![]() If you're interested I recommend you to read the paper of W. This was just a very short introduction in the Mathematics of the Rubik's Cube, you can find much more interesting material about this topic. If you don't understand what the letters mean in the previous explication please read the notation. Degree of permutations - this is a number which shows how many times has to execute a permutation to get back to the original position of the cube.Commutative - Commutativity is not a necessary condition of the permutation group but notice that FB = BF but FR != RF.Inverse element - every permutation has an inverse permutation: ex.Neutral element - there is a permutation which doesn't rearrange the set: ex.Associative - the permutations in the row can be grouped together: ex.Let's see the properties of this mathematic structure. The group of all permutations of a set is the symmetric group. In math a permutation group is a group whose elements are permutations of ordered list, and whose group operation is the permutations which rearrange the set in a certain way. ![]() Now lets see why the Rubik's Cube is a permutation group. On the picture below you can see how a D rotation rearranges the elements of our list. Repeating and combining these permutations we can define new permutations, which rearrange the list in an other way. We can rotate the 6 faces of the cube so we can define 6 basic operations or permutations which rearrange the ordered list in a certain way. It has 6 different colors and each color is repeated exactly 9 times, so the cube can be considered as an ordered list which has 54 elements with numbers between 1 and 6, each number meaning a color being repeated 9 times. Mathematically the Rubik's Cube is a permutation group. We've written a separate article about the God's Number. Then this number was gradually decreasing. ![]() The first estimation of the God's Number was 52 moves in July 1981. This means that we have found the solution of the "Superflip" scramble in 20 moves: This number was calculated thanks to Google who donated 36 CPU-years of idle computer time, solving every position of the Rubik's Cube in less than 21 moves. Since July of 2010 we know that this number is 20, so every position can be solved in twenty moves or less, considering one move a 90 o or 180 o twist of any face. God's Number shows the smallest number of moves needed to solve the 3x3x3 Rubik's Cube from any random starting position. To illustrate this number, if we had as many 6 centimeter large Rubik's Cubes as there are permutations, we could cover the surface of the Earth 300 times.įor a 7x7x7 Cube this number is about 1.95 * 10 160 roughly 19.5 duoquinquagintillion. The classic 3x3x3 has much more possible patterns: approximately 43 quintillion (exactly 43 quintillion, 252 quadrillion, 3 trillion, 274 billion, 489 million, 856 thousand). This strategy wouldn't work neither for a 2x2x2 cube. Even a 2x2x2 Pocket Cube has 3.674.160 possible permutations. There are so many possible states of a 3x3x3 Rubik's Cube you could never finish solving it just turning the faces randomly. "I just rotate the faces randomly until it is solved" - he said. He was sure about that he is able to solve it because it seemed so easy for him. Once I met somebody who has never played with the Rubik's Cube. ![]() It took a month to find the first solution of the Magic Cube despite almost every student was searching for it at the Budapest College of Applied Arts where the inventor, Rubik Erno was a professor back in 1974. Fortunately most of these have been answered like what is the minimum amount of moves needed to solve it from any starting position or what is the number of possible permutations, and the list goes on.Īfter it had been invented nobody could solve it and they weren't even sure that a human being is able to unjumble it at all. ![]() There are some serious questions about the mathematics of the Rubik's Cube. Home » Rubik's Cube » Mathematics of the Rubik's Cube - Permutation Group Mathematics of the Rubik's Cube ![]()
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![]() ![]() This $22 phonemic awareness bundle is comparable to Heggerty in quality at less than 1/3 of the price. First Grade Phonemic Awareness Daily Warm Ups Bundle by Sweet for Kindergarten. The bundle is not yet complete, so grab it now before the price goes up!Ĥ. It includes skills like short vowels, digraphs, blends, magic e, and so much more! It takes 2 minutes to show students how to play the games, and then they are set for independent practice with a buddy or group. I created these phonics board games for authentic practice with the phonemes and graphemes we’ve learned. I try to play them each week in my intervention groups. Phonics Board Game Bundle by Campbell Creates Readers. I also like the half-page aspect because they can be easily glued into our phonics notebooks!ģ. I loved the words in isolation at the top of the passage, and then the paragraph at the bottom. I used these passages this year with my intervention groups. FREEBIE! Decodable Passages for Prefixes and Suffixes by Emily Gibbons at The Literacy Nest. I take pride in this bundle, because I use it every day in my own instruction and know the impact it can have on student performance.Ģ. I made it for my own use, and I love getting to see it out in the world. It includes notepages, decodable word lists and sentences, phonics posters, and exit tickets. I know the title says Orton-Gillingham, but this bundle is for anyone looking to teach phonics explicitly and systematically to their students. ![]() Orton-Gillingham Phonics Bundle by Campbell Creates Readers. 10 Free and Paid Resources Aligned to the Science of Readingġ. I want you to feel confident that there ARE quality, affordable resources you can trust. ![]() Today, I want to share with you 10 resources (paid and free!) that I have made or discovered on Teachers Pay Teachers. I make and sell products on TPT that are aligned to the Science of Reading, but I also purchase from other like-minded individuals. For teachers who want to incorporate Science of Reading based materials into their instruction, this can lead you to distrust purchases from these teacher-sellers. Unfortunately, there’s no system in place to monitor the quality of the resources. Teachers Pay Teachers is a marketplace where teachers can find affordable, easy-to-use materials for their classroom. How do we know which materials to trust when it comes to the Science of Reading? 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Masturbation Morning Featuring Raylene & Romi Raylene’s and Romi’s deep voices are just as sexy as every single feature of their bodies. ![]() This is an AMAZING pair/shoot! Still can’t believe they aren’t twins! Two of the most gorgeous FTVx girls with such great attitudes and both of them are 10’s! Love to see them both start un-shaved, then one shave at beginning and the other at the end. The girls are stunning, especially when naked, though we admit that Romi is the hotter of the two. ![]() ![]() Trays that are used for this purpose will often be on the smaller size.Ĭoffee table trays can sometimes be left out on the table to hold books, flowers, and other items to decorate the table. If your goal is use it to help serve guests and others when they come over to visit, then consider how heavy the tray is and how well you would be able to move it and walk around with it. Both will provide you with some usefulness depending on how you want to use it. There are benefits to having a big or a small coffee table tray. Rectangle Coffee Table Tray with Books and a Plant Coffee Table Tray Decor Ideas Big table with a Small Coffee Table Tray Let’s take a closer look at the importance of a coffee table tray and explore some of the different types that you can use and how you can utilize them in your home. For some longer coffee tables, you may be able to put two of these on the table. A good size for most coffee tables will be 22 by 22 inches. ![]() Going with a smaller tray compared to your table will help make sure that it is going to fit there and not fall off. But how big should the coffee table tray be?Ĭoffee table trays come in a wide variety of sizes to suit your needs. These can come in a variety of materials and can even be left on the table to help with decorating without the mess. They allow you to have a nice addition to the table, without having to run back and forth to the kitchen all the time to grab what you need. ![]() A coffee table tray is simply a nice tray that you can use to serve yourself or others when they are in the living room. ![]() ![]() ![]() Once the mission has been completed, the player can gain relationship points from chatting with Fang.ĭespite Fang being indicated as romanceable from the start of the game, he will refuse a Heart Knot until the player progresses with his missions. His default outfit is a worn, long teal coat and poncho over a tucked white button-up shirt, rolled up brown pants with a messily-tied brown belt, high yellow striped socks, brown laced shoes and a dull blue duffle bag with medicine pouches and scrolls peeking out from it.įang considers X as part of the family, and his closest connection within Sandrock.įang is one of the potential bachelors who can be befriended and romanced.Ī unique trait that Fang holds is that he cannot gain relationship points until the player has completed Taste of Your Own Medicine. If X is not around, Fang may sometimes attempt to respond himself.įang has long, dark brown hair, light skin and light blue eyes. If a player interacts with Fang with X on his shoulder, X will respond directly to the player. On Sunday mornings, he can be found sweeping in the nearby graveyard. ![]() When not doing this, he can be found within the clinic working, or wandering the outer parts of Sandrock. While no mentor is specified, Fang is mentioned to have been one of two doctors with direct involvement in the treatment of Logan's father during the incident with Howlett and the church.įang opens up a medicine stall for free experimental medications every Wednesday and Friday. For unspecified reasons, Fang does not talk much, relying on X for the majority of conversations with others. He is the only doctor currently within Sandrock, with a fascination for gifting experimental medicine to unknowing tourists and builders. ![]() ![]() ![]() Updated iStat Menus to reflect the most recent version.The most powerful system monitoring app for macOS, right in your menubar. Updated September 2017: Removed DeskConnect and Browserism. If you like keeping track of all of your devices' batteries in one place, iBetterCharge has you covered. So, if you are working in your office, and want to make sure your iPhone is fully charged for your Pokémon Go walk later that evening, but it's in the living room, you can click on the Menu bar widget to see if it's got enough juice for you to catch 'em all. When they are connected, you will also always be able to tell how full their batteries are, even if they are in another room. That way, the software can communicate with your devices. In order to connect your iPhone and iPad to iBetterCharge, you have to enable Wi-Fi syncing in iTunes. When it drops below your chosen limit, you'll get a ping to remind you to charge up. But, what if you want to be sure that your device is charged up more than that before you walk out the door? iBetterCharge will send a notification to your Mac when your battery capacity drops below a threshold you choose, from five to 50 percent. When your iPhone or iPad reaches 20 percent, you get a notification that your battery is running low. If you find yourself staring at your computer screen early in the morning or late at night, f.lux will save your sight. ![]() "Darkroom" turns your screen red so you can work in, well, a dark room without your computer screen lighting up. ![]() You can set your screen to "Movie Mode" so you can stare at your screen for two hours straight without having the harsh light burn your eyes. It adds a soft yellow tint to the screen, as well, making it more comfortable for you to work in dimly lit rooms. If the sun is setting or rising, so will your screen. makes Jane a cranky girl.į.lux automatically adjusts the brightness of your screen based on where the sun is. I learned about it when I was complaining to my friend about how my eyes hurt when I start working right after I wake up in the morning. Automatic screen dimmer software is not new and f.lux has been available on the Mac for years. I'm sure you've already heard all about Night Shift on iOS, and might even use it regularly. If you're not sure you need this amazing Menu Bar organization tool, you can try it out for a month for free. You can even customize your Menu Bar widgets to appear whenever there is an update or change currently taking place (like when you sync a file in Dropbox). Clicking on the Bartender puts them all behind its menu until I need to access them.īartender even organizes your Mac's default Menu bar widgets, like WiFi, Bluetooth, the date, and your account user name (which I've never really liked having there anyway).īartender has been updated with a bunch of cool new features, like hotkey assignments and the ability to search your widgets. I absolutely love this little ditty because I want to use all of my favorite Menu bar Widgets, but kinda hate how messy it looks when I'm tracking my activities, monitoring my Mac's stats, and keeping my calendar close at hand. With Bartender, you can organize your widgets by hiding them within the Bartender Bar. With all of these Menu bar widgets, things start to get out of control and your Menu bar starts to look like a hot mess. If you want to try before you buy, you can check it out for a trial period first. Timing is one of the many dozens of apps you can use with the subscription-based Mac app service Setapp, or you can buy it directly from the developer's website. You can even manually add a specific task with a start and end time if you're working off your computer, but still want to track your activity. If something isn't quite right, you can also manually change activity names. It's truly impressive to see how your activities automatically divide into categories. It all happens automatically and you can see your activities by click on the Menu bar shortcut. Wondering where all of the hours in the day went? Can't figure out how long it took you to write that letter to your client or how much time you spent researching a subject for a project? Do you really know how much time you spend on social sites (Facebook can be such a time suck!)? Timing tracks all of this for you in the background while you just going on about your day. ![]() ![]() ![]() And, M2 delivers the peak performance of the PC chip while using just a quarter of the power. Together, they deliver 18% greater multithreaded performance than M1, so M2 can handle CPU-intensive tasks using very little power, like creating music with layers of effects or applying complex filters to photos.Ĭompared with the latest 10-core PC laptop chip, the CPU in M2 provides nearly twice the performance at the same power level. The new M2 features faster performance cores paired with a larger cache, while the efficiency cores have been significantly enhanced for even greater performance gains. And with up to 24GB of fast unified memory, M2 has been designed to handle even larger and more complex workloads. The additional transistors improve features across the entire chip, including the memory controller that delivers 100GB/s of unified memory bandwidth, 50% more than M1. The system-on-a-chip (SoC) design of M2 is built using enhanced, second-generation 5-nanometer technology, and consists of 20 billion transistors, which is 25% more than M1. And along with higher memory bandwidth and new capabilities like ProRes acceleration, M2 continues the tremendous pace of innovation in Apple silicon for the Mac.” With our relentless focus on power-efficient performance, M2 delivers a faster CPU, GPU, and Neural Engine. “M2 starts the second generation of M-series chips and goes beyond the remarkable features of M1. Johny Srouji, Apple’s senior vice president of Hardware Technologies told us… M2 brings all of this, plus new custom technologies and greater efficiency, to the completely redesigned MacBook Air and updated 13-inch MacBook Pro. It also delivers 50% more memory bandwidth compared to M1, and up to 24GB of fast unified memory. The M2 SoC has 24GB of unified LPDDR5 memory, up 8GB from the M1.īuilt using second-generation 5-nanometer technology, Apple’s M2 takes their performance per watt of M1 even further with an 18% faster CPU, a 35% more powerful GPU, and a 40% faster Neural Engine. ![]() If any Mac devices at all, really, seeing as WWDC has become less about hardware launches in recent times.The M2 System on a Chip (SoC) has the same number of cores as the original M1 chip (8) with 4 high-performance cores and 4 high-efficiency cores as does the M1 but runs faster and comes with more graphics cores, with 9 or 10 GPU cores, up from the 7 or 8 that came with the M. ![]() We’re thinking a fall launch for the MacBook Air (which will supposedly come with Apple’s new M2 chip inside) seems a better bet, especially as Apple’s conference is, of course, targeted at developers – so you’d expect a Pro product (Mac Pro, MacBook Pro) to be more likely to make an appearance. Namely that the MacBook Air will most likely go into mass production late in Q2 or in Q3, though if it’s the latter, the notebook might not be on shelves until quite a while after June. Gurman further reaffirms that Apple has a low-end MacBook Pro 13-inch coming to refresh the current offering, plus a revamped Mac mini, and iMac 24-inch – with Mac Pro and iMac Pro models being developed by Apple, too.Īnalysis: new MacBook Air at WWDC 2022? We're not convincedĭoes WWDC really seem like a sound prospect to see the MacBook Air launch?ĭebuting a MacBook Air here would seem like an odd choice, really, even if it does potentially line up with what we’ve heard elsewhere from another heavyweight Apple rumor peddler, Ming-Chi Kuo. If Apple does reveal this new Mac hardware at WWDC, it won’t actually go on sale until a little later in the year, as is usually the case with there being a short gap between an announcement and actual shipping of new Macs - adding a degree of uncertainty to anyone shopping for a new Mac. ![]() ![]() The frozen berries didn’t give off too much liquid so it still turned out fine. So thank you for that, because it’s very easy to make and put together! The recipe explicitly says FRESH berries, but I incorporated frozen blueberries and blackberries along with fresh strawberries and this little convenience was a win. We also took it to a picnic and everybody raved how incredible it was. So light and sweet, and the perfect taste of summer. My husband says this is now his favorite dessert of all time. □ Love all the options that make this an easy/often make! This is currently chilling in our fridge for dessert tonight, perfect for the PNW heatwave we're in! I feel like the easiest version of this is pudding mix or store-bought pudding with mixed berries and Nilla wafers, and the fancy easy version is to mix the salted whipped cream into the mix/store-bought pudding then proceed as directed. I have used frozen raspberries, mixed with fresh, but defrosted them on a towel for awhile to let their extra moisture out before addingįirst time making pudding like this and I messed up the recipe a bit BUT it has to be the most delicious thing I have ever made. It’s easy to customize - I’ve used whatever version of mixed fruit I have on hand (berries and/or stone fruit), and whatever jam I have. I, too, hate bananas but love a fancy pudding, and this doesn’t disappoint. I have made this twice (maybe three times?) and it’s turned out so delicious each time. I accidentally whipped instead of stirring and folding and the consistency of my pudding turned to liquid and I had to restart. I just made this and wanted to leave a tip so others don't make the same mistake I did! On Step 4, the key words are stir and fold. The dish was delicious and not overwhelmingly banana-flavored for the rest of us. So I folded one sliced banana into the final third of the pudding mixture for the top. However, one guest loves bananas, almost to excess. This is one of the best desserts I've ever made!! Incredible!! I think with that adjustment it will be fantastic. I will make it again and use only one teaspoon of salt. I would have given it five stars but it was too much salt. I made this dessert for my family and I followed the recipe exactly. Also, make sure you get the whipped cream well folded in since the only flavor in that is salt. Morton is half of diamond, and table salt would be even less than that. Super disappointing.įor Julia, make sure you're using the right salt. Lots of extra work to add something that made this much worse tasting and less healthy. It broke my heart to cover the beautiful berries with more of the cream but I regrettably did it anyway. The pudding tasted fine but it was disgustingly rich and bland, even a little salty, once the cream was added it. I dearly wish I skipped the cream step and just layered vanilla wafers, vanilla pudding, and those lovely shinty berries on top. This pudding was a hit at the summer BBQ - it's not too sweet, and really highlights the farmer's market berries that are in season right now. One of the best desserts I've ever made! "Lush" is an accurate description: the whipped cream + pudding base turned out so light and fluffy. ![]() Cover with silicone lid or plastic wrap chill at least 6 hours before serving.ĭo ahead: Pudding can be assembled 1 day ahead. Lightly crush reserved cookies and sprinkle over. Repeat layers once more, using half of remaining whipped pudding. Top with a layer of cookies, breaking each into 2–3 pieces. Spoon one third of whipped pudding into a medium serving bowl or dish. Stir jam in a medium bowl to loosen gently fold in fresh berries. It will quickly smooth out.) Stir one quarter of whipped cream into pudding to loosen, then fold in remaining whipped cream. (It may look clumpy and broken initially, but don’t worry. Using electric mixer, beat chilled pudding until smooth and glossy, about 45 seconds. Morton kosher salt in medium bowl until stiff peaks form, about 5 minutes. ![]() ![]() Using an electric mixer on medium-high speed, beat cream and remaining 1 tsp. Scrape pudding into a large bowl and cover with plastic wrap, pressing directly onto surface to prevent a skin from forming. Remove from heat add butter and vanilla bean paste and whisk until butter is melted and pudding is smooth. Reduce heat to low and continue to simmer, whisking constantly, 1 minute. Pour in egg yolk mixture, scraping bowl with a spatula to get every last bit, and cook, whisking constantly and making sure to get into corners of pan, until bubbling and thickened, about 3 minutes. ![]() Heat remaining 2 cups milk in a medium saucepan over medium until steaming. Morton kosher salt in a small bowl to combine. Vigorously whisk egg yolks, sugar, cornstarch, ¼ cup milk, and 1 tsp. ![]() ![]() ![]() When you build with a Broan Fresh Air System, your customers will see the energy-savings while having the comfort knowing that every breath taken inside is as fresh as one taken outside. These systems draw fresh air in, filter it, then distribute it throughout your home. Quicksilver Messenger ServiceJust For Love A Capitol Records Release 1970 Capitol Records, LLCRel.Broan Fresh Air Systems are designed to provide a continuous supply of fresh air to each home - and keep the people in it healthy. 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It provides a simple, fast, and user-optimized interface that anyone can use. This small and straightforward application gives access to all the information that is store on your computer or on the Web. You should therefore choose a different shortcut either in the Spotlight pane of System Preferences or in the Keyboard Shortcuts pane of LaunchBar Preferences.LaunchBar 6 is one of the software solutions that can empower you to search and access any kind of information quickly. The Command-Space shortcut conflicts with the OS X default shortcut for the Spotlight menu. Unfortunately the Command-Space shortcut may conflict with other applications that also use this convenient shortcut by default. And once you’ve got used to LaunchBar, you will probably invoke this shortcut very often, therefore convenient access is an important consideration. Command-Space conflicts with other applicationsĬommand-Space is by far the most conveniently accessible shortcut. If you press and hold the “Search in LaunchBar” shortcut for a short moment you can trigger Instant Send in order to send text or files that are selected in other applications to LaunchBar. Pressing the “Search in LaunchBar” keyboard shortcut repeatedly (with the modifier key kept pressed) allows you to quickly switch between running applications. It aborts any pending text input, abbreviation search, sub-search or send-to operation. Pressing this shortcut not only activates LaunchBar, but also reliably resets LaunchBar to an initial state for starting a new abbreviation search. In this documentation “Command-Space” is used as a synonym for the activation shortcut, despite the fact that you can choose a different shortcut in the Keyboard Shortcuts pane of LaunchBar Preferences. ![]() |
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