, KFC, Domino's, etc.

Master Franchise business model

Subway, Domino, Burger King, etc. use this business strategy.

Opening your restaurant vs Frenchies

Many people pick a franchise because opening a restaurant is risky.

18. Ad-based business model

Social media and search engine giants exploit search and interest data to deliver adverts.

Google, Meta, TikTok, and Snapchat are some examples.

Ad-based business model

Users don't pay for the service or product given, e.g. Google users don't pay for searches.

In exchange, they collected data and hyper-personalized adverts to maximize revenue.

19. Business plan for octopuses

Each business unit functions separately but is connected to the main body.

Instance: Oyo

OYO’s Octopus business model

OYO is Asia's Airbnb, operating hotels, co-working, co-living, and vacation houses.

20, Transactional business model, number

Sales to customers produce revenue.

E-commerce sites and online purchases employ SSL.

Goli is an ex-GymShark.

Transactional business model

21. The peer-to-peer (P2P) business model

In P2P, two people buy and sell goods and services without a third party or platform.

Consider OLX.

OLX Business Model

22. P2P lending as a manner of operation

In P2P lending, one private individual (P2P Lender) lends/invests or borrows money from another (P2P Borrower).

Instance: Kabbage

P2P Lending as a business model

Social lending lets people lend and borrow money directly from each other without an intermediary financial institution.

23. A business model for brokers

Brokerages charge a commission or fee for their services.

Examples include eBay, Coinbase, and Robinhood.

Brokerage business model

Brokerage businesses are common in Real estate, finance, and online and operate on this model.

Types of brokerage business model
  1. Buy/sell similar models Examples include financial brokers, insurance brokers, and others who match purchase and sell transactions and charge a commission.

  2. These brokers charge an advertiser a fee based on the date, place, size, or type of an advertisement. This is known as the classified-advertiser model. For instance, Craiglist

24. Drop shipping as an industry

Dropshipping allows stores to sell things without holding physical inventories.

Drop shipping Business model

When a customer orders, use a third-party supplier and logistic partners.

Retailer product portfolio and customer experience Fulfiller The consumer places the order.

Dropshipping advantages

  • Less money is needed (Low overhead-No Inventory or warehousing)

  • Simple to start (costs under $100)

  • flexible work environment

  • New product testing is simpler

25. Business Model for Space as a Service

It's centered on a shared economy that lets millennials live or work in communal areas without ownership or lease.

Consider WeWork and Airbnb.

WeWork business model

WeWork helps businesses with real estate, legal compliance, maintenance, and repair.

Space as a Service Business Model

26. The business model for third-party logistics (3PL)

In 3PL, a business outsources product delivery, warehousing, and fulfillment to an external logistics company.

Examples include Ship Bob, Amazon Fulfillment, and more.

Third-Party Logistics (3PL)

3PL partners warehouse, fulfill, and return inbound and outbound items for a charge.

Inbound logistics involves bringing products from suppliers to your warehouse.

Outbound logistics refers to a company's production line, warehouse, and customer.

Inbound and outbound in 3PL

27. The last-mile delivery paradigm as a commercial strategy

Last-mile delivery is the collection of supply chain actions that reach the end client.

Examples include Rappi, Gojek, and Postmates.

gojek business model

Last-mile is tied to on-demand and has a nighttime peak.

28. The use of affiliate marketing

Affiliate marketing involves promoting other companies' products and charging commissions.

Examples include Hubspot, Amazon, and Skillshare.

Affiliate business model

Your favorite youtube channel probably uses these short amazon links to get 5% of sales.

affiliate link from a youtube video.

Affiliate marketing's benefits

  • In exchange for a success fee or commission, it enables numerous independent marketers to promote on its behalf.

  • Ensure system transparency by giving the influencers a specific tracking link and an online dashboard to view their profits.

  • Learn about the newest bargains and have access to promotional materials.

29. The business model for virtual goods

This is an in-app purchase for an intangible product.

Examples include PubG, Roblox, Candy Crush, etc.

virtual goods business model

Consumables are like gaming cash that runs out. Non-consumable products provide a permanent advantage without repeated purchases.

30. Business Models for Cloud Kitchens

Ghost, Dark, Black Box, etc.

Delivery-only restaurant.

These restaurants don't provide dine-in, only delivery.

For instance, NextBite and Faasos

Cloud kitchen business model

31. Crowdsourcing as a Business Model

Crowdsourcing = Using the crowd as a platform's source.

In crowdsourcing, you get support from people around the world without hiring them.

Crowdsourcing Business model

Crowdsourcing sites

  1. Open-Source Software gives access to the software's source code so that developers can edit or enhance it. Examples include Firefox browsers and Linux operating systems.

  2. Crowdfunding The oculus headgear would be an example of crowdfunding in essence, with no expectations.

Alex Mathers

Alex Mathers

2 years ago

400 articles later, nobody bothered to read them.

Writing for readers:

14 years of daily writing.

I post practically everything on social media. I authored hundreds of articles, thousands of tweets, and numerous volumes to almost no one.

Tens of thousands of readers regularly praise me.

I despised writing. I'm stuck now.

I've learned what readers like and what doesn't.

Here are some essential guidelines for writing with impact:

Readers won't understand your work if you can't.

Though obvious, this slipped me up. Share your truths.

Stories engage human brains.

Showing the journey of a person from worm to butterfly inspires the human spirit.

Overthinking hinders powerful writing.

The best ideas come from inner understanding in between thoughts.

Avoid writing to find it. Write.

Writing a masterpiece isn't motivating.

Write for five minutes to simplify. Step-by-step, entertaining, easy steps.

Good writing requires a willingness to make mistakes.

So write loads of garbage that you can edit into a good piece.

Courageous writing.

A courageous story will move readers. Personal experience is best.

Go where few dare.

Templates, outlines, and boundaries help.

Limitations enhance writing.

Excellent writing is straightforward and readable, removing all the unnecessary fat.

Use five words instead of nine.

Use ordinary words instead of uncommon ones.

Readers desire relatability.

Too much perfection will turn it off.

Write to solve an issue if you can't think of anything to write.

Instead, read to inspire. Best authors read.

Every tweet, thread, and novel must have a central idea.

What's its point?

This can make writing confusing.

️ Don't direct your reader.

Readers quit reading. Demonstrate, describe, and relate.

Even if no one responds, have fun. If you hate writing it, the reader will too.

Keagan Stokoe

Keagan Stokoe

3 years ago

Generalists Create Startups; Specialists Scale Them

There’s a funny part of ‘Steve Jobs’ by Walter Isaacson where Jobs says that Bill Gates was more a copier than an innovator:

“Bill is basically unimaginative and has never invented anything, which is why I think he’s more comfortable now in philanthropy than technology. He just shamelessly ripped off other people’s ideas….He’d be a broader guy if he had dropped acid once or gone off to an ashram when he was younger.”

Gates lacked flavor. Nobody ever got excited about a Microsoft launch, despite their good products. Jobs had the world's best product taste. Apple vs. Microsoft.

A CEO's core job functions are all driven by taste: recruiting, vision, and company culture all require good taste. Depending on the type of company you want to build, know where you stand between Microsoft and Apple.

How can you improve your product judgment? How to acquire taste?

Test and refine

Product development follows two parallel paths: the ‘customer obsession’ path and the ‘taste and iterate’ path.

The customer obsession path involves solving customer problems. Lean Startup frameworks show you what to build at each step.

Taste-and-iterate doesn't involve the customer. You iterate internally and rely on product leaders' taste and judgment.

Creative Selection by Ken Kocienda explains this method. In Creative Selection, demos are iterated and presented to product leaders. Your boss presents to their boss, and so on up to Steve Jobs. If you have good product taste, you can be a panelist.

The iPhone follows this path. Before seeing an iPhone, consumers couldn't want one. Customer obsession wouldn't have gotten you far because iPhone buyers didn't know they wanted one.

In The Hard Thing About Hard Things, Ben Horowitz writes:

“It turns out that is exactly what product strategy is all about — figuring out the right product is the innovator’s job, not the customer’s job. The customer only knows what she thinks she wants based on her experience with the current product. The innovator can take into account everything that’s possible, but often must go against what she knows to be true. As a result, innovation requires a combination of knowledge, skill, and courage.“

One path solves a problem the customer knows they have, and the other doesn't. Instead of asking a person what they want, observe them and give them something they didn't know they needed.

It's much harder. Apple is the world's most valuable company because it's more valuable. It changes industries permanently.

If you want to build superior products, use the iPhone of your industry.

How to Improve Your Taste

I. Work for a company that has taste.

People with the best taste in products, markets, and people are rewarded for building great companies. Tasteful people know quality even when they can't describe it. Taste isn't writable. It's feel-based.

Moving into a community that's already doing what you want to do may be the best way to develop entrepreneurial taste. Most company-building knowledge is tacit.

Joining a company you want to emulate allows you to learn its inner workings. It reveals internal patterns intuitively. Many successful founders come from successful companies.

Consumption determines taste. Excellence will refine you. This is why restauranteurs visit the world's best restaurants and serious painters visit Paris or New York. Joining a company with good taste is beneficial.

2. Possess a wide range of interests

“Edwin Land of Polaroid talked about the intersection of the humanities and science. I like that intersection. There’s something magical about that place… The reason Apple resonates with people is that there’s a deep current of humanity in our innovation. I think great artists and great engineers are similar, in that they both have a desire to express themselves.” — Steve Jobs

I recently discovered Edwin Land. Jobs modeled much of his career after Land's. It makes sense that Apple was inspired by Land.

A Triumph of Genius: Edwin Land, Polaroid, and the Kodak Patent War notes:

“Land was introverted in person, but supremely confident when he came to his ideas… Alongside his scientific passions, lay knowledge of art, music, and literature. He was a cultured person growing even more so as he got older, and his interests filtered into the ethos of Polaroid.”

Founders' philosophies shape companies. Jobs and Land were invested. It showed in the products their companies made. Different. His obsession was spreading Microsoft software worldwide. Microsoft's success is why their products are bland and boring.

Experience is important. It's probably why startups are built by generalists and scaled by specialists.

Jobs combined design, typography, storytelling, and product taste at Apple. Some of the best original Mac developers were poets and musicians. Edwin Land liked broad-minded people, according to his biography. Physicist-musicians or physicist-photographers.

Da Vinci was a master of art, engineering, architecture, anatomy, and more. He wrote and drew at the same desk. His genius is remembered centuries after his death. Da Vinci's statue would stand at the intersection of humanities and science.

We find incredibly creative people here. Superhumans. Designers, creators, and world-improvers. These are the people we need to navigate technology and lead world-changing companies. Generalists lead.

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Clive Thompson

Clive Thompson

3 years ago

Small Pieces of Code That Revolutionized the World

Few sentences can have global significance.

Photo by Chris Ried on Unsplash

Ethan Zuckerman invented the pop-up commercial in 1997.

He was working for Tripod.com, an online service that let people make little web pages for free. Tripod offered advertising to make money. Advertisers didn't enjoy seeing their advertising next to filthy content, like a user's anal sex website.

Zuckerman's boss wanted a solution. Wasn't there a way to move the ads away from user-generated content?

When you visited a Tripod page, a pop-up ad page appeared. So, the ad isn't officially tied to any user page. It'd float onscreen.

Here’s the thing, though: Zuckerman’s bit of Javascript, that created the popup ad? It was incredibly short — a single line of code:

window.open('http://tripod.com/navbar.html'
"width=200, height=400, toolbar=no, scrollbars=no, resizable=no, target=_top");

Javascript tells the browser to open a 200-by-400-pixel window on top of any other open web pages, without a scrollbar or toolbar.

Simple yet harmful! Soon, commercial websites mimicked Zuckerman's concept, infesting the Internet with pop-up advertising. In the early 2000s, a coder for a download site told me that most of their revenue came from porn pop-up ads.

Pop-up advertising are everywhere. You despise them. Hopefully, your browser blocks them.

Zuckerman wrote a single line of code that made the world worse.

A photo of the cover of “You Are Not Expected To Understand This”; it is blue and lying on its side, with the spine facing the viewer. The editor’s name, Torie Bosch, is in a green monospaced font; the title is in a white monospaced font

I read Zuckerman's story in How 26 Lines of Code Changed the World. Torie Bosch compiled a humorous anthology of short writings about code that tipped the world.

Most of these samples are quite short. Pop-cultural preconceptions about coding say that important code is vast and expansive. Hollywood depicts programmers as blurs spouting out Niagaras of code. Google's success was formerly attributed to its 2 billion lines of code.

It's usually not true. Google's original breakthrough, the piece of code that propelled Google above its search-engine counterparts, was its PageRank algorithm, which determined a web page's value based on how many other pages connected to it and the quality of those connecting pages. People have written their own Python versions; it's only a few dozen lines.

Google's operations, like any large tech company's, comprise thousands of procedures. So their code base grows. The most impactful code can be brief.

The examples are fascinating and wide-ranging, so read the whole book (or give it to nerds as a present). Charlton McIlwain wrote a chapter on the police beat algorithm developed in the late 1960s to anticipate crime hotspots so law enforcement could dispatch more officers there. It created a racial feedback loop. Since poor Black neighborhoods were already overpoliced compared to white ones, the algorithm directed more policing there, resulting in more arrests, which convinced it to send more police; rinse and repeat.

Kelly Chudler's You Are Not Expected To Understand This depicts the police-beat algorithm.

About 25 lines of code that includes several mathematical formula. Alas, it’s hard to redact it in plain text here, since it uses mathematical notation

Even shorter code changed the world: the tracking pixel.

Lily Hay Newman's chapter on monitoring pixels says you probably interact with this code every day. It's a snippet of HTML that embeds a single tiny pixel in an email. Getting an email with a tracking code spies on me. As follows: My browser requests the single-pixel image as soon as I open the mail. My email sender checks to see if Clives browser has requested that pixel. My email sender can tell when I open it.

Adding a tracking pixel to an email is easy:

<img src="URL LINKING TO THE PIXEL ONLINE" width="0" height="0">

An older example: Ellen R. Stofan and Nick Partridge wrote a chapter on Apollo 11's lunar module bailout code. This bailout code operated on the lunar module's tiny on-board computer and was designed to prioritize: If the computer grew overloaded, it would discard all but the most vital work.

When the lunar module approached the moon, the computer became overloaded. The bailout code shut down anything non-essential to landing the module. It shut down certain lunar module display systems, scaring the astronauts. Module landed safely.

22-line code

POODOO    INHINT
    CA  Q
    TS  ALMCADR

    TC  BANKCALL
    CADR  VAC5STOR  # STORE ERASABLES FOR DEBUGGING PURPOSES.

    INDEX  ALMCADR
    CAF  0
ABORT2    TC  BORTENT

OCT77770  OCT  77770    # DONT MOVE
    CA  V37FLBIT  # IS AVERAGE G ON
    MASK  FLAGWRD7
    CCS  A
    TC  WHIMPER -1  # YES.  DONT DO POODOO.  DO BAILOUT.

    TC  DOWNFLAG
    ADRES  STATEFLG

    TC  DOWNFLAG
    ADRES  REINTFLG

    TC  DOWNFLAG
    ADRES  NODOFLAG

    TC  BANKCALL
    CADR  MR.KLEAN
    TC  WHIMPER

This fun book is worth reading.

I'm a contributor to the New York Times Magazine, Wired, and Mother Jones. I've also written Coders: The Making of a New Tribe and the Remaking of the World and Smarter Than You Think: How Technology is Changing Our Minds. Twitter and Instagram: @pomeranian99; Mastodon: @clive@saturation.social.

Vitalik

Vitalik

4 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

  • x+5
  • x^4
  • x^3+3x^2+3x+1
  • 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

  • A(0)+B(0)=C(0)
  • A(1)+B(1)=C(1)
  • A(2)+B(2)=C(2)
  • A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

  • 12+1=13
  • 10+8=18
  • 15+8=23
  • 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

  • A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
  • B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
  • C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3

Victoria Kurichenko

Victoria Kurichenko

3 years ago

My Blog Is in Google's Top 10—Here's How to Compete

"Competition" is beautiful and hateful.

Some people bury their dreams because they are afraid of competition. Others challenge themselves, shaping our world.

Competition is normal.

It spurs innovation and progress.

I wish more people agreed.

As a marketer, content writer, and solopreneur, my readers often ask:

"I want to create a niche website, but I have no ideas. Everything's done"

"Is a website worthwhile?"

I can't count how many times I said, "Yes, it makes sense, and you can succeed in a competitive market."

I encourage and share examples, but it's not enough to overcome competition anxiety.

I launched an SEO writing website for content creators a year ago, knowing it wouldn't beat Ahrefs, Semrush, Backlinko, etc.

Not needed.

Many of my website's pages rank highly on Google.

Everyone can eat the pie.

In a competitive niche, I took a different approach.

Look farther

When chatting with bloggers that want a website, I discovered something fascinating.

They want to launch a website but have no ideas. As a next step, they start listing the interests they believe they should work on, like wellness, lifestyle, investments, etc. I could keep going.

Too many generalists who claim to know everything confuse many.

Generalists aren't trusted.

We want someone to fix our problems immediately.

I don't think broad-spectrum experts are undervalued. People have many demands that go beyond generalists' work. Narrow-niche experts can help.

I've done SEO for three years. I learned from experts and courses. I couldn't find a comprehensive SEO writing resource.

I read tons of articles before realizing that wasn't it. I took courses that covered SEO basics eventually.

I had a demand for learning SEO writing, but there was no solution on the market. My website fills this micro-niche.

Have you ever had trouble online?

Professional courses too general, boring, etc.?

You've bought off-topic books, right?

You're not alone.

Niche ideas!

Big players often disregard new opportunities. Too small. Individual content creators can succeed here.

In a competitive market:

  • Never choose wide subjects

  • Think about issues you can relate to and have direct experience with.

  • Be a consumer to discover both the positive and negative aspects of a good or service.

  • Merchandise your annoyances.

  • Consider ways to transform your frustrations into opportunities.

The right niche is half-success. Here is what else I did to hit the Google front page with my website.

An innovative method for choosing subjects

Why publish on social media and websites?

Want likes, shares, followers, or fame?

Some people do it for fun. No judgment.

I bet you want more.

You want to make decent money from blogging.

Writing about random topics, even if they are related to your niche, won’t help you attract an audience from organic search. I'm a marketer and writer.

I worked at companies with dead blogs because they posted for themselves, not readers. They did not follow SEO writing rules; that’s why most of their content flopped.

I learned these hard lessons and grew my website from 0 to 3,000+ visitors per month while working on it a few hours a week only. Evidence:

I choose website topics using these criteria:

- Business potential. The information should benefit my audience and generate revenue. There would be no use in having it otherwise.

My topics should help me:

Attract organic search traffic with my "fluff-free" content -> Subscribers > SEO ebook sales.

Simple and effective.

- traffic on search engines. The number of monthly searches reveals how popular my topic is all across the world. If I find that no one is interested in my suggested topic, I don't write a blog article.

- Competition. Every search term is up against rivals. Some are more popular (thus competitive) since more websites target them in organic search. A new website won't score highly for keywords that are too competitive. On the other side, keywords with moderate to light competition can help you rank higher on Google more quickly.

- Search purpose. The "why" underlying users' search requests is revealed. I analyze search intent to understand what users need when they plug various queries in the search bar and what content can perfectly meet their needs.

My specialty website produces money, ranks well, and attracts the target audience because I handpick high-traffic themes.

Following these guidelines, even a new website can stand out.

I wrote a 50-page SEO writing guide where I detailed topic selection and share my front-page Google strategy.

My guide can help you run a successful niche website.

In summary

You're not late to the niche-website party.

The Internet offers many untapped opportunities.

We need new solutions and are willing to listen.

There are unexplored niches in any topic.

Don't fight giants. They have their piece of the pie. They might overlook new opportunities while trying to keep that piece of the pie. You should act now.