The winner for a given board cannot have less cells than the loser since that means the loser just moved, despite the fact the winner had already won on the last move. . Indefinite article before noun starting with "the", "ERROR: column "a" does not exist" when referencing column alias, Vanishing of a product of cyclotomic polynomials in characteristic 2. [1] A collection of open problems in this area is provided at the end of the book. Tic-tac-toe is a two-player game, that is played on a 33 square grid. Boards with 1 non-winning move: ${9 \choose 1} = 9$, Boards with 3 non-winning moves: ${9 \choose 2} {7 \choose 1} = 252$, Boards with 5 non-winning moves: ${9 \choose 3} {6 \choose 2} - 8 {6 \choose 2} = 1140$, Boards with 7 non-winning moves: ${9 \choose 4} {5 \choose 3} - 8 {6 \choose 1} {5 \choose 3} - 2 {6 \choose 4} - 6 {3 \choose 2} {3 \choose 2} = 696$. The best answers are voted up and rise to the top, Not the answer you're looking for? The most common of these is "Noughts and Crosses" or . It also includes an extensive analysis of tic-tac-toe-like symmetric line-forming games, and discusses the ErdsSelfridge theorem according to which sparse-enough sets of winning configurations lead to drawn maker-breaker games. And I can agree that this is not a research-level question because of the game is completely resolved. It's a very simple game - the first person to get 3 in a row wins. How does the case $n=7$ reduce to ordinary tic-tac-toe? but it's not giving me the answer that I want. How to save a selection of features, temporary in QGIS? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. rev2023.1.18.43176. Below is a proof of concept in Python, but first the output of time when run on the process sending output to /dev/null to show how fast it is: As one commenter has pointed out, there is one more restriction. Seven participants of the first excursion also participated in the second, 8 participants of . They chooose $-n$. Background checks for UK/US government research jobs, and mental health difficulties. . In a 3-by-3 grid game, the player who is playing "X" always goes first. Level up your programming skills with exercises across 52 languages . Simply select whether you want to be X or O and choose the difficulty level and you'll be all set. I've done this by analyzing game combinatorics, generating "win" and "fork" patterns, and applying algorithm below: 1 - Win This may be a more subtle question than it seems at first sight. A fun tic tac toe game. Note that ALL 504 do not contain any winning sets as there are not yet 5 pieces on the board. TABLE TOP TIC TAC TOE BRAIN GAMES FOR KIDS 8-12 - Tic tac toe board games for teens and tic tac toe game for kids in small size, the one tabletop games of the most delightful children's games, and this uniquely designed board game offers a convenient place to store the noughts and crosses keeping them safe and in one place. Tic-Tac-Toe Mickey and Minnie Mouse pieces. To learn more, see our tips on writing great answers. Contact Us | Sitemap | Privacy Policy>. hWUwuu}|VWwWv_;xne1?y4YM"??}]+&:/sonjybY-/E_]? Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Nice catch, quid. So now suppose you're X, the first player, and you're strategy-stealing; and suppose you come across a moment where the square you're 'supposed' to make your move in vs. the opponent's given plays is already taken. Wall shelves, hooks, other wall-mounted things, without drilling? Thanks for contributing an answer to Stack Overflow! Maintain two lists: one that will contain numbers encoding board positions, and one that will contain the AI's data about each position. P,`7'6UV2g4vJ.3):UZ|J%gXQN}OY$PsIx6o If opponent can't make another one-move-to-win position himself, forking player has a certain win. Update the question so it's on-topic for Theoretical Computer Science Stack Exchange. For example, the data can be compressed into a different base depending on which slots are available in the gameboard and such. If it's there, use the data from the corresponding element of the second list. 13680+504+9= 14193 boards. The way I once tried to explain this was. It is one of most widespread pen-and-paper based game for two players. In Pac-Man, you've probably noticed that if you run off the edge of the screen, you reemerge on the opposite side. There are only 3 9, or 19,683 possible combinations of placing x, o or <blank> in the grid, and not all of those are valid. %PDF-1.7
There are only 3**9, or 19,683 possible combinations of placing x, o, or in the grid, and not all of those are valid. The game is still going on. Another way could be to start with each of the eight winning positions. We start with $0$. But we can actually ignore that since there's no way to win two ways without a common cell without having already violated the "maximum difference of one" rule, since you need six cells for that, with the opponent only having three. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. In the Pern series, what are the "zebeedees"? Level up your programming skills with exercises across 52 languages, and insightful discussion with our dedicated team of welcoming mentors. ###Tic Tac Toe Counter is an experiment in combinatorics/discrete math. The easy answer might be that the board is a $3\times 3$ square and so you are looking at the symmetry group of a square. The first player to collect three cards that sum to zero wins the game. How could one outsmart a tracking implant? what are all the possible states after the change in that rule? Christian Science Monitor: a socially acceptable source among conservative Christians? Repeat for if the player played 2, 3, etc up to 9. Choose from 3 fun and vibrant themes that everyone can enjoy. The goal of tic-tac-toe is to be the first player to get three in a row on a 3-by-3 grid or four in a row in a 4-by-4 grid. Why did it take so long for Europeans to adopt the moldboard plow? I won't change the code to take that into account but it would be a simple matter of checking who has the most cells (the last person that moved) and ensuring the winning line belonged to them. ENTER FOR A CHANCE TO WIN $10,000. Then we choose $2$. If both have three in a row, then one of them would have won in the previous move. So for example a game which looked like: (with positions 1-9). You could apply this logic to determine if a board is valid or not: If there are multiple three-in-a-rows for a certain symbol, make sure they all overlap at the same cell. The online multiplayer game option for Tic Tac Toe is fantastic. An adverb which means "doing without understanding", Two parallel diagonal lines on a Schengen passport stamp. How does the number of copies affect the diamond distance? There's actually another limitation in that it's impossible for one side to have won in two different ways without a common cell (again, they would have won in a previous move), meaning that: can be. [1] Similarly, reviewer Kyle Burke complains that "many definitions and explanations are awkwardly 'math heavy'; undefined terms from advanced mathematics abound in small examples, where simpler descriptions would suffice". @Trenly Do you consider for example 132 and 231 to be the same? How many possible game boards(game states) of tic tac toe n x n is possible? ]dy3E?;Ka}?omo}TRHm.Pr|Wq4>J},)v+x0"J2oX They have to choose $-1$. I still can't figure out an efficient way to handle this. combinatorics combinatorial-game-theory tic-tac-toe . First, a valid game position is one where the difference between x and o counts is no more than one, since they have to alternate moves. To avoid this, if $1 9f"QGk({fi^Q&?tsbew!9?zq:kTR]WSWW=W;/@cl|"YH`'H
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Assuming only one of them has a win, that's a legal, winning game. A fun tic tac toe game. This will naturally create a list (well, two lists) of no more than $304$ elements, because we only allocate memory to positions we actually encounter - but we never have to explicitly figure out which positions those are. In addition, it's impossible to have a state where both sides have three in a row, so they can be discounted as well. That would be 9*8*7, or 9!/6! All Possible Tic Tac Toe Winning Combinations. rev2023.1.18.43176. Moreover, the threshold between a weak win and a strong draw can often be determined precisely. Assuming that X makes the first move every time: If board symmetries are taken into account, two games are considered the same if rotating and/or reflecting the board makes one of the games into a copy of the other game. Scribd is the world's largest social reading and publishing site. I won't change the code to take that into account but it would be a simple matter of checking who has the most cells (the last person that moved) and ensuring the winning line belonged to them. The original Tic Tac Toe challenges you to place three symbols in a row, on a three by three grid, switching turns with your opponent. I am assuming I am asking a naive question here, but why isn't the answer $9! Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. He focuses on ``Tic-Tac-Toe like games'' . How much does the variation in distance from center of milky way as earth orbits sun effect gravity? By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. With Tic Tac Toe Deluxe, you can play against the computer or with your friends and family. Why lexigraphic sorting implemented in apex in a different way than in other languages? This discovers all possible combinations for tic tac toe (255,168) -- written in JavaScript using recursion. This game is decided and there is no winning strategy. Play tic tac toe in two modes, either against a tough artificial intelligence or against a friend who is with to you. Ah, now I see. Either I do not understand this description or it is incomplete/wrong. Here is the scenario: 2 players play a game of Tic-Tac-Toe. Obviously that is not a perfect-playing method. Is there a standard algorithm or common that should be applied to similar questions like this that I'm not aware of? Then name $b$, forcing them to name $-b$, then $c$, forcing them to name $-c$, then $-b-c$, winning. Then 132, 134, 135, 136, 137, 138, 139. Quantum tic-tac-toe with alpha-beta pruning - best representation of states? Why is sending so few tanks to Ukraine considered significant? All Possible Tic Tac Toe Winning Combinations, Possible winning combinations of the TIC TAC TOE game, Microsoft Azure joins Collectives on Stack Overflow. Play the classic Tic-Tac-Toe game (also called Noughts and Crosses) for free online with one or two players. I couldn't think well at that moment. You can always choose two such numbers, since each positive number is missed by one of the following triples: $1+2=3, 1+3=4, 1+4=5, 2+3=5$. Then we choose $1$. The only issue I forsee with that could be large counts of beads. Tic Tac Toe Deluxe. Part B of the book discusses the potential-based method by which the ErdsSelfridge theorem was proven, and extends it to additional examples, including some in which the maker wins. Levels of Understanding There are different levels of understanding based on Benjamin Bloom's taxonomy, also known as Bloom's Taxonomy.. Update the spot with the respective player sign. Books in which disembodied brains in blue fluid try to enslave humanity. Will all turbine blades stop moving in the event of a emergency shutdown. Despite its apparent simplicity, Tic-tac-toe requires detailed analysis to determine even some elementary combinatory facts, the most interesting of which are the number of possible games and the number of possible positions. Making statements based on opinion; back them up with references or personal experience. I asked to put my answer check the answers section. I'm interested in AI as an area to study on in MSc. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. And I'm figuring things out by myself. <>/Metadata 462 0 R/ViewerPreferences 463 0 R>>
Neave Interactive. endobj
By the argument in the previous paragraph, this is at least as good for you as position $P_0$ is; but since we knew (by strategy) that $P_0$ was a winning position for you, then the new position $P_0$+X is winning too. Also, I personally think that it would have been easier to just build a TicTacToe class to deal with the game logic. This gives a total of $255168$ possible games. Combinatorics, and Game Theory close to each other in a novel, unexpected way. Just head over to Google.com and type the words tic tac toe into the search box, then press the Enter key on your keyboard. Copyright 2012. The main challenge of combinatorial game theory is to handle combinatorial chaos, where brute force study is . Play a retro version of tic-tac-toe (noughts and crosses, tres en raya) against the computer or with two players. Each player occupies a cell in turns, with the objective of placing three marks in a horizontal, vertical, or diagonal pattern. Because tic tac toe is a game with a finite amount of possible grid positions and finite amount of ways to fill up the grid with different combinations of 5 Xs and 4 Os, this makes tic tac toe game that can be completely analyzed for a perfect strategy in which neither player wins, concluding in a draw. You match 4 instead of 3 in a row in these larger board sizes, changing the game entirely. 2 0 obj
That might work, since the AI takes an input of the partial probabilities for each layout. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. After analysis, it is Card trick: guessing the suit if you see the remaining three cards (important is that you can't move or turn the cards). Both players continue alternating until all the cells have . 551 13 : 06. -pv{@`3^&U^-2z)dennmkw[i;zt ?VjWk9 Add to Favorites . This is how I got the 504 possible board layouts after 3 moves. - siuman Jan 13, 2013 at 13:22 It seemed to me that it is inefficiently complex to handle these further positions. I know that there is 9 boards after the first move, and 504 after the third move. And I suggest a "Double Fork" position which requires to calculate three moves after. It only takes a minute to sign up. Tic-tac-toe (American English), noughts and crosses (Commonwealth English), or Xs and Os (Canadian or Irish English) is a paper-and-pencil game for two players who take turns marking the spaces in a three-by-three grid with X or O.The player who succeeds in placing three of their marks in a horizontal, vertical, or diagonal row is the winner. It was published in 2008 by the Cambridge University Press as volume 114 of their Encyclopedia of Mathematics and its Applications book series (.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}ISBN978-0-521-46100-9). Flutter change focus color and icon color but not works. @Pham, possibly, but to what end? So for N=1 we just have a line, with each point having two neighbors. extra moves. To learn more, see our tips on writing great answers. Generating all these variations exhaustively would be useless for such an easy game. Now (according to strategy) we choose b=1 and c=2 so neither 1, 2, nor 1+2 is 4. The first player to collect three cards that sum to zero wins the game. Wooden Dog and Bone Tic Tac Toe. If you want to train an AI to play the optimal move, it does not matter in which order the tokens were placed: only the current state of the board matters. First, a valid game position is one where the difference between x and o counts is no more than one, since they have to alternate moves. These positions are, as I name them, "win and fork" and "double fork". What did it sound like when you played the cassette tape with programs on it? . So, to get to the finish line as quickly as possible, attempt to catch the appropriate direction. Wall shelves, hooks, other wall-mounted things, without drilling? Has natural gas "reduced carbon emissions from power generation by 38%" in Ohio? 34,389 Solution 1. probabilistic method in combinatorics. You . Looking to protect enchantment in Mono Black. Heuristics Development of Tic-Tac-Toe Game Using Heuristic Search Authors: Azlan Mohd Zain Universiti Teknologi Malaysia CW Chai CC Goh BJ Lim Abstract and Figures Available via license: CC BY. @Trenly As Jaap Scherphuis remarked above, my solution assumes that playing 1-3-2 would be the same as playing 2-3-1. How to generate all the permutations of a multiset? They need to choose $-n$, which does not create any winning move for them, so we can choose $2$, creating two winning options ($-2$ and $-3$) [note due to $n \ge 5$ there is no interference with the earlier moves], and thus win. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. Does the LM317 voltage regulator have a minimum current output of 1.5 A? What non-academic job options are there for a PhD in algebraic topology? Combinatorial Games: Tic-Tac-Toe Theory (Encyclopedia of Mathematics and its Applications, Series Number 114) Illustrated Edition by Jzsef Beck (Author) ISBN-13: 978-0521461009 ISBN-10: 0521461006 Why is ISBN important? Since I am not good at combinatorics, I thouggt I would ask here: How many unique gameboards are there in tic-tac-toe which contain 1, 3, 5, or 7 moves and no winning pattern? Abstract The game of tic-tac-toe is a well-known paper-and-pencil game that is played by two players. Misha, I understand it may be more than what many people think a calculator can handle, but there are ways to compress data. This problem is trivial as you could even manually write down every possible game play in a tree. What are the disadvantages of using a charging station with power banks? tic-tac-toe noun tik-tak-t variants or tick-tack-toe : a game in which two players alternately put Xs and Os in compartments of a figure formed by two vertical lines crossing two horizontal lines and each tries to get a row of three Xs or three Os before the opponent does Word History Etymology Add to Favorites 136, 137, 138, 139 base depending on which are... Science Monitor: a socially acceptable source among conservative Christians strategy board games in gameboard! Are boards which have winning combinations after 6 moves up with references or personal experience U^-2z ) dennmkw [ ;... The LM317 voltage regulator have a line, with each of the partial probabilities each! The game logic P_0 $ +X boards ( game states ) of tic tac toe ( wild combinatorics... Positions are, as I name them, `` win and a strong can! Then one of them has a win, that 's a legal, winning game LM317 voltage have... Intelligence or against a friend who is with to you 137, 138, 139 this is... Color but not works, I personally think that it would have been easier to build! Means `` doing without understanding '', two parallel diagonal lines on a Schengen stamp. Of the second, 8 participants of the first person to get to the finish line as quickly tic tac toe combinatorics... Played the cassette tape with programs on it logo 2023 Stack Exchange Inc ; user contributions licensed under CC.... Other wall-mounted things, without drilling it you will have learned a great of... Of placing three marks in a horizontal, vertical, or 9 /6. Winning sets as there are more than 9 beads of one type, but how likely is?... Lines on a Schengen passport stamp $ b=1 $ and $ c=a+1 $ deal. To Ukraine considered significant 504 do not contain any winning sets as there are yet. 138, 139 playing 2-3-1 has natural gas `` reduced carbon emissions from power generation by 38 % in., that 's a legal, winning game category of combinatorial game Theory close to other... ( wild tic-tac-toe combinatorics ), Microsoft Azure joins Collectives on Stack Overflow featured on Meta 2022 Community-a-thon Linked... Player to collect three cards that sum to zero wins the game.! Government research jobs, and game Theory close to each other in different! Like this that I want $ c=a+1 $ I were to make exactly. Is correct ; Noughts and Crosses ) for free online with one or two players emergency.... With your friends and family could be large counts of beads whole world not understand description! Common that should be applied to similar questions like this that I want but it & # x27.... All the inputs, and mental health difficulties excursion also participated in the previous move is.. Stack Exchange Inc ; user contributions licensed under CC BY-SA with each point two! A game of tic-tac-toe is a well-known paper-and-pencil game that is played on a 33 grid. Add to Favorites how I got the 504 possible board layouts with 7 given. Have learned a great deal of mathematics obj assuming only one of them would have easier..., unexpected way 9 beads of one type, but how likely is that go about explaining Science... Long for Europeans to adopt the moldboard plow in distance from center milky... A weak win and a strong draw can often be determined precisely in as., 135, 136, 137, 138, 139 data can be compressed into different. Checks for UK/US government research jobs, and mental health difficulties be applied to similar questions like this that 'm! It exactly like the original it would have been easier to just a! Stack Exchange is a two-player game, that 's awkward if there are more than 9 beads one. V+X0 '' J2oX They have to choose $ -1 $ toe is one of them would have won in Pern..., privacy policy and cookie policy avoid this, if $ 1 < n-2... Beads of one type, but how likely is that know that there are only 138 terminal board.. The partial probabilities for each position the case $ n=7 $ reduce to ordinary tic-tac-toe a 33 square grid of! Large counts of beads # x27 ; s not giving me the answer $ 9 /6. Largest social reading and publishing site flutter change focus color and icon color but not works get in! Vibrant themes that everyone can enjoy three marks in a row, then one of them has a,... Contain any winning sets as there are more than 9 beads of one type, but what! Me the answer that I 'm not aware of up and rise to the top not! Agree that this is not a research-level question because of the most popular board. Into account symmetry, how many possible games placing Xs and Os on board! Monitor: a socially acceptable source among conservative Christians tac toe n X n is possible Add Favorites. To zero wins the game in this area is provided at the end of the most of! Possible games } TRHm.Pr|Wq4 > J }, ) v+x0 '' J2oX They have to $... The question so it 's there, use the data can be compressed into a different way than other... $ reduce to ordinary tic-tac-toe think that it is one of most widespread pen-and-paper based game for players. Neave Interactive * 8 * 7, or diagonal pattern tic tac toe combinatorics this power banks $!. Trenly do you consider for example a game which looked like: ( positions... Way I once tried to explain this was on-topic for Theoretical computer Science Stack Exchange Inc ; user contributions under. Class to deal with the objective of placing three marks in a row.... Game boards ( game states ) of tic tac toe is one of them has a,. Do not understand this description or it is one of them has a win, that awkward. In AI as an area to study on in MSc, other wall-mounted things, without?! Rise to the top, not the answer $ 9! /6 wild tic-tac-toe combinatorics,... For two players everything is made of fabrics and craft supplies dennmkw [ I ; zt? VjWk9 to! Are, as I name them, `` win and fork '' and `` Double fork '' and Double... 9 boards after the change in that rule tic tac toe combinatorics are more than beads... End of the partial probabilities for each position check the answers section J2oX They have to choose $ a+1 defending! ) against the computer or with your friends and family tres en raya ) against the computer or two... Novel, unexpected way is played by two players the cassette tape with on! Quick input if my reasoning is correct 2022 Community-a-thon Recap Linked 3 tic-tac-toe game 7 Taking into symmetry! With power banks with power banks, other wall-mounted things, without drilling feed. Of general $ n $ been studied end of the game entirely } |VWwWv_ ; xne1 y4YM... Either against a friend who is with to you which means `` without! Be 9 * 8 * 7, or 9! /6 to our terms tic tac toe combinatorics service, privacy policy cookie. Online with one or two players all design styles tic-tac-toe all comes under the category of combinatorial game Theory to. Go about explaining the Science of a multiset eight winning positions be for!, 137, 138, 139 is playing & quot ; Noughts Crosses... Exercises across 52 languages either against a friend who is with to you easier! Which have winning combinations after 6 moves can often be determined precisely your,. Other languages been studied learn more, see our tips on writing answers. Diagonal pattern use the data can be compressed into a different way in. To study on in MSc /Metadata 462 0 R/ViewerPreferences 463 0 R > > Neave Interactive policy cookie! All turbine blades stop moving in the previous move solution assumes that playing 1-3-2 would the.? } ] + &: /sonjybY-/E_ ] 7, or diagonal pattern b=1 and so! Xs and Os on the board until into a different way than in other?... Cards that sum to zero wins the game of tic-tac-toe is a question and answer site for studying. $ and $ c=a+1 $ about explaining the Science of a world where everything is of. Terminal board positions, the data from the corresponding element of the first player to collect three cards sum! Skills with exercises across 52 languages, and have started the logic games & # ;! ; & # x27 ; s a very simple game - the first excursion participated. Participants of 2, nor 1+2 is 4 possible game boards ( game states ) of tac!, see our tips on writing great answers placing three marks in row! That sum to zero wins the game entirely 2 0 obj that might work, since AI. Game - the first player to collect three cards that sum to zero wins the game hooks other! Draw can often be determined precisely in algebraic topology for UK/US government research jobs, and after! Solution assumes that playing 1-3-2 would be 9 * 8 * 7, or diagonal pattern focuses! On `` tic-tac-toe like games & # x27 ; s not giving me the answer that 'm! Temporary in QGIS on writing great answers my answer check the answers section welcoming mentors each player a... 504 possible board layouts with 7 moves given that there are more than 9 of. Tanks to Ukraine considered significant changing the game is completely resolved design styles a different base depending on which are... Vjwk9 tic tac toe combinatorics to Favorites ; s not giving me the answer $!.
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