5 Bored Apes borrowed to claim $1.1 million in APE tokens
Takeaway
Unknown user took advantage of the ApeCoin airdrop to earn $1.1 million.
He used a flash loan to borrow five BAYC NFTs, claim the airdrop, and repay the NFTs.
Yuga Labs, the creators of BAYC, airdropped ApeCoin (APE) to anyone who owns one of their NFTs yesterday.
For the Bored Ape Yacht Club and Mutant Ape Yacht Club collections, the team allocated 150 million tokens, or 15% of the total ApeCoin supply, worth over $800 million. Each BAYC holder received 10,094 tokens worth $80,000 to $200,000.
But someone managed to claim the airdrop using NFTs they didn't own. They used the airdrop's specific features to carry it out. And it worked, earning them $1.1 million in ApeCoin.
The trick was that the ApeCoin airdrop wasn't based on who owned which Bored Ape at a given time. Instead, anyone with a Bored Ape at the time of the airdrop could claim it. So if you gave someone your Bored Ape and you hadn't claimed your tokens, they could claim them.
The person only needed to get hold of some Bored Apes that hadn't had their tokens claimed to claim the airdrop. They could be returned immediately.
So, what happened?
The person found a vault with five Bored Ape NFTs that hadn't been used to claim the airdrop.
A vault tokenizes an NFT or a group of NFTs. You put a bunch of NFTs in a vault and make a token. This token can then be staked for rewards or sold (representing part of the value of the collection of NFTs). Anyone with enough tokens can exchange them for NFTs.
This vault uses the NFTX protocol. In total, it contained five Bored Apes: #7594, #8214, #9915, #8167, and #4755. Nobody had claimed the airdrop because the NFTs were locked up in the vault and not controlled by anyone.
The person wanted to unlock the NFTs to claim the airdrop but didn't want to buy them outright s o they used a flash loan, a common tool for large DeFi hacks. Flash loans are a low-cost way to borrow large amounts of crypto that are repaid in the same transaction and block (meaning that the funds are never at risk of not being repaid).
With a flash loan of under $300,000 they bought a Bored Ape on NFT marketplace OpenSea. A large amount of the vault's token was then purchased, allowing them to redeem the five NFTs. The NFTs were used to claim the airdrop, before being returned, the tokens sold back, and the loan repaid.
During this process, they claimed 60,564 ApeCoin airdrops. They then sold them on Uniswap for 399 ETH ($1.1 million). Then they returned the Bored Ape NFT used as collateral to the same NFTX vault.
Attack or arbitrage?
However, security firm BlockSecTeam disagreed with many social media commentators. A flaw in the airdrop-claiming mechanism was exploited, it said.
According to BlockSecTeam's analysis, the user took advantage of a "vulnerability" in the airdrop.
"We suspect a hack due to a flaw in the airdrop mechanism. The attacker exploited this vulnerability to profit from the airdrop claim" said BlockSecTeam.
For example, the airdrop could have taken into account how long a person owned the NFT before claiming the reward.
Because Yuga Labs didn't take a snapshot, anyone could buy the NFT in real time and claim it. This is probably why BAYC sales exploded so soon after the airdrop announcement.
More on NFTs & Art

Tora Northman
3 years ago
Pixelmon NFTs are so bad, they are almost good!
Bored Apes prices continue to rise, HAPEBEAST launches, Invisible Friends hype continues to grow. Sadly, not all projects are as successful.
Of course, there are many factors to consider when buying an NFT. Is the project a scam? Will the reveal derail the project? Possibly, but when Pixelmon first teased its launch, it generated a lot of buzz.
With a primary sale mint price of 3 ETH ($8,100 USD), it started as an expensive project, with plenty of fans willing to invest in what was sold as a game. After it was revealed, it fell rapidly.
Why? It was overpromised and under delivered.
According to the project's creator[^1], the funds generated will be used to develop the artwork. "The Pixelmon reveal was wrong. This is what our Pixelmon look like in-game. "Despite the fud, I will not go anywhere," he wrote on Twitter. The goal remains. The funds will still be used to build our game. I will finish this project."
The project raised $70 million USD, but the NFTs buyers received were not the project's original teasers. Some call it "the worst NFT project ever," while others call it a complete scam.
But there's hope for some buyers. Kevin emerged from the ashes as the project was roasted over the fire.
A Minecraft character meets Salad Fingers - that's Kevin. He's a frog-like creature whose reveal was such a terrible NFT that it became part of history – and a meme.
If you're laughing at people paying $8K for a silly pixelated image, you might need to take it back. Precisely because of this, lucky holders who minted Kevin have been able to sell the now-memed NFT for over 8 ETH (around $24,000 USD), with some currently listed for 100 ETH.
Of course, Twitter has been awash in memes mocking those who invested in the project, because what else can you do when so many people lose money?
It's still unclear if the NFT project is a scam, but the team behind it was hired on Upwork. There's still hope for redemption, but Kevin's rise to fame appears to be the only positive outcome so far.
[^1] This is not the first time the creator (A 20-yo New Zealanders) has sought money via an online platform and had people claiming he under-delivered. He raised $74,000 on Kickstarter for a card game called Psycho Chicken. There are hundreds of comments on the Kickstarter project saying they haven't received the product and pleading for a refund or an update.

Steffan Morris Hernandez
2 years ago
10 types of cognitive bias to watch out for in UX research & design
10 biases in 10 visuals
Cognitive biases are crucial for UX research, design, and daily life. Our biases distort reality.
After learning about biases at my UX Research bootcamp, I studied Erika Hall's Just Enough Research and used the Nielsen Norman Group's wealth of information. 10 images show my findings.
1. Bias in sampling
Misselection of target population members causes sampling bias. For example, you are building an app to help people with food intolerances log their meals and are targeting adult males (years 20-30), adult females (ages 20-30), and teenage males and females (ages 15-19) with food intolerances. However, a sample of only adult males and teenage females is biased and unrepresentative.
2. Sponsor Disparity
Sponsor bias occurs when a study's findings favor an organization's goals. Beware if X organization promises to drive you to their HQ, compensate you for your time, provide food, beverages, discounts, and warmth. Participants may endeavor to be neutral, but incentives and prizes may bias their evaluations and responses in favor of X organization.
In Just Enough Research, Erika Hall suggests describing the company's aims without naming it.
Third, False-Consensus Bias
False-consensus bias is when a person thinks others think and act the same way. For instance, if a start-up designs an app without researching end users' needs, it could fail since end users may have different wants. https://www.nngroup.com/videos/false-consensus-effect/
Working directly with the end user and employing many research methodologies to improve validity helps lessen this prejudice. When analyzing data, triangulation can boost believability.
Bias of the interviewer
I struggled with this bias during my UX research bootcamp interviews. Interviewing neutrally takes practice and patience. Avoid leading questions that structure the story since the interviewee must interpret them. Nodding or smiling throughout the interview may subconsciously influence the interviewee's responses.
The Curse of Knowledge
The curse of knowledge occurs when someone expects others understand a subject as well as they do. UX research interviews and surveys should reduce this bias because technical language might confuse participants and harm the research. Interviewing participants as though you are new to the topic may help them expand on their replies without being influenced by the researcher's knowledge.
Confirmation Bias
Most prevalent bias. People highlight evidence that supports their ideas and ignore data that doesn't. The echo chamber of social media creates polarization by promoting similar perspectives.
A researcher with confirmation bias may dismiss data that contradicts their research goals. Thus, the research or product may not serve end users.
Design biases
UX Research design bias pertains to study construction and execution. Design bias occurs when data is excluded or magnified based on human aims, assumptions, and preferences.
The Hawthorne Impact
Remember when you behaved differently while the teacher wasn't looking? When you behaved differently without your parents watching? A UX research study's Hawthorne Effect occurs when people modify their behavior because you're watching. To escape judgment, participants may act and speak differently.
To avoid this, researchers should blend into the background and urge subjects to act alone.
The bias against social desire
People want to belong to escape rejection and hatred. Research interviewees may mislead or slant their answers to avoid embarrassment. Researchers should encourage honesty and confidentiality in studies to address this. Observational research may reduce bias better than interviews because participants behave more organically.
Relative Time Bias
Humans tend to appreciate recent experiences more. Consider school. Say you failed a recent exam but did well in the previous 7 exams. Instead, you may vividly recall the last terrible exam outcome.
If a UX researcher relies their conclusions on the most recent findings instead of all the data and results, recency bias might occur.
I hope you liked learning about UX design, research, and real-world biases.

Alex Carter
3 years ago
Metaverse, Web 3, and NFTs are BS
Most crypto is probably too.
The goals of Web 3 and the metaverse are admirable and attractive. Who doesn't want an internet owned by users? Who wouldn't want a digital realm where anything is possible? A better way to collaborate and visit pals.
Companies pursue profits endlessly. Infinite growth and revenue are expected, and if a corporation needs to sacrifice profits to safeguard users, the CEO, board of directors, and any executives will lose to the system of incentives that (1) retains workers with shares and (2) makes a company answerable to all of its shareholders. Only the government can guarantee user protections, but we know how successful that is. This is nothing new, just a problem with modern capitalism and tech platforms that a user-owned internet might remedy. Moxie, the founder of Signal, has a good articulation of some of these current Web 2 tech platform problems (but I forget the timestamp); thoughts on JRE aside, this episode is worth listening to (it’s about a bunch of other stuff too).
Moxie Marlinspike, founder of Signal, on the Joe Rogan Experience podcast.
Source: https://open.spotify.com/episode/2uVHiMqqJxy8iR2YB63aeP?si=4962b5ecb1854288
Web 3 champions are premature. There was so much spectacular growth during Web 2 that the next wave of founders want to make an even bigger impact, while investors old and new want a chance to get a piece of the moonshot action. Worse, crypto enthusiasts believe — and financially need — the fact of its success to be true, whether or not it is.
I’m doubtful that it will play out like current proponents say. Crypto has been the white-hot focus of SV’s best and brightest for a long time yet still struggles to come up any mainstream use case other than ‘buy, HODL, and believe’: a store of value for your financial goals and wishes. Some kind of the metaverse is likely, but will it be decentralized, mostly in VR, or will Meta (previously FB) play a big role? Unlikely.
METAVERSE
The metaverse exists already. Our digital lives span apps, platforms, and games. I can design a 3D house, invite people, use Discord, and hang around in an artificial environment. Millions of gamers do this in Rust, Minecraft, Valheim, and Animal Crossing, among other games. Discord's voice chat and Slack-like servers/channels are the present social anchor, but the interface, integrations, and data portability will improve. Soon you can stream YouTube videos on digital house walls. You can doodle, create art, play Jackbox, and walk through a door to play Apex Legends, Fortnite, etc. Not just gaming. Digital whiteboards and screen sharing enable real-time collaboration. They’ll review code and operate enterprises. Music is played and made. In digital living rooms, they'll watch movies, sports, comedy, and Twitch. They'll tweet, laugh, learn, and shittalk.
The metaverse is the evolution of our digital life at home, the third place. The closest analog would be Discord and the integration of Facebook, Slack, YouTube, etc. into a single, 3D, customizable hangout space.
I'm not certain this experience can be hugely decentralized and smoothly choreographed, managed, and run, or that VR — a luxury, cumbersome, and questionably relevant technology — must be part of it. Eventually, VR will be pragmatic, achievable, and superior to real life in many ways. A total sensory experience like the Matrix or Sword Art Online, where we're physically hooked into the Internet yet in our imaginations we're jumping, flying, and achieving athletic feats we never could in reality; exploring realms far grander than our own (as grand as it is). That VR is different from today's.
Ben Thompson released an episode of Exponent after Facebook changed its name to Meta. Ben was suspicious about many metaverse champion claims, but he made a good analogy between Oculus and the PC. The PC was initially far too pricey for the ordinary family to afford. It began as a business tool. It got so powerful and pervasive that it affected our personal life. Price continues to plummet and so much consumer software was produced that it's impossible to envision life without a home computer (or in our pockets). If Facebook shows product market fit with VR in business, through use cases like remote work and collaboration, maybe VR will become practical in our personal lives at home.
Before PCs, we relied on Blockbuster, the Yellow Pages, cabs to get to the airport, handwritten taxes, landline phones to schedule social events, and other archaic methods. It is impossible for me to conceive what VR, in the form of headsets and hand controllers, stands to give both professional and especially personal digital experiences that is an order of magnitude better than what we have today. Is looking around better than using a mouse to examine a 3D landscape? Do the hand controls make x10 or x100 work or gaming more fun or efficient? Will VR replace scalable Web 2 methods and applications like Web 1 and Web 2 did for analog? I don't know.
My guess is that the metaverse will arrive slowly, initially on displays we presently use, with more app interoperability. I doubt that it will be controlled by the people or by Facebook, a corporation that struggles to properly innovate internally, as practically every large digital company does. Large tech organizations are lousy at hiring product-savvy employees, and if they do, they rarely let them explore new things.
These companies act like business schools when they seek founders' results, with bureaucracy and dependency. Which company launched the last popular consumer software product that wasn't a clone or acquisition? Recent examples are scarce.
Web 3
Investors and entrepreneurs of Web 3 firms are declaring victory: 'Web 3 is here!' Web 3 is the future! Many profitable Web 2 enterprises existed when Web 2 was defined. The word was created to explain user behavior shifts, not a personal pipe dream.
Origins of Web 2: http://www.oreilly.com/pub/a/web2/archive/what-is-web-20.html
One of these Web 3 startups may provide the connecting tissue to link all these experiences or become one of the major new digital locations. Even so, successful players will likely use centralized power arrangements, as Web 2 businesses do now. Some Web 2 startups integrated our digital lives. Rockmelt (2010–2013) was a customizable browser with bespoke connectors to every program a user wanted; imagine seeing Facebook, Twitter, Discord, Netflix, YouTube, etc. all in one location. Failure. Who knows what Opera's doing?
Silicon Valley and tech Twitter in general have a history of jumping on dumb bandwagons that go nowhere. Dot-com crash in 2000? The huge deployment of capital into bad ideas and businesses is well-documented. And live video. It was the future until it became a niche sector for gamers. Live audio will play out a similar reality as CEOs with little comprehension of audio and no awareness of lasting new user behavior deceive each other into making more and bigger investments on fool's gold. Twitter trying to buy Clubhouse for $4B, Spotify buying Greenroom, Facebook exploring live audio and 'Tiktok for audio,' and now Amazon developing a live audio platform. This live audio frenzy won't be worth their time or energy. Blind guides blind. Instead of learning from prior failures like Twitter buying Periscope for $100M pre-launch and pre-product market fit, they're betting on unproven and uncompelling experiences.
NFTs
NFTs are also nonsense. Take Loot, a time-limited bag drop of "things" (text on the blockchain) for a game that didn't exist, bought by rich techies too busy to play video games and foolish enough to think they're getting in early on something with a big reward. What gaming studio is incentivized to use these items? Who's encouraged to join? No one cares besides Loot owners who don't have NFTs. Skill, merit, and effort should be rewarded with rare things for gamers. Even if a small minority of gamers can make a living playing, the average game's major appeal has never been to make actual money - that's a profession.
No game stays popular forever, so how is this objective sustainable? Once popularity and usage drop, exclusive crypto or NFTs will fall. And if NFTs are designed to have cross-game appeal, incentives apart, 30 years from now any new game will need millions of pre-existing objects to build around before they start. It doesn’t work.
Many games already feature item economies based on real in-game scarcity, generally for cosmetic things to avoid pay-to-win, which undermines scaled gaming incentives for huge player bases. Counter-Strike, Rust, etc. may be bought and sold on Steam with real money. Since the 1990s, unofficial cross-game marketplaces have sold in-game objects and currencies. NFTs aren't needed. Making a popular, enjoyable, durable game is already difficult.
With NFTs, certain JPEGs on the internet went from useless to selling for $69 million. Why? Crypto, Web 3, early Internet collectibles. NFTs are digital Beanie Babies (unlike NFTs, Beanie Babies were a popular children's toy; their destinies are the same). NFTs are worthless and scarce. They appeal to crypto enthusiasts seeking for a practical use case to support their theory and boost their own fortune. They also attract to SV insiders desperate not to miss the next big thing, not knowing what it will be. NFTs aren't about paying artists and creators who don't get credit for their work.
South Park's Underpants Gnomes
NFTs are a benign, foolish plan to earn money on par with South Park's underpants gnomes. At worst, they're the world of hucksterism and poor performers. Or those with money and enormous followings who, like everyone, don't completely grasp cryptocurrencies but are motivated by greed and status and believe Gary Vee's claim that CryptoPunks are the next Facebook. Gary's watertight logic: if NFT prices dip, they're on the same path as the most successful corporation in human history; buy the dip! NFTs aren't businesses or museum-worthy art. They're bs.
Gary Vee compares NFTs to Amazon.com. vm.tiktok.com/TTPdA9TyH2
We grew up collecting: Magic: The Gathering (MTG) cards printed in the 90s are now worth over $30,000. Imagine buying a digital Magic card with no underlying foundation. No one plays the game because it doesn't exist. An NFT is a contextless image someone conned you into buying a certificate for, but anyone may copy, paste, and use. Replace MTG with Pokemon for younger readers.
When Gary Vee strongarms 30 tech billionaires and YouTube influencers into buying CryptoPunks, they'll talk about it on Twitch, YouTube, podcasts, Twitter, etc. That will convince average folks that the product has value. These guys are smart and/or rich, so I'll get in early like them. Cryptography is similar. No solid, scaled, mainstream use case exists, and no one knows where it's headed, but since the global crypto financial bubble hasn't burst and many people have made insane fortunes, regular people are putting real money into something that is highly speculative and could be nothing because they want a piece of the action. Who doesn’t want free money? Rich techies and influencers won't be affected; normal folks will.
Imagine removing every $1 invested in Bitcoin instantly. What would happen? How far would Bitcoin fall? Over 90%, maybe even 95%, and Bitcoin would be dead. Bitcoin as an investment is the only scalable widespread use case: it's confidence that a better use case will arise and that being early pays handsomely. It's like pouring a trillion dollars into a company with no business strategy or users and a CEO who makes vague future references.
New tech and efforts may provoke a 'get off my lawn' mentality as you approach 40, but I've always prided myself on having a decent bullshit detector, and it's flying off the handle at this foolishness. If we can accomplish a functional, responsible, equitable, and ethical user-owned internet, I'm for it.
Postscript:
I wanted to summarize my opinions because I've been angry about this for a while but just sporadically tweeted about it. A friend handed me a Dan Olson YouTube video just before publication. He's more knowledgeable, articulate, and convincing about crypto. It's worth seeing:
This post is a summary. See the original one here.
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Sam Hickmann
3 years ago
What is headline inflation?
Headline inflation is the raw Consumer price index (CPI) reported monthly by the Bureau of labour statistics (BLS). CPI measures inflation by calculating the cost of a fixed basket of goods. The CPI uses a base year to index the current year's prices.
Explaining Inflation
As it includes all aspects of an economy that experience inflation, headline inflation is not adjusted to remove volatile figures. Headline inflation is often linked to cost-of-living changes, which is useful for consumers.
The headline figure doesn't account for seasonality or volatile food and energy prices, which are removed from the core CPI. Headline inflation is usually annualized, so a monthly headline figure of 4% inflation would equal 4% inflation for the year if repeated for 12 months. Top-line inflation is compared year-over-year.
Inflation's downsides
Inflation erodes future dollar values, can stifle economic growth, and can raise interest rates. Core inflation is often considered a better metric than headline inflation. Investors and economists use headline and core results to set growth forecasts and monetary policy.
Core Inflation
Core inflation removes volatile CPI components that can distort the headline number. Food and energy costs are commonly removed. Environmental shifts that affect crop growth can affect food prices outside of the economy. Political dissent can affect energy costs, such as oil production.
From 1957 to 2018, the U.S. averaged 3.64 percent core inflation. In June 1980, the rate reached 13.60%. May 1957 had 0% inflation. The Fed's core inflation target for 2022 is 3%.
Central bank:
A central bank has privileged control over a nation's or group's money and credit. Modern central banks are responsible for monetary policy and bank regulation. Central banks are anti-competitive and non-market-based. Many central banks are not government agencies and are therefore considered politically independent. Even if a central bank isn't government-owned, its privileges are protected by law. A central bank's legal monopoly status gives it the right to issue banknotes and cash. Private commercial banks can only issue demand deposits.
What are living costs?
The cost of living is the amount needed to cover housing, food, taxes, and healthcare in a certain place and time. Cost of living is used to compare the cost of living between cities and is tied to wages. If expenses are higher in a city like New York, salaries must be higher so people can live there.
What's U.S. bureau of labor statistics?
BLS collects and distributes economic and labor market data about the U.S. Its reports include the CPI and PPI, both important inflation measures.

Jayden Levitt
3 years ago
The country of El Salvador's Bitcoin-obsessed president lost $61.6 million.
It’s only a loss if you sell, right?
Nayib Bukele proclaimed himself “the world’s coolest dictator”.
His jokes aren't clear.
El Salvador's 43rd president self-proclaimed “CEO of El Salvador” couldn't be less presidential.
His thin jeans, aviator sunglasses, and baseball caps like a cartel lord.
He's popular, though.
Bukele won 53% of the vote by fighting violent crime and opposition party corruption.
El Salvador's 6.4 million inhabitants are riding the cryptocurrency volatility wave.
They were powerless.
Their autocratic leader, a former Yamaha Motors salesperson and Bitcoin believer, wants to help 70% unbanked locals.
He intended to give the citizens a way to save money and cut the country's $200 million remittance cost.
Transfer and deposit costs.
This makes logical sense when the president’s theatrics don’t blind you.
El Salvador's Bukele revealed plans to make bitcoin legal tender.
Remittances total $5.9 billion (23%) of the country's expenses.
Anything that reduces costs could boost the economy.
The country’s unbanked population is staggering. Here’s the data by % of people who either have a bank account (Blue) or a mobile money account (Black).
According to Bukele, 46% of the population has downloaded the Chivo Bitcoin Wallet.
In 2021, 36% of El Salvadorans had bank accounts.
Large rural countries like Kenya seem to have resolved their unbanked dilemma.
An economy surfaced where village locals would sell, trade and store network minutes and data as a store of value.
Kenyan phone networks realized unbanked people needed a safe way to accumulate wealth and have an emergency fund.
96% of Kenyans utilize M-PESA, which doesn't require a bank account.
The software involves human agents who hang out with cash and a phone.
These people are like ATMs.
You offer them cash to deposit money in your mobile money account or withdraw cash.
In a country with a faulty banking system, cash availability and a safe place to deposit it are important.
William Jack and Tavneet Suri found that M-PESA brought 194,000 Kenyan households out of poverty by making transactions cheaper and creating a safe store of value.
Mobile money, a service that allows monetary value to be stored on a mobile phone and sent to other users via text messages, has been adopted by most Kenyan households. We estimate that access to the Kenyan mobile money system M-PESA increased per capita consumption levels and lifted 194,000 households, or 2% of Kenyan households, out of poverty.
The impacts, which are more pronounced for female-headed households, appear to be driven by changes in financial behaviour — in particular, increased financial resilience and saving. Mobile money has therefore increased the efficiency of the allocation of consumption over time while allowing a more efficient allocation of labour, resulting in a meaningful reduction of poverty in Kenya.
Currently, El Salvador has 2,301 Bitcoin.
At publication, it's worth $44 million. That remains 41% of Bukele's original $105.6 million.
Unknown if the country has sold Bitcoin, but Bukeles keeps purchasing the dip.
It's still falling.
This might be a fantastic move for the impoverished country over the next five years, if they can live economically till Bitcoin's price recovers.
The evidence demonstrates that a store of value pulls individuals out of poverty, but others say Bitcoin is premature.
You may regard it as an aggressive endeavor to front run the next wave of adoption, offering El Salvador a financial upside.

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
