5 VC investment criteria

I reviewed 200 startup business concepts last week. Brainache.

The enterprises sold various goods and services. The concepts were achingly similar: give us money, we'll produce a product, then get more to expand. No different from daily plans and pitches.

Most of those 200 plans sounded plausible. But 10% looked venture-worthy. 90% of startups need alternatives to venture finance.

With the success of VC-backed businesses and the growth of venture funds, a common misperception is that investors would fund any decent company idea. Finding investors that believe in the firm and founders is the key to funding.

Incorrect. Venture capital needs investing in certain enterprises. If your startup doesn't match the model, as most early-stage startups don't, you can revise your business plan or locate another source of capital.

Before spending six months pitching angels and VCs, make sure your startup fits these criteria.

Likely to generate $100 million in sales

First, I check the income predictions in a pitch deck. If it doesn't display $100M, don't bother.

The math doesn't work for venture financing in smaller businesses.

Say a fund invests $1 million in a startup valued at $5 million that is later acquired for $20 million. That's a win everyone should celebrate. Most VCs don't care.

Consider a $100M fund. The fund must reach $360M in 7 years with a 20% return. Only 20-30 investments are possible. 90% of the investments will fail, hence the 23 winners must return $100M-$200M apiece. $15M isn't worth the work.

Angel investors and tiny funds use the same ideas as venture funds, but their smaller scale affects the calculations. If a company can support its growth through exit on less than $2M in angel financing, it must have $25M in revenues before large companies will consider acquiring it.

Aiming for Hypergrowth

A startup's size isn't enough. It must expand fast.

Developing a great business takes time. Complex technology must be constructed and tested, a nationwide expansion must be built, or production procedures must go from lab to pilot to factories. These can be enormous, world-changing corporations, but venture investment is difficult.

The normal 10-year venture fund life. Investments are made during first 3–4 years.. 610 years pass between investment and fund dissolution. Funds need their investments to exit within 5 years, 7 at the most, therefore add a safety margin.

Longer exit times reduce ROI. A 2-fold return in a year is excellent. Loss at 2x in 7 years.

Lastly, VCs must prove success to raise their next capital. The 2nd fund is raised from 1st fund portfolio increases. Third fund is raised using 1st fund's cash return. Fund managers must raise new money quickly to keep their jobs.

Branding or technology that is protected

No big firm will buy a startup at a high price if they can produce a competing product for less. Their development teams, consumer base, and sales and marketing channels are large. Who needs you?

Patents, specialist knowledge, or brand name are the only answers. The acquirer buys this, not the thing.

I've heard of several promising startups. It's not a decent investment if there's no exit strategy.

A company that installs EV charging stations in apartments and shopping areas is an example. It's profitable, repeatable, and big. A terrific company. Not a startup.

This building company's operations aren't secret. No technology to protect, no special information competitors can't figure out, no go-to brand name. Despite the immense possibilities, a large construction company would be better off starting their own.

Most venture businesses build products, not services. Services can be profitable but hard to safeguard.

Probable purchase at high multiple

Once a software business proves its value, acquiring it is easy. Pharma and medtech firms have given up on their own research and instead acquire startups after regulatory permission. Many startups, especially in specialized areas, have this weakness.

That doesn't mean any lucrative $25M-plus business won't be acquired. In many businesses, the venture model requires a high exit premium.

A startup invents a new glue. 3M, BASF, Henkel, and others may buy them. Adding more adhesive to their catalogs won't boost commerce. They won't compete to buy the business. They'll only buy a startup at a profitable price. The acquisition price represents a moderate EBITDA multiple.

The company's $100M revenue presumably yields $10m in profits (assuming they’ve reached profitability at all). A $30M-$50M transaction is likely. Not terrible, but not what venture investors want after investing $25M to create a plant and develop the business.

Private equity buys profitable companies for a moderate profit multiple. It's a good exit for entrepreneurs, but not for investors seeking 10x or more what PE firms pay. If a startup offers private equity as an exit, the conversation is over.

Constructed for purchase

The startup wants a high-multiple exit. Unless the company targets $1B in revenue and does an IPO, exit means acquisition.

If they're constructing the business for acquisition or themselves, founders must decide.

If you want an indefinitely-running business, I applaud you. We need more long-term founders. Most successful organizations are founded around consumer demands, not venture capital's urge to grow fast and exit. Not venture funding.

if you don't match the venture model, what to do

VC funds moonshots. The 10% that succeed are extraordinary. Not every firm is a rocketship, and launching the wrong startup into space, even with money, will explode.

But just because your startup won't make $100M in 5 years doesn't mean it's a bad business. Most successful companies don't follow this model. It's not venture capital-friendly.

Although venture capital gets the most attention due to a few spectacular triumphs (and disasters), it's not the only or even most typical option to fund a firm.

Other ways to support your startup:

• Personal and family resources, such as credit cards, second mortgages, and lines of credit

• bootstrapping off of sales

• government funding and honors

• Private equity & project financing

• collaborating with a big business

• Including a business partner

Before pitching angels and VCs, be sure your startup qualifies. If so, include them in your pitch.

From a simple landing where we tested our MVP to a platform that distributes 20,000 codes per month, we learned a lot.

Starting point

Kwotet was my first startup. Everyone might post book quotes online.

I wanted a change.

Kwotet lacked attention, thus I felt stuck. After experiencing the trials of starting Kwotet, I thought of developing a waitlist service, but I required a strong co-founder.

I knew Dries from school, but we weren't close. He was an entrepreneurial programmer who worked a lot outside school. I needed this.

We brainstormed throughout school hours. We developed features to put us first. We worked until 3 am to launch this product.

Putting in the hours is KEY when building a startup

The instant that we lost our spark

In Belgium, college seniors do their internship in their last semester.

As we both made the decision to pick a quite challenging company, little time was left for Lancero.

Eventually, we lost interest. We lost the spark…

The only logical choice was to find someone with the same spark we started with to acquire Lancero.

And we did @ MicroAcquire.

Sell before your product dies. Make sure to profit from all the gains.

What did we do following the sale?

Not far from selling Lancero I lost my dad. I was about to start a new company. It was focused on positivity. I got none left at the time.

We still didn’t let go of the dream of becoming full-time entrepreneurs. As Dries launched the amazing company Plunk, and I’m still in the discovering stages of my next journey!

Dream!

You’re an entrepreneur if:

• You're imaginative.

• You enjoy disassembling and reassembling things.

• You're adept at making new friends.

• YOU HAVE DREAMS.

You don’t need to believe me if I tell you “everything is possible”… I wouldn't believe it myself if anyone told me this 2 years ago.

Until I started doing, living my dreams.

Unexpected Success

Shopify CEO and co-founder Tobias Lutke. Shopify is worth $48 billion.

World-renowned entrepreneur Tobi

Tobi never expected his first online snowboard business to become a multimillion-dollar software corporation.

Tobi founded Shopify to establish a 20-person company.

The publicly traded corporation employs over 10,000 people.

Here's Tobi Lutke's incredible story.

Elon Musk tweeted his admiration for the Shopify creator.

30-October-2019.

Musk praised Shopify founder Tobi Lutke on Twitter.

Happened:

Explore this programmer's journey.

What difficulties did Tobi experience as a young child?

Germany raised Tobi.

Tobi's parents realized he was smart but had trouble learning as a toddler.

Tobi was learning disabled.

Tobi struggled with school tests.

Tobi's learning impairments were undiagnosed.

Tobi struggled to read as a dyslexic.

Tobi also found school boring.

Germany's curriculum didn't inspire Tobi's curiosity.

“The curriculum in Germany was taught like here are all the solutions you might find useful later in life, spending very little time talking about the problem…If I don’t understand the problem I’m trying to solve, it’s very hard for me to learn about a solution to a problem.”

Studying computer programming

After tenth grade, Tobi decided school wasn't for him and joined a German apprenticeship program.

This curriculum taught Tobi software engineering.

He was an apprentice in a small Siemens subsidiary team.

Tobi worked with rebellious Siemens employees.

Team members impressed Tobi.

Tobi joined the team for this reason.

Tobi was pleased to get paid to write programming all day.

His life could not have been better.

Devoted to snowboarding

Tobi loved snowboarding.

He drove 5 hours to ski at his folks' house.

His friends traveled to the US to snowboard when he was older.

However, the cheap dollar conversion rate led them to Canada.

2000.

Tobi originally decided to snowboard instead than ski.

Snowboarding captivated him in Canada.

On the trip to Canada, Tobi encounters his wife.

Tobi meets his wife Fiona McKean on his first Canadian ski trip.

They maintained in touch after the trip.

Fiona moved to Germany after graduating.

Tobi was a startup coder.

Fiona found work in Germany.

Her work included editing, writing, and academics.

“We lived together for 10 months and then she told me that she need to go back for the master's program.”

With Fiona, Tobi immigrated to Canada.

Fiona invites Tobi.

Tobi agreed to move to Canada.

Programming helped Tobi move in with his girlfriend.

Tobi was an excellent programmer, therefore what he did in Germany could be done anywhere.

He worked remotely for his German employer in Canada.

Tobi struggled with remote work.

Due to poor communication.

No slack, so he used email.

Programmers had trouble emailing.

Tobi's startup was developing a browser.

After the dot-com crash, individuals left that startup.

It ended.

Tobi didn't intend to work for any major corporations.

Tobi left his startup.

He believed he had important skills for any huge corporation.

He refused to join a huge corporation.

Because of Siemens.

Tobi learned to write professional code and about himself while working at Siemens in Germany.

Siemens culture was odd.

Employees were distrustful.

Siemens' rigorous dress code implies that the corporation doesn't trust employees' attire.

It wasn't Tobi's place.

“There was so much bad with it that it just felt wrong…20-year-old Tobi would not have a career there.”

Focused only on snowboarding

Tobi lived in Ottawa with his girlfriend.

Canada is frigid in winter.

Ottawa's winters last.

Almost half a year.

Tobi wanted to do something worthwhile now.

So he snowboarded.

Tobi began snowboarding seriously.

He sought every snowboarding knowledge.

He researched the greatest snowboarding gear first.

He created big spreadsheets for snowboard-making technologies.

Tobi grew interested in selling snowboards while researching.

He intended to sell snowboards online.

He had no choice but to start his own company.

A small local company offered Tobi a job.

Interested.

He must sign papers to join the local company.

He needed a work permit when he signed the documents.

Tobi had no work permit.

He was allowed to stay in Canada while applying for permanent residency.

“I wasn’t illegal in the country, but my state didn’t give me a work permit. I talked to a lawyer and he told me it’s going to take a while until I get a permanent residency.”

Tobi's lawyer told him he cannot get a work visa without permanent residence.

His lawyer said something else intriguing.

Tobis lawyer advised him to start a business.

Tobi declined this local company's job offer because of this.

Tobi considered opening an internet store with his technical skills.

He sold snowboards online.

“I was thinking of setting up an online store software because I figured that would exist and use it as a way to sell snowboards…make money while snowboarding and hopefully have a good life.”

What brought Tobi and his co-founder together, and how did he support Tobi?

Tobi lived with his girlfriend's parents.

In Ottawa, Tobi encounters Scott Lake.

Scott was Tobis girlfriend's family friend and worked for Tobi's future employer.

Scott and Tobi snowboarded.

Tobi pitched Scott his snowboard sales software idea.

Scott liked the idea.

They planned a business together.

“I was looking after the technology and Scott was dealing with the business side…It was Scott who ended up developing relationships with vendors and doing all the business set-up.”

Issues they ran into when attempting to launch their business online

Neither could afford a long-term lease.

That prompted their online business idea.

They would open a store.

Tobi anticipated opening an internet store in a week.

Tobi seeks open-source software.

Most existing software was pricey.

Tobi and Scott couldn't afford pricey software.

“In 2004, I was sitting in front of my computer absolutely stunned realising that we hadn’t figured out how to create software for online stores.”

They required software to:

• to upload snowboard images to the website.

• people to look up the types of snowboards that were offered on the website. There must be a search feature in the software.

• Online users transmit payments, and the merchant must receive them.

• notifying vendors of the recently received order.

No online selling software existed at the time.

Online credit card payments were difficult.

How did they advance the software while keeping expenses down?

Tobi and Scott needed money to start selling snowboards.

Tobi and Scott funded their firm with savings.

“We both put money into the company…I think the capital we had was around CAD 20,000(Canadian Dollars).”

Despite investing their savings.

They minimized costs.

They tried to conserve.

No office rental.

They worked in several coffee shops.

Tobi lived rent-free at his girlfriend's parents.

He installed software in coffee cafes.

How were the software issues handled?

Tobi found no online snowboard sales software.

Two choices remained:

1. Change your mind and try something else.

2. Use his programming expertise to produce something that will aid in the expansion of this company.

Tobi knew he was the sole programmer working on such a project from the start.

“I had this realisation that I’m going to be the only programmer who has ever worked on this, so I don’t have to choose something that lots of people know. I can choose just the best tool for the job…There is been this programming language called Ruby which I just absolutely loved ”

Ruby was open-source and only had Japanese documentation.

Latin is the source code.

Tobi used Ruby twice.

He assumed he could pick the tool this time.

Why not build with Ruby?

How did they find their first time operating a business?

Tobi writes applications in Ruby.

He wrote the initial software version in 2.5 months.

Tobi and Scott founded Snowdevil to sell snowboards.

Tobi coded for 16 hours a day.

His lifestyle was unhealthy.

He enjoyed pizza and coke.

“I would never recommend this to anyone, but at the time there was nothing more interesting to me in the world.”

Their initial purchase and encounter with it

Tobi worked in cafes then.

“I was working in a coffee shop at this time and I remember everything about that day…At some time, while I was writing the software, I had to type the email that the software would send to tell me about the order.”

Tobi recalls everything.

He checked the order on his laptop at the coffee shop.

Pennsylvanian ordered snowboard.

Tobi walked home and called Scott. Tobi told Scott their first order.

They loved the order.

How were people made aware about Snowdevil?

2004 was very different.

Tobi and Scott attempted simple website advertising.

Google AdWords was new.

Ad clicks cost 20 cents.

Online snowboard stores were scarce at the time.

Google ads propelled the snowdevil brand.

Snowdevil prospered.

They swiftly recouped their original investment in the snowboard business because to its high profit margin.

Tobi and Scott struggled with inventories.

“Snowboards had really good profit margins…Our biggest problem was keeping inventory and getting it back…We were out of stock all the time.”

Selling snowboards returned their investment and saved them money.

They did not appoint a business manager.

They accomplished everything alone.

Sales dipped in the spring, but something magical happened.

Spring sales plummeted.

They considered stocking different boards.

They naturally wanted to add boards and grow the business.

However, magic occurred.

Tobi coded and improved software while running Snowdevil.

He modified software constantly. He wanted speedier software.

He experimented to make the software more resilient.

Tobi received emails requesting the Snowdevil license.

They intended to create something similar.

“I didn’t stop programming, I was just like Ok now let me try things, let me make it faster and try different approaches…Increasingly I got people sending me emails and asking me If I would like to licence snowdevil to them. People wanted to start something similar.”

Software or skateboards, your choice

Scott and Tobi had to choose a hobby in 2005.

They might sell alternative boards or use software.

The software was a no-brainer from demand.

Daniel Weinand is invited to join Tobi's business.

Tobis German best friend is Daniel.

Tobi and Scott chose to use the software.

Tobi and Scott kept the software service.

Tobi called Daniel to invite him to Canada to collaborate.

Scott and Tobi had quit snowboarding until then.

How was Shopify launched, and whence did the name come from?

The three chose Shopify.

Named from two words.

First:

• Shop

Final part:

• Simplify

Shopify

Shopify's crew has always had one goal:

• creating software that would make it simple and easy for people to launch online storefronts.

Launched Shopify after raising money for the first time.

Shopify began fundraising in 2005.

First, they borrowed from family and friends.

They needed roughly $200k to run the company efficiently.

$200k was a lot then.

When questioned why they require so much money. Tobi told them to trust him with their goals. The team raised seed money from family and friends.

Shopify.com has a landing page. A demo of their goal was on the landing page.

In 2006, Shopify had about 4,000 emails.

Shopify rented an Ottawa office.

“We sent a blast of emails…Some people signed up just to try it out, which was exciting.”

How things developed after Scott left the company

Shopify co-founder Scott Lake left in 2008.

Scott was CEO.

“He(Scott) realized at some point that where the software industry was going, most of the people who were the CEOs were actually the highly technical person on the founding team.”

Scott leaving the company worried Tobi.

Tobis worried about finding a new CEO.

To Tobi:

A great VC will have the network to identify the perfect CEO for your firm.

Tobi started visiting Silicon Valley to meet with venture capitalists to recruit a CEO.

Initially visiting Silicon Valley

Tobi came to Silicon Valley to start a 20-person company.

This company creates eCommerce store software.

Tobi never wanted a big corporation. He desired a fulfilling existence.

“I stayed in a hostel in the Bay Area. I had one roommate who was also a computer programmer. I bought a bicycle on Craiglist. I was there for a week, but ended up staying two and a half weeks.”

Tobi arrived unprepared.

When venture capitalists asked him business questions.

He answered few queries.

Tobi didn't comprehend VC meetings' terminology.

He wrote the terms down and looked them up.

Some were fascinated after he couldn't answer all these queries.

“I ended up getting the kind of term sheets people dream about…All the offers were conditional on moving our company to Silicon Valley.”

Canada received Tobi.

He wanted to consult his team before deciding. Shopify had five employees at the time.

2008.

A global recession greeted Tobi in Canada. The recession hurt the market.

His term sheets were useless.

The economic downturn in the world provided Shopify with a fantastic opportunity.

The global recession caused significant job losses.

Fired employees had several ideas.

They wanted online stores.

Entrepreneurship was desired. They wanted to quit work.

People took risks and tried new things during the global slump.

Shopify subscribers skyrocketed during the recession.

“In 2009, the company reached neutral cash flow for the first time…We were in a position to think about long-term investments, such as infrastructure projects.”

Then, Tobi Lutke became CEO.

How did Tobi perform as the company's CEO?

“I wasn’t good. My team was very patient with me, but I had a lot to learn…It’s a very subtle job.”

2009–2010.

Tobi limited the company's potential.

He deliberately restrained company growth.

Tobi had one costly problem:

• Whether Shopify is a venture or a lifestyle business.

The company's annual revenue approached $1 million.

Tobi battled with the firm and himself despite good revenue.

His wife was supportive, but the responsibility was crushing him.

“It’s a crushing responsibility…People had families and kids…I just couldn’t believe what was going on…My father-in-law gave me money to cover the payroll and it was his life-saving.”

Throughout this trip, everyone supported Tobi.

They believed it.

$7 million in donations received

Tobi couldn't decide if this was a lifestyle or a business.

Shopify struggled with marketing then.

Later, Tobi tried 5 marketing methods.

He told himself that if any marketing method greatly increased their growth, he would call it a venture, otherwise a lifestyle.

The Shopify crew brainstormed and voted on marketing concepts.

Tested.

“Every single idea worked…We did Adwords, published a book on the concept, sponsored a podcast and all the ones we tracked worked.”

To Silicon Valley once more

Shopify marketing concepts worked once.

Tobi returned to Silicon Valley to pitch investors.

He raised $7 million, valuing Shopify at $25 million.

All investors had board seats.

“I find it very helpful…I always had a fantastic relationship with everyone who’s invested in my company…I told them straight that I am not going to pretend I know things, I want you to help me.”

Tobi developed skills via running Shopify.

Shopify had 20 employees.

Leaving his wife's parents' home

Tobi left his wife's parents in 2014.

Tobi had a child.

Shopify has 80,000 customers and 300 staff in 2013.

Public offering in 2015

Shopify investors went public in 2015.

Shopify powers 4.1 million e-Commerce sites.

Shopify stores are 65% US-based.

It is currently valued at $48 billion.

In 8 months, I doubled my audience with them.

LinkedIn's buzz isn't over.

People dream of social proof every day. They want clients, interesting jobs, and field recognition.

LinkedIn coaches will benefit greatly. Sell learning? Probably. Can you use it?

Consistency has been key in my eight-month study of LinkedIn. However, I'll share seven of my tips. 700 to 4500 people followed me.

1. Communication, communication, communication

LinkedIn is a social network. I like to think of it as a cafe. Here, you can share your thoughts, meet friends, and discuss life and work.

Do not treat LinkedIn as if it were a board for your post-its.

More socializing improves relationships. It's about people, like any network.

Consider interactions. Three main areas:

• Respond to criticism left on your posts.

• Comment on other people's posts

• Start and maintain conversations through direct messages.

Engage people. You spend too much time on Facebook if you only read your wall. Keeping in touch and having meaningful conversations helps build your network.

Every day, start a new conversation to make new friends.

2. Stick with those you admire

Interact thoughtfully.

Choose your contacts. Build your tribe is a term. Respectful networking.

I only had past colleagues, family, and friends in my network at the start of this year. Not business-friendly. Since then, I've sought out people I admire or can learn from.

Finding a few will help you. As they connect you to their networks. Friendships can lead to clients.

Don't underestimate network power. Cafe-style. Meet people at each table. But avoid people who sell SEO, web redesign, VAs, mysterious job opportunities, etc.

3. Share eye-catching infographics

Daily infographics flood LinkedIn. Visuals are popular. Use Canva's free templates if you can't draw them.

Last week's:

It's a fun way to visualize your topic.

You can repost and comment on infographics. Involve your network. I prefer making my own because I build my brand around certain designs.

My friend posted infographics consistently for four months and grew his network to 30,000.

If you start, credit the authors. As you steal someone's work.

4. Invite some friends over.

LinkedIn alone can be lonely. Having a few friends who support your work daily will boost your growth.

I was lucky to be invited to a group of networkers. We share knowledge and advice.

Having a few regulars who can discuss your posts is helpful. It's artificial, but it works and engages others.

Consider who you'd support if they were in your shoes.

You can pay for an engagement group, but you risk supporting unrelated people with rubbish posts.

Help each other out.

5. Don't let your feed or algorithm divert you.

LinkedIn's algorithm is magical.

Which time is best? How fast do you need to comment? Which days are best?

Overemphasize algorithms. Consider the user. No need to worry about the best time.

Remember to spend time on LinkedIn actively. Not passively. That is what Facebook is for.

Surely someone would find a LinkedIn recipe. Don't beat the algorithm yet. Consider your audience.

6. The more personal, the better

Personalization isn't limited to selfies. Share your successes and failures.

The more personality you show, the better.

People relate to others, not theories or quotes. Why should they follow you? Everyone posts the same content?

Consider your friends. What's their appeal?

Because they show their work and identity. It's simple. Medium and Linkedin are your platforms. Find out what works.

You can copy others' hooks and structures. You decide how simple to make it, though.

7. Have fun with those who have various post structures.

I like writing, infographics, videos, and carousels. Because you can:

Repurpose your content!

Out of one blog post I make:

• Newsletter

• Infographics (positive and negative points of view)

• Carousel

• Personal stories

• Listicle

Create less but more variety. Since LinkedIn posts last 24 hours, you can rotate the same topics for weeks without anyone noticing.

Effective!

The final LI snippet to think about

LinkedIn is about consistency. Some say 15 minutes. If you're serious about networking, spend more time there.

The good news is that it is worth it. The bad news is that it takes time.

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.