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Some Strategy For Dots and Boxes

Dots and Boxes at Cool Math Games: Add some color to the pencil-and-paper classic! Can you outsmart the computer and fill the board with your boxes? Play the classic game of strategy. You can challenge the computer, a friend, or join a match against another online player.

Game Pigeon Plus! It’s wayyyyy overpriced. You have to pay $3 for custom skins, accessories, and ad free play? That’s a bit much considering the look of your props and avatar don’t matter to the game. And (most of the time) the ads don’t pop up in the middle of the game, they pop up when you’re waiting for the opponent. Some Strategy For Dots and Boxes In the game of Dots and Boxes, the winner is generally the player who makes the last move. The reason for this is that at the end of the game, there are usually a few long corridors or chains of boxes left to be taken.

In the game of Dots and Boxes, the winner is generally the player who makes the last move. The reason for this is that at the end of the game, there are usually a few long corridors or chains of boxes left to be taken.If your opponent is forced to play in one of these chains, then you can take all but two of the boxes and, by sacrificing the last two boxes, make certain that it is his turn to play into the next long chain. You will thus win all but two boxes in each long chain, and of course you will win all boxes in the last chain. We say a chain is long if it contains at least three boxes.

The above program for playing Dots and Boxes uses an algorithm that is not very good, but it will play well once there are only long chains left. You may use it to improve your play at the next level of understanding. This next level requires determining which player will move last. This is most usefully done using the following rule.

The Long Chain Rule: Suppose the playing field is a rectangle of m rows and n columns and so has mn boxes. If both m and n are even, then the first player should play to make the number of long chains odd. If either m or n is odd, then the first player should play to make the number of long chains even.

Of course then the second player wants an even number of long chains if both m and n are even, and an odd number of long chains otherwise.

It must be pointed out that in this rule, loops do not count as long chains.

Here is the reason this rule works. There are (m+1)n horizontal edge moves and m(n+1) vertical edge moves for a total of 2mn+m+n moves. Without the rule that the player who completes a box moves again, we could say that the player who moves first also moves last if and only if 2mn+m+n is odd.

With the rule that the player who completes a box moves again, we must subtract one for each time at least one box is filled, except for the last box. Some moves complete two boxes simultaneously. Let us call these moves double-box moves. If there are no double-box moves, then since there are mn boxes and since completing the last box doesn't change things, we must subtract mn-1 from the total number of moves to get the number of move changes. This gives 2mn+m+n-(mn-1)=mn+m+n+1=(m+1)(n+1). Thus if there are no double-box moves, then the player who moves first also moves last if and only if (m+1)(n+1) is odd. The same is therefore true if there is an even number of double-box moves in the game.

Another way of putting this is to say that if (and only if) (m+1)(n+1) is odd, the first player wants to arrange things so that there is an even number of double-box moves in the game. For a chain of length 1 or 2, neither player need allow a double-box move to be made. However, in each chain of length 3 or more, either player may take all boxes but two, providing the opponent with a single double-box move. For a loop of four or more boxes, either player may take all but four boxes, providing the opponent with two double-box moves. Thus, in a well played game, the number of double-box moves is equal to the number of long chains, plus twice the number of loops, minus one because the player to move in the last long chain will take all of the boxes. So if (m+1)(n+1) is odd, the first player wants an odd number of long chains in the game. Moreover, (m+1)(n+1) is odd if and only if both m and n are even.

To go beyond this level of understanding of the game, read the book of Berlekamp.

Dots and Boxes is a pencil-and-paper game for two players (sometimes more). It was first published in the 19th century by French mathematician Édouard Lucas, who called it la pipopipette.^[1] It has gone by many other names,^[2] including the game of dots,^[3]dot to dot grid,^[4]boxes,^[5] and pigs in a pen.^[6]

The game starts with an empty grid of dots. Usually two players take turns adding a single horizontal or vertical line between two unjoined adjacent dots. A player who completes the fourth side of a 1×1 box earns one point and takes another turn. (A point is typically recorded by placing a mark that identifies the player in the box, such as an initial.) The game ends when no more lines can be placed. The winner is the player with the most points.^[2][7] The board may be of any size grid. When short on time, or to learn the game, a 2×2 board (3×3 dots) is suitable.^[8] A 5×5 board, on the other hand, is good for experts.^[9]

The diagram on the right shows a game being played on a 2×2 board (3×3 dots). The second player ('B') plays a rotated mirror image of the first player's moves, hoping to divide the board into two pieces and tie the game. But the first player ('A') makes a sacrifice at move 7 and B accepts the sacrifice, getting one box. However, B must now add another line, and so B connects the center dot to the center-right dot, causing the remaining unscored boxes to be joined together in a chain (shown at the end of move 8). With A's next move, A gets all three of them and ends the game, winning 3–1.

Strategy[edit]

For most novice players, the game begins with a phase of more-or-less randomly connecting dots, where the only strategy is to avoid adding the third side to any box. This continues until all the remaining (potential) boxes are joined together into chains – groups of one or more adjacent boxes in which any move gives all the boxes in the chain to the opponent. At this point, players typically take all available boxes, then open the smallest available chain to their opponent. For example, a novice player faced with a situation like position 1 in the diagram on the right, in which some boxes can be captured, may take all the boxes in the chain, resulting in position 2. But, with their last move, they have to open the next, larger chain, and the novice loses the game.^[2][10]

A more experienced player faced with position 1 will instead play the double-cross strategy, taking all but 2 of the boxes in the chain and leaving position 3. The opponent will take these two boxes and then be forced to open the next chain. By achieving position 3, player A wins. The same double-cross strategy applies no matter how many long chains there are: a player using this strategy will take all but two boxes in each chain and take all the boxes in the last chain. If the chains are long enough, then this player will win.

The next level of strategic complexity, between experts who would both use the double-cross strategy (if they were allowed to), is a battle for control: An expert player tries to force their opponent to open the first long chain, because the player who first opens a long chain usually loses.^[2][10] Against a player who does not understand the concept of a sacrifice, the expert simply has to make the correct number of sacrifices to encourage the opponent to hand him the first chain long enough to ensure a win. If the other player also sacrifices, the expert has to additionally manipulate the number of available sacrifices through earlier play.

In combinatorial game theory, dots and boxes is an impartial game and many positions can be analyzed using Sprague–Grundy theory. However, Dots and Boxes lacks the normal play convention of most impartial games (where the last player to move wins), which complicates the analysis considerably.^[2][10]

Unusual grids and variants[edit]

Dots and Boxes need not be played on a rectangular grid – it can be played on a triangular grid or a hexagonal grid.^[2]

Dots and Boxes has a dual graph form called 'Strings-and-Coins'. This game is played on a network of coins (vertices) joined by strings (edges). Players take turns cutting a string. When a cut leaves a coin with no strings, the player 'pockets' the coin and takes another turn. The winner is the player who pockets the most coins. Strings-and-Coins can be played on an arbitrary graph.^[2]

A variant Kropki played in Poland allows a player to claim a region of several squares as soon as its boundary is completed.^[11]

In analyses of Dots and Boxes, a game board that starts with outer lines already drawn is called a Swedish board while the standard version that starts fully blank is called an American board. An intermediate version with only the left and bottom sides starting with drawn lines is called an Icelandic board.^[12]

A game called Trxilt combines some elements of Dots and Boxes with some elements of Chess.

References[edit]

1. ^Lucas, Édouard (1895), 'La Pipopipette: nouveau jeu de combinaisons', L'arithmétique amusante, Paris: Gauthier-Villars et fils, pp. 204–209.
2. ^ ^abcdefgBerlekamp, Elwyn R.; Conway, John H.; Guy, Richard K. (1982), 'Chapter 16: Dots-and-Boxes', Winning Ways for your Mathematical Plays, Volume 2: Games in Particular, Academic Press, pp. 507–550.
3. ^Holladay, J. C. (1966), 'A note on the game of dots', American Mathematical Monthly, 73 (7): 717–720, doi:10.2307/2313978, JSTOR2313978, MR0200068.
4. ^Swain, Heather (2012), Play These Games: 101 Delightful Diversions Using Everyday Items, Penguin, pp. 160–162, ISBN9781101585030.
5. ^Solomon, Eric (1993), 'Boxes: an enclosing game', Games with Pencil and Paper, Dover Publications, Inc., pp. 37–39, ISBN9780486278728. Reprint of 1973 publication by Thomas Nelson and Sons.
6. ^King, David C. (1999), Civil War Days: Discover the Past with Exciting Projects, Games, Activities, and Recipes, American Kids in History, 4, Wiley, pp. 29–30, ISBN9780471246121.
7. ^Berlekamp, Elwyn (2000), The Dots-and-Boxes Game: Sophisticated Child's Play, AK Peters, Ltd, ISBN1-56881-129-2.
8. ^Berlekamp, Conway & Guy (1982), 'the 4-box game', pp. 513–514.
9. ^Berlekamp (2000), p. xi: [the 5×5 board] 'is big enough to be quite challenging, and yet small enough to keep the game reasonably short'.
10. ^ ^abcWest, Julian (1996), 'Championship-level play of dots-and-boxes'(PDF), in Nowakowski, Richard (ed.), Games of No Chance, Berkeley: MSRI Publications, pp. 79–84.
11. ^Grzegorzka, Jakub; Dyda. 'Dots - rules of the game'. zagram.org. Retrieved 2017-11-27.
12. ^Wilson, David, Dots-and-Boxes Analysis Results, University of Wisconsin, retrieved 2016-04-07.

Strategy Game Pc

External links[edit]

• Barile, Margherita. 'Dots and Boxes'. MathWorld.
• Ilan Vardi, Dots Strategies.

Dots And Boxes Strategy Game Pigeon Free