Improving collaboration with the Six Thinking Hats
Six Thinking Hats was written by Dr. Edward de Bono. "Six Thinking Hats" and parallel thinking allow groups to plan thinking processes in a detailed and cohesive way, improving collaboration.
Fundamental ideas
In order to develop strategies for thinking about specific issues, the method assumes that the human brain thinks in a variety of ways that can be intentionally challenged. De Bono identifies six brain-challenging directions. In each direction, the brain brings certain issues into conscious thought (e.g. gut instinct, pessimistic judgement, neutral facts). Some may find wearing hats unnatural, uncomfortable, or counterproductive.
The example of "mismatch" sensitivity is compelling. In the natural world, something out of the ordinary may be dangerous. This mode causes negative judgment and critical thinking.
Colored hats represent each direction. Putting on a colored hat symbolizes changing direction, either literally or metaphorically. De Bono first used this metaphor in his 1971 book "Lateral Thinking for Management" to describe a brainstorming framework. These metaphors allow more complete and elaborate thought separation. Six thinking hats indicate ideas' problems and solutions.
Similarly, his CoRT Thinking Programme introduced "The Five Stages of Thinking" method in 1973.
| HAT | OVERVIEW | TECHNIQUE |
|---|---|---|
| BLUE | "The Big Picture" & Managing | CAF (Consider All Factors); FIP (First Important Priorities) |
| WHITE | "Facts & Information" | Information |
| RED | "Feelings & Emotions" | Emotions and Ego |
| BLACK | "Negative" | PMI (Plus, Minus, Interesting); Evaluation |
| YELLOW | "Positive" | PMI |
| GREEN | "New Ideas" | Concept Challenge; Yes, No, Po |
Strategies and programs
After identifying the six thinking modes, programs can be created. These are groups of hats that encompass and structure the thinking process. Several of these are included in the materials for franchised six hats training, but they must often be adapted. Programs are often "emergent," meaning the group plans the first few hats and the facilitator decides what to do next.
The group agrees on how to think, then thinks, then evaluates the results and decides what to do next. Individuals or groups can use sequences (and indeed hats). Each hat is typically used for 2 minutes at a time, although an extended white hat session is common at the start of a process to get everyone on the same page. The red hat is recommended to be used for a very short period to get a visceral gut reaction – about 30 seconds, and in practice often takes the form of dot-voting.
| ACTIVITY | HAT SEQUENCE |
|---|---|
| Initial Ideas | Blue, White, Green, Blue |
| Choosing between alternatives | Blue, White, (Green), Yellow, Black, Red, Blue |
| Identifying Solutions | Blue, White, Black, Green, Blue |
| Quick Feedback | Blue, Black, Green, Blue |
| Strategic Planning | Blue, Yellow, Black, White, Blue, Green, Blue |
| Process Improvement | Blue, White, White (Other People's Views), Yellow, Black, Green, Red, Blue |
| Solving Problems | Blue, White, Green, Red, Yellow, Black, Green, Blue |
| Performance Review | Blue, Red, White, Yellow, Black, Green, Blue |
Use
Speedo's swimsuit designers reportedly used the six thinking hats. "They used the "Six Thinking Hats" method to brainstorm, with a green hat for creative ideas and a black one for feasibility.
Typically, a project begins with extensive white hat research. Each hat is used for a few minutes at a time, except the red hat, which is limited to 30 seconds to ensure an instinctive gut reaction, not judgement. According to Malcolm Gladwell's "blink" theory, this pace improves thinking.
De Bono believed that the key to a successful Six Thinking Hats session was focusing the discussion on a particular approach. A meeting may be called to review and solve a problem. The Six Thinking Hats method can be used in sequence to explore the problem, develop a set of solutions, and choose a solution through critical examination.
Everyone may don the Blue hat to discuss the meeting's goals and objectives. The discussion may then shift to Red hat thinking to gather opinions and reactions. This phase may also be used to determine who will be affected by the problem and/or solutions. The discussion may then shift to the (Yellow then) Green hat to generate solutions and ideas. The discussion may move from White hat thinking to Black hat thinking to develop solution set criticisms.
Because everyone is focused on one approach at a time, the group is more collaborative than if one person is reacting emotionally (Red hat), another is trying to be objective (White hat), and another is critical of the points which emerge from the discussion (Black hat). The hats help people approach problems from different angles and highlight problem-solving flaws.
(Edited)
More on Leadership
William Anderson
3 years ago
When My Remote Leadership Skills Took Off
4 Ways To Manage Remote Teams & Employees
The wheels hit the ground as I landed in Rochester.
Our six-person satellite office was now part of my team.
Their manager only reported to me the day before, but I had my ticket booked ahead of time.
I had managed remote employees before but this was different. Engineers dialed into headquarters for every meeting.
So when I learned about the org chart change, I knew a strong first impression would set the tone for everything else.
I was either their boss, or their boss's boss, and I needed them to know I was committed.
Managing a fleet of satellite freelancers or multiple offices requires treating others as more than just a face behind a screen.
You must comprehend each remote team member's perspective and daily interactions.
The good news is that you can start using these techniques right now to better understand and elevate virtual team members.
1. Make Visits To Other Offices
If budgeted, visit and work from offices where teams and employees report to you. Only by living alongside them can one truly comprehend their problems with communication and other aspects of modern life.
2. Have Others Come to You
• Having remote, distributed, or satellite employees and teams visit headquarters every quarter or semi-quarterly allows the main office culture to rub off on them.
When remote team members visit, more people get to meet them, which builds empathy.
If you can't afford to fly everyone, at least bring remote managers or leaders. Hopefully they can resurrect some culture.
3. Weekly Work From Home
No home office policy?
Make one.
WFH is a team-building, problem-solving, and office-viewing opportunity.
For dial-in meetings, I started working from home on occasion.
It also taught me which teams “forget” or “skip” calls.
As a remote team member, you experience all the issues first hand.
This isn't as accurate for understanding teams in other offices, but it can be done at any time.
4. Increase Contact Even If It’s Just To Chat
Don't underestimate office banter.
Sometimes it's about bonding and trust, other times it's about business.
If you get all this information in real-time, please forward it.
Even if nothing critical is happening, call remote team members to check in and chat.
I guarantee that building relationships and rapport will increase both their job satisfaction and yours.
Jano le Roux
3 years ago
The Real Reason Adobe Just Paid $20 billion for Figma
Sketch or Figma?
Designers are pissed.
The beast ate the beauty.
Figma deserves $20B.
Do designers deserve Adobe?
Adobe devours new creative tools and spits them out with a slimy Adobe aftertaste.
Frame.io — $1.3B
Magento — $1.7B
Macromedia — $3.6B
Nothing compares to the risky $20B acquisition.
If they can't be beaten, buy them.
And then make them boring.
Adobe's everywhere.
Like that friend who dabbles in everything creatively, there's not enough time to master one thing.
Figma was Adobe's thigh-mounted battle axe.
a UX design instrument with a sizable free tier.
a UX design tool with a simple and quick user interface.
a tool for fluid collaboration in user experience design.
a web-based UX design tool that functions well.
a UX design tool with a singular goal of perfection.
UX design software that replaced Adobe XD.
Adobe XD could do many of Figma's things, but it didn't focus on the details. This is a major issue when working with detail-oriented professionals.
UX designers.
Design enthusiasts first used Figma. More professionals used it. Institutions taught it. Finally, major brands adopted Figma.
Adobe hated that.
Adobe dispatched a team of lawyers to resolve the Figma issue, as big companies do. Figma didn’t bite for months.
Oh no.
Figma resisted.
Figma helped designers leave Adobe. Figma couldn't replace Photoshop, but most designers used it to remove backgrounds.
Online background removal tools improved.
The Figma problem grew into a thorn, a knife, and a battle ax in Adobe's soft inner thigh.
Figma appeared to be going public. Adobe couldn’t allow that. It bought Figma for $20B during the IPO drought.
Adobe has a new issue—investors are upset.
The actual cause of investors' ire toward Adobe
Spoiler: The math just doesn’t add up.
According to Adobe's press release, Figma's annual recurring revenue (ARR) is $400M and growing rapidly.
The $20B valuation requires a 50X revenue multiple, which is unheard of.
Venture capitalists typically use:
10% to 29% growth per year: ARR multiplied by 1 to 5
30% to 99% growth per year: ARR multiplied by 6 to 10
100% to 400% growth per year: ARR multiplied by 10 to 20
Showing an investor a 50x multiple is like telling friends you saw a UFO. They'll think you're crazy.
Adobe's stock fell immediately after the acquisition because it didn't make sense to a number-cruncher.
Designers started a Tweet storm in the digital town hall where VCs and designers often meet.
Adobe acquired Workfront for $1.5 billion at the end of 2020. This purchase made sense for investors.
Many investors missed the fact that Adobe is acquiring Figma not only for its ARR but also for its brilliant collaboration tech.
Adobe could use Figmas web app technology to make more products web-based to compete with Canva.
Figma's high-profile clients could switch to Adobe's enterprise software.
However, questions arise:
Will Adobe make Figma boring?
Will Adobe tone down Figma to boost XD?
Would you ditch Adobe and Figma for Sketch?
Will Lockett
3 years ago
Tesla recently disclosed its greatest secret.
The VP has revealed a secret that should frighten the rest of the EV world.
Tesla led the EV revolution. Elon Musk's invention offers a viable alternative to gas-guzzlers. Tesla has lost ground in recent years. VW, BMW, Mercedes, and Ford offer EVs with similar ranges, charging speeds, performance, and cost. Tesla's next-generation 4680 battery pack, Roadster, Cybertruck, and Semi were all delayed. CATL offers superior batteries than the 4680. Martin Viecha, Tesla's Vice President, recently told Business Insider something that startled the EV world and will establish Tesla as the EV king.
Viecha mentioned that Tesla's production costs have dropped 57% since 2017. This isn't due to cheaper batteries or devices like Model 3. No, this is due to amazing factory efficiency gains.
Musk wasn't crazy to want a nearly 100% automated production line, and Tesla's strategy of sticking with one model and improving it has paid off. Others change models every several years. This implies they must spend on new R&D, set up factories, and modernize service and parts systems. All of this costs a ton of money and prevents them from refining production to cut expenses.
Meanwhile, Tesla updates its vehicles progressively. Everything from the backseats to the screen has been enhanced in a 2022 Model 3. Tesla can refine, standardize, and cheaply produce every part without changing the production line.
In 2017, Tesla's automobile production averaged $84,000. In 2022, it'll be $36,000.
Mr. Viecha also claimed that new factories in Shanghai and Berlin will be significantly cheaper to operate once fully operating.
Tesla's hand is visible. Tesla selling $36,000 cars for $60,000 This barely beats the competition. Model Y long-range costs just over $60,000. Tesla makes $24,000+ every sale, giving it a 40% profit margin, one of the best in the auto business.
VW I.D4 costs about the same but makes no profit. Tesla's rivals face similar challenges. Their EVs make little or no profit.
Tesla costs the same as other EVs, but they're in a different league.
But don't forget that the battery pack accounts for 40% of an EV's cost. Tesla may soon fully utilize its 4680 battery pack.
The 4680 battery pack has larger cells and a unique internal design. This means fewer cells are needed for a car, making it cheaper to assemble and produce (per kWh). Energy density and charge speeds increase slightly.
Tesla underestimated the difficulty of making this revolutionary new cell. Each time they try to scale up production, quality drops and rejected cells rise.
Tesla recently installed this battery pack in Model Ys and is scaling production. If they succeed, Tesla battery prices will plummet.
Tesla's Model Ys 2170 battery costs $11,000. The same size pack with 4680 cells costs $3,400 less. Once scaled, it could be $5,500 (50%) less. The 4680 battery pack could reduce Tesla production costs by 20%.
With these cost savings, Tesla could sell Model Ys for $40,000 while still making a profit. They could offer a $25,000 car.
Even with new battery technology, it seems like other manufacturers will struggle to make EVs profitable.
Teslas cost about the same as competitors, so don't be fooled. Behind the scenes, they're still years ahead, and the 4680 battery pack and new factories will only increase that lead. Musk faces a first. He could sell Teslas at current prices and make billions while other manufacturers struggle. Or, he could massively undercut everyone and crush the competition once and for all. Tesla and Elon win.
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caroline sinders
3 years ago
Holographic concerts are the AI of the Future.
A few days ago, I was discussing dall-e with two art and tech pals. One artist acquaintance said she knew a frightened illustrator. Would the ability to create anything with a click derail her career? The artist feared this. My curator friend smiled and said this has always been a dread among artists. When the camera was invented, didn't painters say this? Even in the Instagram era, painting exists.
When art and technology collide, there's room for innovation, experimentation, and fear — especially if the technology replicates or replaces art making. What is art's future with dall-e? How does technology affect music, beyond visual art? Recently, I saw "ABBA Voyage," a holographic ABBA concert in London.
"Abba voyage?" my phone asked in early March. A Gen X friend I met through a fashion blogging ring texted me.
"What's abba Voyage?" I asked while opening my front door with keys and coffee.
We're going! Marti, visiting London, took me to a show.
"Absolutely no ABBA songs here." I responded.
My parents didn't play ABBA much, so I don't know much about them. Dad liked Jimi Hendrix, Cream, Deep Purple, and New Orleans jazz. Marti told me ABBA Voyage was a holographic ABBA show with a live band.
The show was fun, extraordinary fun. Nearly everyone on the dance floor wore wigs, ankle-breaking platforms, sequins, and bellbottoms. I saw some millennials and Zoomers among the boomers.
I was intoxicated by the experience.
Automatons date back to the 18th-century mechanical turk. The mechanical turk was a chess automaton operated by a person. The mechanical turk seemed to perform like a human without human intervention, but it required a human in the loop to work properly.
Humans have used non-humans in entertainment for centuries, such as puppets, shadow play, and smoke and mirrors. A show can have animatronic, technological, and non-technological elements, and a live show can blur real and illusion. From medieval puppet shows to mechanical turks to AI filters, bots, and holograms, entertainment has evolved over time.
I'm not a hologram skeptic, but I'm skeptical of technology, especially since I work with it. I love live performances, I love hearing singers breathe, forget lines, and make jokes. Live shows are my favorite because I love watching performers make mistakes or interact with the audience. ABBA Voyage was different.
Marti and I traveled to Manchester after ABBA Voyage to see Liam Gallagher. Similar but different vibe. Similar in that thousands dressed up for the show. ABBA's energy was dizzying. 90s chic replaced sequins in the crowd. Doc Martens, nylon jackets, bucket hats, shaggy hair. The Charlatans and Liam Gallagher opened and closed, respectively. Fireworks. Incredible. People went crazy. Yelling exhausted my voice.
This week in music featured AI-enabled holograms and a decades-old rocker. Both are warm and gooey in our memories.
After seeing both, I'm wondering if we need AI hologram shows. Why? Is it good?
Like everything tech-related, my answer is "maybe." Because context and performance matter. Liam Gallagher and ABBA both had great, different shows.
For a hologram to work, it must be impossible and big. It must be big, showy, and improbable to justify a hologram. It must feel...expensive, like a stadium pop show. According to a quick search, ABBA broke up on bad terms. Reuniting is unlikely. This is also why Prince or Tupac hologram shows work. We can only engage with their legacy through covers or...holograms.
I drove around listening to the radio a few weeks ago. "Dreaming of You" by Selena played. Selena's music defined my childhood. I sang along and turned up the volume (or as loud as my husband would allow me while driving on the highway).
I discovered Selena's music six months after her death, so I never saw her perform live. My babysitter Melissa played me her album after I moved to Houston. Melissa took me to see the Selena movie five times when it came out. I quickly wore out my VHS copy. I constantly sang "Bibi Bibi Bom Bom" and "Como la Flor." I love Selena. A Selena hologram? Yes, probably.
Instagram advertised a cellist's Arthur Russell tribute show. Russell is another deceased artist I love. I almost walked down the aisle to "This is How We Walk on the Moon," but our cellist couldn't find it. Instead, I walked to Magnetic Fields' "The Book of Love." I "discovered" Russell after a friend introduced me to his music a few years ago.
I use these as analogies for the Liam Gallagher and ABBA concerts.
You have no idea how much I'd pay to see a hologram of Selena's 1995 Houston Livestock Show and Rodeo concert. Arthur Russell's hologram is unnecessary. Russell's work was intimate and performance-based. We can't separate his life from his legacy; popular audiences overlooked his genius. He died of AIDS broke. Like Selena, he died prematurely. Given his music and history, another performer would be a better choice than a hologram. He's no Selena. Selena could have rivaled Beyonce.
Pop shows' size works for holograms. Along with ABBA holograms, there was an anime movie and a light show that would put Tron to shame. ABBA created a tourable stadium show. The event was lavish, expensive, and well-planned. Pop, unlike rock, isn't gritty. Liam Gallagher hologram? No longer impossible, it wouldn't work. He's touring. I'm not sure if a rockstar alone should be rendered as a hologram; it was the show that made ABBA a hologram.
Holograms, like AI, are part of the future of entertainment, but not all of it. Because only modern interpretations of Arthur Russell's work reveal his legacy. That's his legacy.
Large-scale arena performers may use holograms in the future, but the experience must be impossible. A teacher once said that the only way to convey emotion in opera is through song, and I feel the same way about holograms, AR, VR, and mixed reality. A story's impossibility must make sense, like in opera. Impossibility and bombastic performance must be present for an immersive element to "work." ABBA was an impossible and improbable experience, which made it magical. It helped the holographic show work.
Marti told me about ABBA Voyage. She said it was a great concert. Marti has worked in music since the 1990s. She's a music expert; she's seen many shows.
Ai isn't a god or sentient, and the ABBA holograms aren't real. The renderings were glassy-eyed, flat, and robotic, like the Polar Express or the Jaws shark. Even today, the uncanny valley is insurmountable. We know it's not real because it's not about reality. It was about a suspended moment and performance feelings.
I knew this was impossible, an 'unreal' experience, but the emotions I felt were real, like watching a movie or tv show. Perhaps this is one of the better uses of AI, like CGI and special effects, like the beauty of entertainment- we were enraptured and entertained for hours. I've been playing ABBA since then.
Sam Warain
3 years ago
Sam Altman, CEO of Open AI, foresees the next trillion-dollar AI company
“I think if I had time to do something else, I would be so excited to go after this company right now.”
Sam Altman, CEO of Open AI, recently discussed AI's present and future.
Open AI is important. They're creating the cyberpunk and sci-fi worlds.
They use the most advanced algorithms and data sets.
GPT-3...sound familiar? Open AI built most copyrighting software. Peppertype, Jasper AI, Rytr. If you've used any, you'll be shocked by the quality.
Open AI isn't only GPT-3. They created DallE-2 and Whisper (a speech recognition software released last week).
What will they do next? What's the next great chance?
Sam Altman, CEO of Open AI, recently gave a lecture about the next trillion-dollar AI opportunity.
Who is the organization behind Open AI?
Open AI first. If you know, skip it.
Open AI is one of the earliest private AI startups. Elon Musk, Greg Brockman, and Rebekah Mercer established OpenAI in December 2015.
OpenAI has helped its citizens and AI since its birth.
They have scary-good algorithms.
Their GPT-3 natural language processing program is excellent.
The algorithm's exponential growth is astounding. GPT-2 came out in November 2019. May 2020 brought GPT-3.
Massive computation and datasets improved the technique in just a year. New York Times said GPT-3 could write like a human.
Same for Dall-E. Dall-E 2 was announced in April 2022. Dall-E 2 won a Colorado art contest.
Open AI's algorithms challenge jobs we thought required human innovation.
So what does Sam Altman think?
The Present Situation and AI's Limitations
During the interview, Sam states that we are still at the tip of the iceberg.
So I think so far, we’ve been in the realm where you can do an incredible copywriting business or you can do an education service or whatever. But I don’t think we’ve yet seen the people go after the trillion dollar take on Google.
He's right that AI can't generate net new human knowledge. It can train and synthesize vast amounts of knowledge, but it simply reproduces human work.
“It’s not going to cure cancer. It’s not going to add to the sum total of human scientific knowledge.”
But the key word is yet.
And that is what I think will turn out to be wrong that most surprises the current experts in the field.
Reinforcing his point that massive innovations are yet to come.
But where?
The Next $1 Trillion AI Company
Sam predicts a bio or genomic breakthrough.
There’s been some promising work in genomics, but stuff on a bench top hasn’t really impacted it. I think that’s going to change. And I think this is one of these areas where there will be these new $100 billion to $1 trillion companies started, and those areas are rare.
Avoid human trials since they take time. Bio-materials or simulators are suitable beginning points.
AI may have a breakthrough. DeepMind, an OpenAI competitor, has developed AlphaFold to predict protein 3D structures.
It could change how we see proteins and their function. AlphaFold could provide fresh understanding into how proteins work and diseases originate by revealing their structure. This could lead to Alzheimer's and cancer treatments. AlphaFold could speed up medication development by revealing how proteins interact with medicines.
Deep Mind offered 200 million protein structures for scientists to download (including sustainability, food insecurity, and neglected diseases).
Being in AI for 4+ years, I'm amazed at the progress. We're past the hype cycle, as evidenced by the collapse of AI startups like C3 AI, and have entered a productive phase.
We'll see innovative enterprises that could replace Google and other trillion-dollar companies.
What happens after AI adoption is scary and unpredictable. How will AGI (Artificial General Intelligence) affect us? Highly autonomous systems that exceed humans at valuable work (Open AI)
My guess is that the things that we’ll have to figure out are how we think about fairly distributing wealth, access to AGI systems, which will be the commodity of the realm, and governance, how we collectively decide what they can do, what they don’t do, things like that. And I think figuring out the answer to those questions is going to just be huge. — Sam Altman CEO
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