The Bored Ape Yacht Club is one of the most prestigious NFT collections in the world. A collection of 10,000 NFTs, each depicting an ape with different traits and visual attributes, Jimmy Fallon, Steph Curry and Post Malone are among their star-studded owners. Right now the price of entry is 52 ether, or $210,000.

Which is why it's so painful to see that someone accidentally sold their Bored Ape NFT for $3,066.

Unusual trades are often a sign of funny business, as in the case of the person who spent $530 million to buy an NFT from themselves. In Saturday's case, the cause was a simple, devastating "fat-finger error." That's when people make a trade online for the wrong thing, or for the wrong amount. Here the owner, real name Max or username maxnaut, meant to list his Bored Ape for 75 ether, or around $300,000. Instead he accidentally listed it for 0.75. One hundredth the intended price.

It was bought instantaneously. The buyer paid an extra $34,000 to speed up the transaction, ensuring no one could snap it up before them. The Bored Ape was then promptly listed for $248,000. The transaction appears to have been done by a bot, which can be coded to immediately buy NFTs listed below a certain price on behalf of their owners in order to take advantage of these exact situations.

"How'd it happen? A lapse of concentration I guess," Max told me. "I list a lot of items every day and just wasn't paying attention properly. I instantly saw the error as my finger clicked the mouse but a bot sent a transaction with over 8 eth [$34,000] of gas fees so it was instantly sniped before I could click cancel, and just like that, $250k was gone."

"And here within the beauty of the Blockchain you can see that it is both honest and unforgiving," he added.

Fat finger trades happen sporadically in traditional finance -- like the Japanese trader who almost bought 57% of Toyota's stock in 2014 -- but most financial institutions will stop those transactions if alerted quickly enough. Since cryptocurrency and NFTs are designed to be decentralized, you essentially have to rely on the goodwill of the buyer to reverse the transaction.

Fat finger errors in cryptocurrency trades have made many a headline over the past few years. Back in 2019, the company behind Tether, a cryptocurrency pegged to the US dollar, nearly doubled its own coin supply when it accidentally created $5 billion-worth of new coins. In March, BlockFi meant to send 700 Gemini Dollars to a set of customers, worth roughly $1 each, but mistakenly sent out millions of dollars worth of bitcoin instead. Last month a company erroneously paid a $24 million fee on a $100,000 transaction.

Similar incidents are increasingly being seen in NFTs, now that many collections have accumulated in market value over the past year. Last month someone tried selling a CryptoPunk NFT for $19 million, but accidentally listed it for $19,000 instead. Back in August, someone fat finger listed their Bored Ape for $26,000, an error that someone else immediately capitalized on. The original owner offered $50,000 to the buyer to return the Bored Ape -- but instead the opportunistic buyer sold it for the then-market price of $150,000.

"The industry is so new, bad things are going to happen whether it's your fault or the tech," Max said. "Once you no longer have control of the outcome, forget and move on."

The Bored Ape Yacht Club launched back in April 2021, with 10,000 NFTs being sold for 0.08 ether each -- about $190 at the time. While NFTs are often associated with individual digital art pieces, collections like the Bored Ape Yacht Club, which allow owners to flaunt their NFTs by using them as profile pictures on social media, are becoming increasingly prevalent. The Bored Ape Yacht Club has since become the second biggest NFT collection in the world, second only to CryptoPunks, which launched in 2017 and is considered the "original" NFT collection.

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.

WWDC 2022 began early Tuesday morning. WWDC brought a ton of new features (which went for just shy of two hours).

With so many announcements, we thought we'd compile them. And now...

WWDC?

WWDC is Apple's developer conference. This includes iOS, macOS, watchOS, and iPadOS (all of its iPads). It's where Apple announces new features for developers to use. It's also where Apple previews new software.

Virtual WWDC runs June 6-10.  You can rewatch the stream on Apple's website.

WWDC 2022 news:

Completely everything. Really. iOS 16 first.

iOS 16.

iOS 16 is a major iPhone update. iOS 16 adds the ability to customize the Lock Screen's color/theme. And widgets. It also organizes notifications and pairs Lock Screen with Focus themes. Edit or recall recently sent messages, recover recently deleted messages, and mark conversations as unread. Apple gives us yet another reason to stay in its walled garden with iMessage.

New iOS includes family sharing. Parents can set up a child's account with parental controls to restrict apps, movies, books, and music. iOS 16 lets large families and friend pods share iCloud photos. Up to six people can contribute photos to a separate iCloud library.

Live Text is getting creepier. Users can interact with text in any video frame. Touch and hold an image's subject to remove it from its background and place it in apps like messages. Dictation offers a new on-device voice-and-touch experience. Siri can run app shortcuts without setup in iOS 16. Apple also unveiled a new iOS 16 feature to help people break up with abusive partners who track their locations or read their messages. Safety Check.

Apple Pay Later allows iPhone users to buy products and pay for them later. iOS 16 pushes Mail. Users can schedule emails and cancel delivery before it reaches a recipient's inbox (be quick!). Mail now detects if you forgot an attachment, as Gmail has for years. iOS 16's Maps app gets "Multi-Stop Routing," .

Apple News also gets an iOS 16 update. Apple News adds My Sports. With iOS 16, the Apple Watch's Fitness app is also coming to iOS and the iPhone, using motion-sensing tech to track metrics and performance (as long as an athlete is wearing or carrying the device on their person). 

iOS 16 includes accessibility updates like Door Detection.

watchOS9

Many of Apple's software updates are designed to take advantage of the larger screens in recent models, but they also improve health and fitness tracking.

The most obvious reason to upgrade watchOS every year is to get new watch faces from Apple. WatchOS 9 will add four new faces.

Runners' workout metrics improve.
Apple quickly realized that fitness tracking would be the Apple Watch's main feature, even though it's been the killer app for wearables since their debut. For watchOS 9, the Apple Watch will use its accelerometer and gyroscope to track a runner's form, stride length, and ground contact time. It also introduces the ability to specify heart rate zones, distance, and time intervals, with vibrating haptic feedback and voice alerts.

The Apple Watch's Fitness app is coming to iOS and the iPhone, using the smartphone's motion-sensing tech to track metrics and performance (as long as an athlete is wearing or carrying the device on their person).

We'll get sleep tracking, medication reminders, and drug interaction alerts. Your watch can create calendar events. A new Week view shows what meetings or responsibilities stand between you and the weekend.

iPadOS16

WWDC 2022 introduced iPad updates. iPadOS 16 is similar to iOS for the iPhone, but has features for larger screens and tablet accessories. The software update gives it many iPhone-like features.

iPadOS 16's Home app, like iOS 16, will have a new design language. iPad users who want to blame it on the rain finally have a Weather app. iPadOS 16 will have iCloud's Shared Photo Library, Live Text and Visual Look Up upgrades, and FaceTime Handoff, so you can switch between devices during a call.

Apple highlighted iPadOS 16's multitasking at WWDC 2022. iPad's Stage Manager sounds like a community theater app. It's a powerful multitasking tool for tablets and brings them closer to emulating laptops. Apple's iPadOS 16 supports multi-user collaboration. You can share content from Files, Keynote, Numbers, Pages, Notes, Reminders, Safari, and other third-party apps in Apple Messages.

M2-chip

WWDC 2022 revealed Apple's M2 chip. Apple has started the next generation of Apple Silicon for the Mac with M2. Apple says this device improves M1's performance.

M2's second-generation 5nm chip has 25% more transistors than M1's. 100GB/s memory bandwidth (50 per cent more than M1). M2 has 24GB of unified memory, up from 16GB but less than some ultraportable PCs' 32GB. The M2 chip has 10% better multi-core CPU performance than the M2, and it's nearly twice as fast as the latest 10-core PC laptop chip at the same power level (CPU performance is 18 per cent greater than M1).

New MacBooks

Apple introduced the M2-powered MacBook Air. Apple's entry-level laptop has a larger display, a new processor, new colors, and a notch.

M2 also powers the 13-inch MacBook Pro. The 13-inch MacBook Pro has 24GB of unified memory and 50% more memory bandwidth. New MacBook Pro batteries last 20 hours. As I type on the 2021 MacBook Pro, I can only imagine how much power the M2 will add.

macOS 13.0 (or, macOS Ventura)

macOS Ventura will take full advantage of M2 with new features like Stage Manager and Continuity Camera and Handoff for FaceTime. Safari, Mail, Messages, Spotlight, and more get updates in macOS Ventura.

Apple hasn't run out of California landmarks to name its OS after yet. macOS 13 will be called Ventura when it's released in a few months, but it's more than a name change and new wallpapers. 

Stage Manager organizes windows

Stage Manager is a new macOS tool that organizes open windows and applications so they're still visible while focusing on a specific task. The main app sits in the middle of the desktop, while other apps and documents are organized and piled up to the side.

Improved Searching

Spotlight is one of macOS's least appreciated features, but with Ventura, it's becoming even more useful. Live Text lets you extract text from Spotlight results without leaving the window, including images from the photo library and the web.

Mail lets you schedule or unsend emails.

We've all sent an email we regret, whether it contained regrettable words or was sent at the wrong time. In macOS Ventura, Mail users can cancel or reschedule a message after sending it. Mail will now intelligently determine if a person was forgotten from a CC list or if a promised attachment wasn't included. Procrastinators can set a reminder to read a message later.

Safari adds tab sharing and password passkeys

Apple is updating Safari to make it more user-friendly... mostly. Users can share a group of tabs with friends or family, a useful feature when researching a topic with too many tabs. Passkeys will replace passwords in Safari's next version. Instead of entering random gibberish when creating a new account, macOS users can use TouchID to create an on-device passkey. Using an iPhone's camera and a QR system, Passkey syncs and works across all Apple devices and Windows computers.

Continuity adds Facetime device switching and iPhone webcam.

With macOS Ventura, iPhone users can transfer a FaceTime call from their phone to their desktop or laptop using Handoff, or vice versa if they started a call at their desk and need to continue it elsewhere. Apple finally admits its laptop and monitor webcams aren't the best. Continuity makes the iPhone a webcam. Apple demonstrated a feature where the wide-angle lens could provide a live stream of the desk below, while the standard zoom lens could focus on the speaker's face. New iPhone laptop mounts are coming.

System Preferences

System Preferences is Now System Settings and Looks Like iOS
Ventura's System Preferences has been renamed System Settings and is much more similar in appearance to iOS and iPadOS. As the iPhone and iPad are gateway devices into Apple's hardware ecosystem, new Mac users should find it easier to adjust.

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