Dots-and-boxes - how to play ◮Played on a rectangular grid with m rows, each containing n dots. ◮A move consists of drawing a horizontal or vertical line connecting two dots. ◮Upon completing a box, a player ‘claims’ it and moves again. This site contains user submitted content, comments and opinions and is for informational purposes only. Apple may provide or recommend responses as a possible solution based on the information provided; every potential issue may involve several factors not detailed in the conversations captured in an electronic forum and Apple can therefore provide no guarantee as to the. Combinatorial Game Theory This is the branch of mathematics which studies the theory of games such as Dots-and-Boxes. That is, games in which all the information is available to the players, there are no secret factors,such as cards to hide from your opponent, and no element of chance, such as dice.
Dots and Boxes has a rich mathematical theory which can be divided into three separate parts. The first is the mathematical proof of the chain rule, the second is the application of combinatorial game theory to resolve Dots and Boxes endgames, and the third is computer analysis.
The chain rule was discovered by Elwyn Berlekamp while he was in Junior High School and an outline of the proof is given in the chapter on Dots and Boxes in volume 2 of Winning Ways.
The application of combinatorial game theory to Dots and Boxes is also due to Berlekamp, and is fully discussed both in Winning Ways and in Berlekamp's book on Dots and Boxes.
Computer analysis consists of an exhaustive analysis of all positions to provide a rigorous evaluation of a game. It has been provided for 5x5 dots as well as other special positions by David Wilson. This type of analysis was first used with great success by Ken Thompson for analyzing chess endgames.
A normal game is one in which both players take turns without moving into any chains or cycles until all that is left is chains and cycles. At this point, a player will have to move into a chain or cycle. A normal game will continue with the other player responding by taking control and keeping it. A normal game ends when the player having control takes all the boxes in the last chain.
More precisely, the chain rule states that if a player wants to force his opponent to be the first to move into a chain or cycle in a normal game, then
To understand the proof of this result, one works backwards, by understanding how a normal game ends.
In a normal game, the last player is the one who took all the boxes in the last chain on his last turn. But this is the player who kept control in order to take all the boxes of the last chain, so this is also the player who got control when the first chain was given away. One can conclude from this that
Lemma 1: In a normal game the player who took the last turn is the one who forced his opponent to first move into a chain or cycle.
Since the first player always takes the odd turns and the second player always takes the even turns (recall the definition of turn) one can restate this as follows.
Lemma 2: A normal game in which the first player forces his opponent to first move into a chain will have an odd total number of turns. A normal game in which the second player forces his opponent to first move into a chain will have an even total number of turns.
The chain rule now follows from the following result, due to Berlekamp.
Theorem: No matter what the shape of the initial Dots board, one always has the formula
where DOTS is the total number of dots, DOUBLECROSSES is the total number of doublecrosses, and TURNS is the total number of turns.
This theorem immediately implies the chain rule, so I will do this part before turning to the proof of the theorem itself.
First, let's assume that the total number of dots is odd. The formula of the Theorem shows that
This is due to the simple results
Since the parity of TURNS determines the chain rule, it remains to understand DOUBLECROSSES, the total number of doublecrosses.
In the final phase of a normal game, the player in control always leaves one doublecross for each chain, except the last, and two doublecrosses for each cycle. Since there have been no doublecrosses before the final phase (by definition), one gets the formula
where CHAINS is the total number of chains and CYCLES is the total number of cycles. One therefore gets
Combining this with the previous formula, one concludes The escape game pigeon forge the island.
The reader will note that, due to the opening remarks, this is exactly the first part of the chain rule. The second part of the chain rule is proved in exactly the same way. This completes the proof of the chain rule, assuming that the Theorem is true.
Escape room games in pigeon forge. If there are no doublecrosses, placing an edge has either just created a single box or has given the turn to the other player (but not both). This gives
where the -1 is needed because the first turn was not included in this count: It is not preceded by placing an edge. But, there is a known formula from graph theory that states that
is always true for any shape of Dots board. I will discuss this important and deep result in the next section. Combining these two equations gives the result
that is, the number of turns equals the number of total dots, when there are no doublecrosses, and the Theorem is true in this case.
For the general case, let C be the total number of doublecrosses. Now, consider the same counting argument as above, but without taking into the count any edge which fills in a doublecross. There are C such edges, and note that such edge captures two boxes but does not change the turn. Therefore, one gets
which simplifies to
Elementary algebra gives
which is the required result.
where E is the total number of edges, B is the total number of boxes, and D is the total number of dots, which I said was true for any dots board.
In fact, this is a special case of a general formula in graph theory due to the famous mathematician Leonhard Euler (1707-1783). This formula states that for any planar graph (one that you can draw without edges crossing each other) one has
where V is the total number of vertices, E is the total number of edges, and F is the total number of faces. I won't prove this here because there are many proofs available online. I should just say that the formula is usually written with a '2' on the right, because it is really about graphs on a sphere, so that the outside of the graph counts as an extra face.
To understand what I just wrote, you should think of Euler's formula three dimensionally. For example, consider the cube. It has 8 vertices (V = 8), 12 edges (E = 12), and 6 faces (F = 6), and so V-E+ F = 8 - 12 + 6 = 2. You can check that the same holds true for a pyramid with square base (like the great pyramid of egypt), where the base is counted as a face, or any other polyhedron.
The formula for polyhedra was first noted by mathematician and philosopher Rene Descartes, but Euler was the first to state it explicitly for all graphs.
This formula is a first step in algebraic topology, the area of mathematics which tries to find numerical characteristics of geometrical objects which don't change if you deform the object smoothly. It is the simplest example of the Euler-Poincare characteristic.
DIAGRAM
The chain rule will still be valid, except that it will have to be modifed because Euler's formula says that
In fact, there is no longer any boundary to the dots board, so making chains is much harder since they have to be rooted to cycles. So, the chain rule for toroidal Dots should probably be called the cycle rule since the game will be dominated by cycles.
The iMessages need to be sent through Apple’s servers, and the only way to do this legitimately is to use an Apple device.
Using an app running on a Mac computer as a server that relays messages to the Android device is a very smart way to make iMessage work on Android, where it is not technically supported.
Thankfully, you can simply use the number one app to run Apple IOS apps on Android using IOS emulator so no harm no foul. Here’s how you can get CIDER installed and ready to roll: After it is installed, simply go to the App drawer and launch it. That’s it, now you can easily run iOS apps and games on Android.
Why You Can’t Normally Use iMessage on Android. You usually can’t use iMessage on Android because Apple uses a special end-to-end encryption system in iMessage that secures the messages from the device they’re sent on, through Apple’s servers, to the device receiving them.
How to Send and Receive iMessage on Android Phone
Android and iPhone play together??? Android uses Google Play and iOS uses Game Center. We may in a future update change multiplayer services and if so one of the main criteria will be that iOS and Android are able to play multiplayer.
This app is capable of sending both iMessage and SMS messages. iMessages are in blue and text messages are green. iMessages only work between iPhones (and other Apple devices such as iPads). If you are using an iPhone and you send a message to a friend on Android, it will be sent as a SMS message and will be green.
Open the app once it is installed and the emulator will then install padoid and all you have to do is tap on the padoid icon on your homepage and your Android will easily run any iOS app. Cider Apk is one of the Best iOS Emulator for Android in which you can run iOS Apps and Games on your Android Smartphone.
Because iOS apps are generally better than Android counterparts (for the reasons I said above), they generate a greater appeal. Even Google’s own apps behave faster, smoother and have a better UI on iOS than Android. iOS APIs have been much more consistent than that of Google.
Your phone can now run both iOS and Android—whenever you want. Right now.
Installation Steps
How to send SMS/MMS on your iPad
With FaceTime’s popularity, Android users may wonder if they can get FaceTime for Android to host their own video and audio chats. Sorry, Android fans, but the answer is no: You cannot use FaceTime on Android. The same thing goes for FaceTime on Windows. But there is good news: FaceTime is just one video-calling app.
iMessage For Android – Best Alternatives
Problem #4: Galaxy S6 re-sends previously sent text messages and photos
You might not be able to get SMS or text messages that someone sends you from an iPhone because they’re still being sent as iMessage. This can happen if you used iMessage on your iPhone and then transferred your SIM card or phone number to a non-Apple phone (like an Android, Windows, or BlackBerry phone).
Complete these steps from your iPhone before starting to use your new smartphone:
it can be download in iOS or Android. so multiply scan but you can either play with your friends. if you want there you go so some of the games it has it has three basic games. but now the phone you can also use uber system.
These iOS Android cross platform games are not only popular but also wanted the most by the users.
Top 16 iOS Android Cross Platform Games
If Android has an updated version of PUBG Mobile that iOS does not, for example, cross-platform play is temporarily disabled. Cross-platform play in PUBG Mobile may not be exactly what you want, but it does work in a limited capacity. PUBG Mobile is available now on iOS and Android devices.
This app allows you to send text messages and photos from your iPad to another iOS device over Wi-Fi or cellular data. And the text –messaging is free. Activate your iMessage with your Apple ID on your iPad by taping Settings > Messages > swipe iMessage to ON. Tap Send & Receive > tap Use your Apple ID for iMessage.
When an iPhone user sends a text message to a non-iPhone user like an Android phone, the message is sent through SMS, as is indicated by a green message bubble. Sending text messages by SMS is also the fallback when an iMessage doesn’t send for whatever reason.
If you have an iPhone and another iOS device, like an iPad, your iMessage settings might be set to receive and start messages from your Apple ID instead of your phone number. To check if your phone number is set to send and receive messages, go to Settings > Messages, and tap Send & Receive.
Nina, iPhone and Android are two different flavors of smartphones, in fact the iPhone is just Apple’s name for the phone they happen to make, but their operating system, iOS, is Android’s main competitor. Manufacturers put Android on some very cheap phones and you do get what you pay for.
Why iOS is safer than Android (for now) We’ve long expected Apple’s iOS to become a bigger target for hackers. However, it’s safe to assume that since Apple does not make APIs available to developers, the iOS operating system has fewer vulnerabilities. However, iOS isn’t 100% invulnerable.
Top 8 Most Popular Mobile Operating Systems
Apple’s iPhone is closely associated with iOS, Apple’s mobile software that lets users trade iMessages, snap Live Photos, and run over a million apps made for the platform. Apple could very well offer its smartphone with the Android operating system—Google’s OS that’s free for phone makers to install on their devices.
And many articles shows that you can install Android on an iPhone or iPad if you don’t mind have your ios device jailbroken. To install Android on iPad, you should firstly jailbreak your iPad mini and then use some apps to install android system.
Head to Settings then tap Security and toggle the Unknown sources switch to On. With that done, you simply need to get an APK (Android Application Package) on your device in whatever way you prefer: you can download it from the web, transfer it via USB, use a third-party file manager app, and so on.
The good news is that weMessage is a new app that brings Apple’s closed messaging system iMessage to Android, and it works surprisingly well. I have recently published my first app to the Play Store, called weMessage. weMessage is an app that allows you to use iMessage on your Android phones or tablets.
iMessage is so good that many smartphone users would love to see an Android version come out, although it’s something Apple will probably never do. Android Messages, not to be confused with Hangouts or Allo, is Google’s texting app, and a new version of the app will soon be available on your Android device.
Despite the differences, WhatsApp and iMessage are useful messaging apps with a large user base. The apps allow you to communicate with family and friends for free via your smart phone. WhatsApp allows users to mass text 256 people at once. You can keep the conversation going on larger devices too.
Photo in the article by “Cecyl GILLET” https://www.cecylgillet.com/blog/comments.php?y=11&m=09&entry=entry110902-110511