21

Accurding to Zermelo's theorem, in finite two-person games of perfect information in which the players move alternately and no affect of chance, one of those three possibilities is true: First player can always win. Second player can always win. Both players can force a draw. For example, Tic-Tac-Toe is known to have a strategy by both players that will ...


21

One of the following is true: There is a dominant strategy for White. There is a dominant strategy for Black. There are strategies for both players that guarantee they don't lose, i.e. perfect play results in a draw (e.g. as in Tic-Tac-Toe). No one knows which is true. Most experts guess that perfect play leads to a draw, and a few believe White can always ...


20

In Solving Quantum Tic-Tac-Toe, Ishizeki and Matsuura use a computer to search the game space to find solutions, but don't specifically speak to strategy. However, there are a few strategies we can glean from their results: Go for a half-point victory, not a full point victory. In their search they found the first player, X, cannot guarantee a win by being ...


13

Note: This answer below is using 1st Edition scoring rules (Cities only scored once). Although parts of the analysis are valuable to calculating the highest score possible (36 Scoring opportunities, try to score with Meeples that have a high point value per the 36 turns, the highest score of 278 is not correct. I will have to do further analysis to determine ...


9

I suspect that you're right that in principle and possibly even in practice an optimal AI could be built for this game to determine who can win from the starting position, though it would likely take more computing power than you're likely to have readily available. The best approach for brute-force solving a game like this is something close to what ...


9

As Hackworth notes, testing all possible layouts is obviously infeasible. However, it might be possible to get a decent upper bound on the points. I think we can safely assume that meeples will not be a scarce resource when playing in this way (since we can choose to complete the map in any order, it should not be difficult to ensure we always have meeples ...


8

Is there an optimal way of placing the tiles in order to maximize one player's score? Yes With a finite amount of tiles, a finite number of legal placements for any tile at any point in the game, and a finite amount of meeple plays after placing a tile, it should be obvious that there are a finite number of possible games. Every game has a definite result ...


7

No - there need not always be a tie in the general case. Even in games of perfect complete information there may still be a bias towards one player. For example the game of nim cannot end in a tie, and depending on the starting position gives an advantage to either the first or second player - e.g. Size of heaps | Result with A | B | perfect play --...


6

This is a generalized version of @Guvante's answer. O (player 2) always wins in 4. The diagonals are not necessary to achieve a win. Examples are included in bold, but this strategy works for all choices made by X. X places a piece at (a,b): (1,1) O chooses a row that is not a, and places two pieces in that row, but not in column b: (2,2) and (2,5) Now, O ...


6

My co-worker found what I believe is a winning strategy for the first player, but now I find earlier evidence by others as well. This is for the original version of the game, where you can send your opponent to an already won field and he has to place his mark there. It seems that the question is still open for the updated version where he can then choose ...


5

The maximum theoretical score is 338. I previously answered the question for 1st Edition rules (cities only scored once by farmers), so here is my second stab at this question. First, it should be rather clear to everyone that since there are only 72 tiles and one is the starting tile, that the maximum number of scoring opportunities is 36 if you go first. ...


4

The maximum possible score is 342. 2 Farms with 16 cities = 96 5 Farms with 2 cities = 30 7 Roads with a total of 14 end pieces and 32 non end pieces = 46 6 Cloisters = 54 16 Cities worth a total of 116 Total = 342, 36 meeples placed. The answer user1873 gave is quite good, but overlooks that 16 cities are possible. It was also extremely non trivial to ...


4

For the bonus, the shortest possible game that results in all pieces the same colour (9 moves) is here: https://www.youtube.com/watch?v=6ehiWOSp_wk


3

D3C5D6E3B4C3D2C4F4 which is the same sequence Julia gave, not only answers the bonus question, but actually all questions. The least number for both players to not have an option is the 9 move wipe out, so 13 discs including the starting discs. It also is the answer to the first question, unless you want to say "one player of the two players can't make a ...


3

This sounds a lot like a strategy-stealing argument. It's often used to prove that one side of a game has an advantage, even if optimal strategy is not known. As a broad example, consider a game with the following properties: It is a turn-based game The starting positions are identical except Player A goes first Player A has the option of skipping the first ...


3

The winning strategy for such a small Hex board is shown in this basic strategy guide. Like tic-tac-toe, on a 4x4 board white will always win by opening on the main diagonal, because for every counter that black can make, there is another way for white to force the win. Once white can form a "two-bridge" by placing the second piece in a non-adjacent space ...


2

I have flip-flopped a few times, but think that Player 2 has an unbeatable strategy. The trick is that Player 1 needs an X in each row and each column while Player 2 needs a "box". Four O's that form a rectangle with no X's that share a row or column with said rectangle. First two O's sharing a row with self and separated by 1 square. Never put below or ...


2

It is very hard to say it without browsing all possible combinations. Another question is are you asking about moves in game that make sense or any random game. I was playing this game many years and usually one consider move choosing out of 3 maximally 5 possibilities for good move. However in mid-game there are 10-20 legal moves frequently. You can try ...


2

This is actually deceptively simple. A full set includes every card of that type in the game. If you possess a card in a specific set, then nobody else can complete that set. If you possess cards from every set, then no set can be completed without trading with you. The flip side is that if there is any set that you lack cards from, then that set can ...


2

You are missing a couple of important factors in your question that make it impossible to answer as it stands. Mainly that a game with perfect information available to both players doesn't mean that it is not balanced in one players favor. While it may be true that there are games out there where it is balanced between both players it is also true that ...


1

Suppose you're playing a three player game with wheat, barley, and corn. For one of the trading players to win with wheat, the two trading players must have nine wheat between them, which is the same as the shut out player not getting any wheat. To get a hand, we choose 9 cards out of 27. There are C(27,9) ways of doing that, where C is the binomial ...


1

(I'm mostly ignoring Bull and Bear rules here, but I think they can be added in with a little fiddling.) If you just want to know whether it's possible, then it's not a question of probability but of showing that there is at least one arrangement of cards that would allow a player to win without trading. And it's pretty easy to see that this must exist, ...


1

The term "dominant strategy" is generally used for games with incomplete information (or simultaneous moves, which can be analyzed as a game with incomplete information). In a deterministic game where players take turns and there is no secret information, the term "perfect play" is generally used, rather than "dominant strategy". The term "dominant strategy" ...


1

essentially, it seems that in a more complex game, it is more likely that there will be a draw. Connect-Four is the most complex game we have solved so far, and although the first player wins, that is due to the mathematical nature of the game (i.e., it is impossible to go back to a previous position like for one side of chess). There is no proof that either ...


1

The biggest piece of strategy is knowing how to move a pile of cards from one stack to the next. The basic algorithm is a recursive algorithm. To move n cards from stack a to stack b, you need to: move the n-1 cards from stack a to stack c move the nth card to stack b move the n-1 cards from stack c to stack b. To move a stack of three ...


1

Sort of. A computer could brute force the optimal move each round based upon the decisions made by the opponent. You could even construct a matrix(huge) of all possible moves for a game. But there is no single path of one players choices that would guarantee a win. Your opponent can make choices that counter your choices effectively and give them the ...


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