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The following demonstrate proofs of various identities and theorems using pictures, inspired from this gallery. Nichomauss' Theorem: the sum of the first cubes can be written as the square of the sum of the first integers, a statement that can be written as . Here, we use the same re-arrangement as the first proof on this page (the sum of first odd integers is a square). Here's another re-arrangement to see this: This also suggests the following alternative proof:
It’s been 28 years since that vengeful hook-handed killer targeted Julie (Jennifer Love Hewitt), Ray (Freddie Prinze Jr.), and their friends—but 29 since the hit-and-run that set their misery in motion. In the new I Know What You Did Last Summer, a direct sequel to 1998’s I Still Know What You Did Last Summer (the one set in the Bahamas), Julie and Ray are back, though the focus is clearly on the next generation of good-looking kids who haven’t yet learned the importance of taking responsibility
In Brief Amazon has revealed that Prime Day 2025 will take place on July 8-11, extending the annual sale event to four days this year. Amazon notes that consumers will be able to shop for “pantry staples and snacks for summer gatherings to home-improvement must-haves, back-to-school supplies, health and personal care finds, family-favorite toys, best-selling books, pet essentials, and even premium and luxury splurges.” For the first time, Amazon is going to introduce “Today’s Big Deals,” whic
VINCENT FEURAY / Contributor / Getty Last year, Google introduced the ability for Gemini in Drive to interact with PDF files, summarizing a long document into digestible paragraphs. Gemini now has some new tricks related to PDFs stored in Drive, going beyond just a basic summary. Google introduced PDF summary cards for Gemini, Thursday. When you open a PDF stored in Drive, Gemini will proactively summarize the file's contents and suggest next actions like "Draft a sample proposal" or "List int
Introduction Popular programming languages provide methods or functions which locate a substring in a given string. In C it is the function strstr , the C++ class std::string has the method find , Python's string has methods pos and index , and so on, so forth. All these APIs were designed for one-shot searches. During past decades several algorithms to solve this problem were designed, an excellent page by Christian Charras and Thierry Lecroq lists most of them (if not all). Basically these al
We're barely into June, but July-like weather has descended on parts of the country, including the Northeast, where 90-plus temperatures are expected. But you've still got to eat -- and cook -- which means hot kitchens that can cause a domino effect, raising the temperature in the entire home. Read more: Best Foods for Staying Hydrated During Hot Weather Using appliances that don't produce as much heat and planning your meals so that cooking doesn't coincide with the hottest part of the day a
Introduction Popular programming languages provide methods or functions which locate a substring in a given string. In C it is the function strstr , the C++ class std::string has the method find , Python's string has methods pos and index , and so on, so forth. All these APIs were designed for one-shot searches. During past decades several algorithms to solve this problem were designed, an excellent page by Christian Charras and Thierry Lecroq lists most of them (if not all). Basically these al
filedb A key-value store inspired by Bitcask. FileDB is a Zig-implementation of Bitcask by Riak1 paper. FileDB stores record metadata in a log-structured hashtable and parallely keeps 1 disk file open for inserting records in append-only mode. On restarts or MAX_FILE_REACHED , the disk file is rotated and all the oldfiles are kept open for reading only . , the disk file is rotated and all the oldfiles are kept open for reading . A compaction process running every config.compactionInterval se
The job hunt isn't what it used to be. Experts share nine tips that can help you stand out to recruiters. Jeffrey Hazelwood/CNET Since getting laid off from a tech marketing role last November, Stephanie Wandell has applied to hundreds of jobs. She's heard back from only a handful of recruiters, without any offers. "I was a little bit naive going into it, thinking I could do what I always do and depend on applying to as many places as I can," said Wandell. "It became pretty clear that this tim
Introduction Popular programming languages provide methods or functions which locate a substring in a given string. In C it is the function strstr , the C++ class std::string has the method find , Python's string has methods pos and index , and so on, so forth. All these APIs were designed for one-shot searches. During past decades several algorithms to solve this problem were designed, an excellent page by Christian Charras and Thierry Lecroq lists most of them (if not all). Basically these al
filedb A key-value store inspired by Bitcask. FileDB is a Zig-implementation of Bitcask by Riak1 paper. FileDB stores record metadata in a log-structured hashtable and parallely keeps 1 disk file open for inserting records in append-only mode. On restarts or MAX_FILE_REACHED , the disk file is rotated and all the oldfiles are kept open for reading only . , the disk file is rotated and all the oldfiles are kept open for reading . A compaction process running every config.compactionInterval se
For the sake of argument, assume that if your job consists of sitting at a computer, reading on a screen, and typing on a keyboard, then your career will go away. There is always room at the top, and there may always be room for humans at the top of any career. Assume (this is a tough ask, I know) that you are NOT one of those people. What is your fallback job? What skills do you have or would like to acquire that might keep you going? Bicycle mechanic? Teach music to children? Woodworking/car
Wikipedia is backing off a plan to test AI article summaries. Earlier this month, the platform announced plans to trial the feature for about 10 percent of mobile web visitors. To say they weren't well-received by editors would be an understatement. The Wikimedia Foundation (WMF) then changed plans and cancelled the test. The AI summaries would have appeared at the top of articles for 10 percent of mobile users. Readers would have had to opt in to see them. The AI-generated summaries only appea
The PCIe 7.0 specification has now been released, while many of us are still waiting for PCIe 6.0 to materialize in consumer products. The PCI Special Interest Group (PCI-SIG) announced on Wednesday that PCIe 7.0 is now available to members of its organization, delivering a theoretical maximum bandwidth speed of 512GB per second in both directions, across a x16 connection. “PCIe technology has served as the high-bandwidth, low-latency IO interconnect of choice for over two decades and we are pl
The job hunt struggle is real. Jeffrey Hazelwood/CNET Since getting laid off from a tech marketing role last November, Stephanie Wandell has applied to hundreds of jobs. She's heard back from only a handful of recruiters, without any offers. "I was a little bit naive going into it, thinking I could do what I always do and depend on applying to as many places as I can," said Wandell. "It became pretty clear that this time is different." Ask anyone who's unemployed and they'll tell you today's
When Apple’s AI summaries for notifications work, they can be pretty useful. However, with Apple Intelligence limited to select devices, most iPhone users still can’t take advantage of it. Now, WhatsApp is testing its own in-app, AI-powered feature that summarizes unread messages in chats, groups, and channels. Here’s how it works. As uncovered by WABetaInfo (via Tecnoblog), WhatsApp’s AI summaries are being tested on Android, and are opt-in through a feature called Private Processing. Once ac
Wikipedia is backing off AI article summaries… for now. Earlier this month, the platform trialed the feature in its mobile app. To say they weren't well-received by editors would be an understatement. The Wikimedia Foundation (WMF) paused the test a day later. The AI summaries appeared at the top of articles for 10 percent of mobile users. Readers had to opt in to see them. The AI-generated summaries only appeared "on a set of articles" for the two-week trial period. Editor comments in the WMF
Generative AI is permeating the Internet, with chatbots and AI summaries popping up faster than we can keep track. Even Wikipedia, the vast repository of knowledge famously maintained by an army of volunteer human editors, is looking to add robots to the mix. The site began testing AI summaries in some articles over the past week, but the project has been frozen after editors voiced their opinions. And that opinion is: "yuck." The seeds of this project were planted at Wikimedia's 2024 conferenc
In Brief Wikipedia has reportedly paused an experiment that used AI to summarize articles on its platform after editors pushed back. Wikipedia announced earlier this month it was going to run the experiment for users who have the Wikipedia browser extension installed and chose to opt in, according to 404 Media. AI-generated summaries appeared at the top of every Wikipedia article with a yellow “unverified” label. Users had to click to expand and read them. Editors almost immediately criticize