More on Personal Growth

Leah
3 years ago
The Burnout Recovery Secrets Nobody Is Talking About
What works and what’s just more toxic positivity
Just keep at it; you’ll get it.
I closed the Zoom call and immediately dropped my head. Open tabs included material on inspiration, burnout, and recovery.
I searched everywhere for ways to avoid burnout.
It wasn't that I needed to keep going, change my routine, employ 8D audio playlists, or come up with fresh ideas. I had several ideas and a schedule. I knew what to do.
I wasn't interested. I kept reading, changing my self-care and mental health routines, and writing even though it was tiring.
Since burnout became a psychiatric illness in 2019, thousands have shared their experiences. It's spreading rapidly among writers.
What is the actual key to recovering from burnout?
Every A-list burnout story emphasizes prevention. Other lists provide repackaged self-care tips. More discuss mental health.
It's like the mid-2000s, when pink quotes about bubble baths saturated social media.
The self-care mania cost us all. Self-care is crucial, but utilizing it to address everything didn't work then or now.
How can you recover from burnout?
Time
Are extended breaks actually good for you? Most people need a break every 62 days or so to avoid burnout.
Real-life burnout victims all took breaks. Perhaps not a long hiatus, but breaks nonetheless.
Burnout is slow and gradual. It takes little bits of your motivation and passion at a time. Sometimes it’s so slow that you barely notice or blame it on other things like stress and poor sleep.
Burnout doesn't come overnight; neither will recovery.
I don’t care what anyone else says the cure for burnout is. It has to be time because time is what gave us all burnout in the first place.

John Rampton
3 years ago
Ideas for Samples of Retirement Letters
Ready to quit full-time? No worries.
Baby Boomer retirement has accelerated since COVID-19 began. In 2020, 29 million boomers retire. Over 3 million more than in 2019. 75 million Baby Boomers will retire by 2030.
First, quit your work to enjoy retirement. Leave a professional legacy. Your retirement will start well. It all starts with a retirement letter.
Retirement Letter
Retirement letters are formal resignation letters. Different from other resignation letters, these don't tell your employer you're leaving. Instead, you're quitting.
Since you're not departing over grievances or for a better position or higher income, you may usually terminate the relationship amicably. Consulting opportunities are possible.
Thank your employer for their support and give them transition information.
Resignation letters aren't merely a formality. This method handles wages, insurance, and retirement benefits.
Retirement letters often accompany verbal notices to managers. Schedule a meeting before submitting your retirement letter to discuss your plans. The letter will be stored alongside your start date, salary, and benefits in your employee file.
Retirement is typically well-planned. Employers want 6-12 months' notice.
Summary
Guidelines for Giving Retirement Notice
Components of a Successful Retirement Letter
Template for Retirement Letter
Ideas for Samples of Retirement Letters
First Example of Retirement Letter
Second Example of Retirement Letter
Third Example of Retirement Letter
Fourth Example of Retirement Letter
Fifth Example of Retirement Letter
Sixth Example of Retirement Letter
Seventh Example of Retirement Letter
Eighth Example of Retirement Letter
Ninth Example of Retirement Letter
Tenth Example of Retirement Letter
Frequently Asked Questions
1. What is a letter of retirement?
2. Why should you include a letter of retirement?
3. What information ought to be in your retirement letter?
4. Must I provide notice?
5. What is the ideal retirement age?
Guidelines for Giving Retirement Notice
While starting a new phase, you're also leaving a job you were qualified for. You have years of experience. So, it may not be easy to fill a retirement-related vacancy.
Talk to your boss in person before sending a letter. Notice is always appreciated. Properly announcing your retirement helps you and your organization transition.
How to announce retirement:
Learn about the retirement perks and policies offered by the company. The first step in figuring out whether you're eligible for retirement benefits is to research your company's retirement policy.
Don't depart without providing adequate notice. You should give the business plenty of time to replace you if you want to retire in a few months.
Help the transition by offering aid. You could be a useful resource if your replacement needs training.
Contact the appropriate parties. The original copy should go to your boss. Give a copy to HR because they will manage your 401(k), pension, and health insurance.
Investigate the option of working as a consultant or part-time. If you desire, you can continue doing some limited work for the business.
Be nice to others. Describe your achievements and appreciation. Additionally, express your gratitude for giving you the chance to work with such excellent coworkers.
Make a plan for your future move. Simply updating your employer on your goals will help you maintain a good working relationship.
Use a formal letter or email to formalize your plans. The initial step is to speak with your supervisor and HR in person, but you must also give written notice.
Components of a Successful Retirement Letter
To write a good retirement letter, keep in mind the following:
A formal salutation. Here, the voice should be deliberate, succinct, and authoritative.
Be specific about your intentions. The key idea of your retirement letter is resignation. Your decision to depart at this time should be reflected in your letter. Remember that your intention must be clear-cut.
Your deadline. This information must be in resignation letters. Laws and corporate policies may both stipulate a minimum amount of notice.
A kind voice. Your retirement letter shouldn't contain any resentments, insults, or other unpleasantness. Your letter should be a model of professionalism and grace. A straightforward thank you is a terrific approach to accomplish that.
Your ultimate goal. Chaos may start to happen as soon as you turn in your resignation letter. Your position will need to be filled. Additionally, you will have to perform your obligations up until a successor is found. Your availability during the interim period should be stated in your resignation letter.
Give us a way to reach you. Even if you aren't consulting, your company will probably get in touch with you at some point. They might send you tax documents and details on perks. By giving your contact information, you can make this process easier.
Template for Retirement Letter
Identify
Title you held
Address
Supervisor's name
Supervisor’s position
Company name
HQ address
Date
[SUPERVISOR],
1.
Inform that you're retiring. Include your last day worked.
2.
Employer thanks. Mention what you're thankful for. Describe your accomplishments and successes.
3.
Helping moves things ahead. Plan your retirement. Mention your consultancy interest.
Sincerely,
[Signature]
First and last name
Phone number
Personal Email
Ideas for Samples of Retirement Letters
First Example of Retirement Letter
Martin D. Carey
123 Fleming St
Bloomfield, New Jersey 07003
(555) 555-1234
June 6th, 2022
Willie E. Coyote
President
Acme Co
321 Anvil Ave
Fairfield, New Jersey 07004
Dear Mr. Coyote,
This letter notifies Acme Co. of my retirement on August 31, 2022.
There has been no other organization that has given me that sense of belonging and purpose.
My fifteen years at the helm of the Structural Design Division have given me a strong sense of purpose. I’ve been fortunate to have your support, and I’ll be always grateful for the opportunity you offered me.
I had a difficult time making this decision. As a result of finding a small property in Arizona where we will be able to spend our remaining days together, my wife and I have decided to officially retire.
In spite of my regret at being unable to contribute to the firm we’ve built, I believe it is wise to move on.
My heart will always belong to Acme Co. Thank you for the opportunity and best of luck in the years to come.
Sincerely,
Martin D. Carey
Second Example of Retirement Letter
Gustavo Fring
Los Pollas Hermanos
12000–12100 Coors Rd SW,
Albuquerque, New Mexico 87045
Dear Mr. Fring,
I write this letter to announce my formal retirement from Los Pollas Hermanos as manager, effective October 15.
As an employee at Los Pollas Hermanos, I appreciate all the great opportunities you have given me. It has been a pleasure to work with and learn from my colleagues for the past 10 years, and I am looking forward to my next challenge.
If there is anything I can do to assist during this time, please let me know.
Sincerely,
Linda T. Crespo
Third Example of Retirement Letter
William M. Arviso
4387 Parkview Drive
Tustin, CA 92680
May 2, 2023
Tony Stark
Owner
Stark Industries
200 Industrial Avenue
Long Beach, CA 90803
Dear Tony:
I’m writing to inform you that my final day of work at Stark Industries will be May14, 2023. When that time comes, I intend to retire.
As I embark on this new chapter in my life, I would like to thank you and the entire Stark Industries team for providing me with so many opportunities. You have all been a pleasure to work with and I will miss you all when I retire.
I am glad to assist you with the transition in any way I can to ensure your new hire has a seamless experience. All ongoing projects will be completed until my retirement date, and all key information will be handed over to the team.
Once again, thank you for the opportunity to be part of the Stark Industries team. All the best to you and the team in the days to come.
Please do not hesitate to contact me if you require any additional information. In order to finalize my retirement plans, I’ll meet with HR and can provide any details that may be necessary.
Sincerely,
(Signature)
William M. Arviso
Fourth Example of Retirement Letter
Garcia, Barbara
First Street, 5432
New York City, NY 10001
(1234) (555) 123–1234
1 October 2022
Gunther
Owner
Central Perk
199 Lafayette St.
New York City, NY 10001
Mr. Gunther,
The day has finally arrived. As I never imagined, I will be formally retiring from Central Perk on November 1st, 2022.
Considering how satisfied I am with my current position, this may surprise you. It would be best if I retired now since my health has deteriorated, so I think this is a good time to do so.
There is no doubt that the past two decades have been wonderful. Over the years, I have seen a small coffee shop grow into one of the city’s top destinations.
It will be hard for me to leave this firm without wondering what more success we could have achieved. But I’m confident that you and the rest of the Central Perk team will achieve great things.
My family and I will never forget what you’ve done for us, and I am grateful for the chance you’ve given me. My house is always open to you.
Sincerely Yours
Garcia, Barbara
Fifth Example of Retirement Letter
Pat Williams
618 Spooky Place
Monstropolis, 23221
123–555–0031
pwilliams@email.com
Feb. 16, 2022
Mike Wazowski
Co-CEO
Monters, Inc.
324 Scare Road
Monstropolis
Dear Mr. Wazowski,
As a formal notice of my upcoming retirement, I am submitting this letter. I will be leaving Monters, Inc. on April 13.
These past 10 years as a marketing associate have provided me with many opportunities. Since we started our company a decade ago, we have seen the face of harnessing screams change dramatically into harnessing laughter. During my time working with this dynamic marketing team, I learned a lot about customer behavior and marketing strategies. Working closely with some of our long-standing clients, such as Boo, was a particular pleasure.
I would be happy to assist with the transition following my retirement. It would be my pleasure to assist in the hiring or training of my replacement. In order to spend more time with my family, I will also be able to offer part-time consulting services.
After I retire, I plan to cash out the eight unused vacation days I’ve accumulated and take my pension as a lump sum.
Thank you for the opportunity to work with Monters, Inc. In the years to come, I wish you all the best!
Sincerely,
Paul Williams
Sixth Example of Retirement Letter
Dear Micheal,
As In my tenure at Dunder Mifflin Paper Company, I have given everything I had. It has been an honor to work here. But I have decided to move on to new challenges and retire from my position — mainly bears, beets, and Battlestar Galactia.
I appreciate the opportunity to work here and learn so much. During my time at this company, I will always remember the good times and memories we shared. Wishing you all the best in the future.
Sincerely,
Dwight K. Shrute
Your signature
May 16
Seventh Example of Retirement Letter
Greetings, Bill
I am announcing my retirement from Initech, effective March 15, 2023.
Over the course of my career here, I’ve had the privilege of working with so many talented and inspiring people.
In 1999, when I began working as a customer service representative, we were a small organization located in a remote office park.
The fact that we now occupy a floor of the Main Street office building with over 150 employees continues to amaze me.
I am looking forward to spending more time with family and traveling the country in our RV. Although I will be sad to leave.
Please let me know if there are any extra steps I can take to facilitate this transfer.
Sincerely,
Frankin, RenitaEighth Example of Retirement Letter
Height Example of Retirement Letter
Bruce,
Please accept my resignation from Wayne Enterprises as Marketing Communications Director. My last day will be August 1, 2022.
The decision to retire has been made after much deliberation. Now that I have worked in the field for forty years, I believe it is a good time to begin completing my bucket list.
It was not easy for me to decide to leave the company. Having worked at Wayne Enterprises has been rewarding both professionally and personally. There are still a lot of memories associated with my first day as a college intern.
My intention was not to remain with such an innovative company, as you know. I was able to see the big picture with your help, however. Today, we are a force that is recognized both nationally and internationally.
In addition to your guidance, the bold, visionary leadership of our company contributed to the growth of our company.
My departure from the company coincides with a particularly hectic time. Despite my best efforts, I am unable to postpone my exit.
My position would be well served by an internal solution. I have a more than qualified marketing manager in Caroline Crown. It would be a pleasure to speak with you about this.
In case I can be of assistance during the switchover, please let me know. Contact us at (555)555–5555. As part of my responsibilities, I am responsible for making sure all work is completed to Wayne Enterprise’s stringent requirements. Having the opportunity to work with you has been a pleasure. I wish you continued success with your thriving business.
Sincerely,
Cash, Cole
Marketing/Communications
Ninth Example of Retirement Letter
Norman, Jamie
2366 Hanover Street
Whitestone, NY 11357
555–555–5555
15 October 2022
Mr. Lippman
Head of Pendant Publishing
600 Madison Ave.
New York, New York
Respected Mr. Lippman,
Please accept my resignation effective November 1, 2022.
Over the course of my ten years at Pendant Publishing, I’ve had a great deal of fun and I’m quite grateful for all the assistance I’ve received.
It was a pleasure to wake up and go to work every day because of our outstanding corporate culture and the opportunities for promotion and professional advancement available to me.
While I am excited about retiring, I am going to miss being part of our team. It’s my hope that I’ll be able to maintain the friendships I’ve formed here for a long time to come.
In case I can be of assistance prior to or following my departure, please let me know. If I can assist in any way to ensure a smooth transfer to my successor, I would be delighted to do so.
Sincerely,
Signed (hard copy letter)
Norman, Jamie
Tenth Example of Retirement Letter
17 January 2023
Greg S. Jackson
Cyberdyne Systems
18144 El Camino Real,
Sunnyvale, CA
Respected Mrs. Duncan,
I am writing to inform you that I will be resigning from Cyberdyne Systems as of March 1, 2023. I’m grateful to have had this opportunity, and it was a difficult decision to make.
My development as a programmer and as a more seasoned member of the organization has been greatly assisted by your coaching.
I have been proud of Cyberdyne Systems’ ethics and success throughout my 25 years at the company. Starting as a mailroom clerk and currently serving as head programmer.
The portfolios of our clients have always been handled with the greatest care by my colleagues. It is our employees and services that have made Cyberdyne Systems the success it is today.
During my tenure as head of my division, I’ve increased our overall productivity by 800 percent, and I expect that trend to continue after I retire.
In light of the fact that the process of replacing me may take some time, I would like to offer my assistance in any way I can.
The greatest contender for this job is Troy Ledford, my current assistant.
Also, before I leave, I would be willing to teach any partners how to use the programmer I developed to track and manage the development of Skynet.
Over the next few months, I’ll be enjoying vacations with my wife as well as my granddaughter moving to college.
If Cyberdyne Systems has any openings for consultants, please let me know. It has been a pleasure working with you over the last 25 years. I appreciate your concern and care.
Sincerely,
Greg S, Jackson
Questions and Answers
1. What is a letter of retirement?
Retirement letters tell your supervisor you're retiring. This informs your employer that you're departing, like a letter. A resignation letter also requests retirement benefits.
Supervisors frequently receive retirement letters and verbal resignations. Before submitting your retirement letter, meet to discuss your plans. This letter will be filed with your start date, salary, and benefits.
2. Why should you include a letter of retirement?
Your retirement letter should explain why you're leaving. When you quit, your manager and HR department usually know. Regardless, a retirement letter might help you leave on a positive tone. It ensures they have the necessary papers.
In your retirement letter, you tell the firm your plans so they can find your replacement. You may need to stay in touch with your company after sending your retirement letter until a successor is identified.
3. What information ought to be in your retirement letter?
Format it like an official letter. Include your retirement plans and retirement-specific statistics. Date may be most essential.
In some circumstances, benefits depend on when you resign and retire. A date on the letter helps HR or senior management verify when you gave notice and how long.
In addition to your usual salutation, address your letter to your manager or supervisor.
The letter's body should include your retirement date and transition arrangements. Tell them whether you plan to help with the transition or train a new employee. You may have a three-month time limit.
Tell your employer your job title, how long you've worked there, and your biggest successes. Personalize your letter by expressing gratitude for your career and outlining your retirement intentions. Finally, include your contact info.
4. Must I provide notice?
Two-week notice isn't required. Your company may require it. Some state laws contain exceptions.
Check your contract, company handbook, or HR to determine your retirement notice. Resigning may change the policy.
Regardless of your company's policy, notification is standard. Entry-level or junior jobs can be let go so the corporation can replace them.
Middle managers, high-level personnel, and specialists may take months to replace. Two weeks' notice is a courtesy. Start planning months ahead.
You can finish all jobs at that period. Prepare transition documents for coworkers and your replacement.
5. What is the ideal retirement age?
Depends on finances, state, and retirement plan. The average American retires at 62. The average retirement age is 66, according to Gallup's 2021 Economy and Personal Finance Survey.
Remember:
Before the age of 59 1/2, withdrawals from pre-tax retirement accounts, such as 401(k)s and IRAs, are subject to a penalty.
Benefits from Social Security can be accessed as early as age 62.
Medicare isn't available to you till you're 65,
Depending on the year of your birth, your Full Retirement Age (FRA) will be between 66 and 67 years old.
If you haven't taken them already, your Social Security benefits increase by 8% annually between ages 6 and 77.

Suzie Glassman
3 years ago
How I Stay Fit Despite Eating Fast Food and Drinking Alcohol
Here's me. Perfectionism is unnecessary.
This post isn't for people who gag at the prospect of eating french fries. I've been ridiculed for stating you can lose weight eating carbs and six-pack abs aren't good.
My family eats frozen processed meals and quick food most weeks (sometimes more). Clean eaters may think I'm unqualified to give fitness advice. I get it.
Hear me out, though. I’m a 44-year-old raising two busy kids with a weekly-traveling husband. Tutoring, dance, and guitar classes fill weeknights. I'm also juggling my job and freelancing.
I'm as worried and tired as my clients. I wish I ate only kale smoothies and salads. I can’t. Despite my mistakes, I'm fit. I won't promise you something just because it worked for me. But here’s a look at how I manage.
What I largely get right about eating
I have a flexible diet and track my daily intake. I count protein, fat, and carbs. Only on vacation or exceptional occasions do I not track.
My protein goal is 1 g per lb. I consume a lot of chicken breasts, eggs, turkey, and lean ground beef. I also occasionally drink protein shakes.
I eat 220–240 grams of carbs daily. My carb count depends on training volume and goals. I'm trying to lose weight slowly. If I want to lose weight faster, I cut carbs to 150-180.
My carbs include white rice, Daves Killer Bread, fruit, pasta, and veggies. I don't eat enough vegetables, so I take Athletic Greens. Also, V8.
Fat grams over 50 help me control my hormones. Recently, I've reached 70-80 grams. Cooking with olive oil. I eat daily dark chocolate. Eggs, butter, milk, and cheese contribute to the rest.
Those frozen meals? What can I say? Stouffer’s lasagna is sometimes needed. I order the healthiest fast food I can find (although I can never bring myself to order the salad). That's a chicken sandwich or a kid's hamburger. I rarely order fries. I eat slowly and savor each bite to feel full.
Potato chips and sugary cereals are in the pantry, but I'm not tempted. My kids eat them because I'd rather teach them moderation than total avoidance. If I eat them, I only eat one portion.
If you're not hungry and eating enough protein and fat, you won't want to eat everything in sight.
I drink once or twice a week. As a result, I rarely overdo it.
Food tracking is tedious and frustrating for many. Taking breaks and using estimates when eating out help. Not perfect, but realistic.
I practice a prolonged fast to enhance metabolic adaptability
Metabolic flexibility is the ability to switch between fuel sources (fat and carbs) based on activity intensity and time since eating. At rest or during low to moderate exertion, your body burns fat. Your body burns carbs after eating and during intense exercise.
Our metabolic flexibility can be hampered by lack of exercise, overeating, and stress. Our bodies become lousy fat burners, making weight loss difficult.
Once a week, I skip dinner (usually around 24 hours). Long-term fasting teaches my body to burn fat. It provides me one low-calorie day a week (I break the fast with a normal-sized dinner).
Fasting day helps me maintain my weight on weekends, when I typically overeat and drink.
Try an extended fast slowly. Delay breakfast by two hours. Next week, add two hours, etc. It takes practice to go that long without biting off your arm. I also suggest consulting your doctor.
I stay active.
I've always been active. As a child, I danced many nights a week, was on the high school dance team, and ran marathons in my 20s.
Often, I feel driven by an internal engine. Working from home makes it easy to exercise. If that’s not you, I get it. Everyone can benefit from raising their baseline.
After taking the kids to school, I walk two miles around the neighborhood. When I need to think, I switch off podcasts. First thing in the morning, I go for a walk.
I lift weights Monday, Wednesday, and Friday. 45 minutes is typical. I run 45-90 minutes on Tuesday and Thursday. I'm slow but reliable. On Saturdays and Sundays, I walk and add a short spin class if I'm not too tired.
I almost never forgo sleep.
I rarely stay up past 10 p.m., much to my night-owl husband's dismay. My 7-8-hour nights help me recover from workouts and handle stress. Without it, I'm grumpy.
I suppose sleep duration matters more than bedtime. Some people just can't fall asleep early. Internal clock and genetics determine sleep and wake hours.
Prioritize sleep.
Last thoughts
Fitness and diet advice is often useless. Some of the advice is inaccurate, dangerous, or difficult to follow if you have a life. I want to throw a shoe at my screen when I see headlines promising to speed up my metabolism or help me lose fat.
I studied exercise physiology for years. No shortcuts exist. No medications or cleanses reset metabolism. I play the hand I'm dealt. I realize that just because something works for me, it won't for you.
If I wanted 15% body fat and ripped abs, I'd have to be stricter. I occasionally think I’d like to get there. But then I remember I’m happy with my life. I like fast food and beer. Pizza and margaritas are favorites (not every day).
You can get it mostly right and live a healthy life.
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forkast
3 years ago
Three Arrows Capital collapse sends crypto tremors
Three Arrows Capital's Google search volume rose over 5,000%.
Three Arrows Capital, a Singapore-based cryptocurrency hedge fund, filed for Chapter 15 bankruptcy last Friday to protect its U.S. assets from creditors.
Three Arrows filed for bankruptcy on July 1 in New York.
Three Arrows was ordered liquidated by a British Virgin Islands court last week after defaulting on a $670 million loan from Voyager Digital. Three days later, the Singaporean government reprimanded Three Arrows for spreading misleading information and exceeding asset limits.
Three Arrows' troubles began with Terra's collapse in May, after it bought US$200 million worth of Terra's LUNA tokens in February, co-founder Kyle Davies told the Wall Street Journal. Three Arrows has failed to meet multiple margin calls since then, including from BlockFi and Genesis.
Three Arrows Capital, founded by Kyle Davies and Su Zhu in 2012, manages $10 billion in crypto assets.
Bitcoin's price fell from US$20,600 to below US$19,200 after Three Arrows' bankruptcy petition. According to CoinMarketCap, BTC is now above US$20,000.
What does it mean?
Every action causes an equal and opposite reaction, per Newton's third law. Newtonian physics won't comfort Three Arrows investors, but future investors will thank them for their overconfidence.
Regulators are taking notice of crypto's meteoric rise and subsequent fall. Historically, authorities labeled the industry "high risk" to warn traditional investors against entering it. That attitude is changing. Regulators are moving quickly to regulate crypto to protect investors and prevent broader asset market busts.
The EU has reached a landmark deal that will regulate crypto asset sales and crypto markets across the 27-member bloc. The U.S. is close behind with a similar ruling, and smaller markets are also looking to improve safeguards.
For many, regulation is the only way to ensure the crypto industry survives the current winter.

Vitalik
3 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

Nir Zicherman
3 years ago
The Great Organizational Conundrum
Only two of the following three options can be achieved: consistency, availability, and partition tolerance
Someone told me that growing from 30 to 60 is the biggest adjustment for a team or business.
I remember thinking, That's random. Each company is unique. I've seen teams of all types confront the same issues during development periods. With new enterprises starting every year, we should be better at navigating growing difficulties.
As a team grows, its processes and systems break down, requiring reorganization or declining results. Why always? Why isn't there a perfect scaling model? Why hasn't that been found?
The Three Things Productive Organizations Must Have
Any company should be efficient and productive. Three items are needed:
First, it must verify that no two team members have conflicting information about the roadmap, strategy, or any input that could affect execution. Teamwork is required.
Second, it must ensure that everyone can receive the information they need from everyone else quickly, especially as teams become more specialized (an inevitability in a developing organization). It requires everyone's accessibility.
Third, it must ensure that the organization can operate efficiently even if a piece is unavailable. It's partition-tolerant.
From my experience with the many teams I've been on, invested in, or advised, achieving all three is nearly impossible. Why a perfect organization model cannot exist is clear after analysis.
The CAP Theorem: What is it?
Eric Brewer of Berkeley discovered the CAP Theorem, which argues that a distributed data storage should have three benefits. One can only have two at once.
The three benefits are consistency, availability, and partition tolerance, which implies that even if part of the system is offline, the remainder continues to work.
This notion is usually applied to computer science, but I've realized it's also true for human organizations. In a post-COVID world, many organizations are hiring non-co-located staff as they grow. CAP Theorem is more important than ever. Growing teams sometimes think they can develop ways to bypass this law, dooming themselves to a less-than-optimal team dynamic. They should adopt CAP to maximize productivity.
Path 1: Consistency and availability equal no tolerance for partitions
Let's imagine you want your team to always be in sync (i.e., for someone to be the source of truth for the latest information) and to be able to share information with each other. Only division into domains will do.
Numerous developing organizations do this, especially after the early stage (say, 30 people) when everyone may wear many hats and be aware of all the moving elements. After a certain point, it's tougher to keep generalists aligned than to divide them into specialized tasks.
In a specialized, segmented team, leaders optimize consistency and availability (i.e. every function is up-to-speed on the latest strategy, no one is out of sync, and everyone is able to unblock and inform everyone else).
Partition tolerance suffers. If any component of the organization breaks down (someone goes on vacation, quits, underperforms, or Gmail or Slack goes down), productivity stops. There's no way to give the team stability, availability, and smooth operation during a hiccup.
Path 2: Partition Tolerance and Availability = No Consistency
Some businesses avoid relying too heavily on any one person or sub-team by maximizing availability and partition tolerance (the organization continues to function as a whole even if particular components fail). Only redundancy can do that. Instead of specializing each member, the team spreads expertise so people can work in parallel. I switched from Path 1 to Path 2 because I realized too much reliance on one person is risky.
What happens after redundancy? Unreliable. The more people may run independently and in parallel, the less anyone can be the truth. Lack of alignment or updated information can lead to people executing slightly different strategies. So, resources are squandered on the wrong work.
Path 3: Partition and Consistency "Tolerance" equates to "absence"
The third, least-used path stresses partition tolerance and consistency (meaning answers are always correct and up-to-date). In this organizational style, it's most critical to maintain the system operating and keep everyone aligned. No one is allowed to read anything without an assurance that it's up-to-date (i.e. there’s no availability).
Always short-lived. In my experience, a business that prioritizes quality and scalability over speedy information transmission can get bogged down in heavy processes that hinder production. Large-scale, this is unsustainable.
Accepting CAP
When two puzzle pieces fit, the third won't. I've watched developing teams try to tackle these difficulties, only to find, as their ancestors did, that they can never be entirely solved. Idealized solutions fail in reality, causing lost effort, confusion, and lower production.
As teams develop and change, they should embrace CAP, acknowledge there is a limit to productivity in a scaling business, and choose the best two-out-of-three path.
