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Dots and Boxes (Game): Solved? Fair? Rigged? Impartial? Let’s Explain

You must be wondering, what type of game is Dot-and-Boxes? Is it a solved or fair game, or would you classify it as a rigged game? Well, here is an in-depth analysis of the game.

In Dots-and-Boxes, players win fair and square. This 2-player fair game ensures that all players have the relevant information. Also, no aspect of play or position on the board is hidden from any player. Both players have equal opportunities to win. As long as both players play optimally, you can predict how the game will turn out from any given position.

What Type of Game is Dots-and-Boxes?

Dots-and-Boxes is a 2-player combinatorial pencil and paper game that, although it seems easy to play, does require strategy

This classic game begins with an unfilled grid of dots. Each player, in sequence, connects adjacent dots using either vertical or horizontal lines. The player who draws the fourth line that closes a box gets to claim that box.

The game ends when all the boxes are completed, and the winner is the player with the highest number of squares.

Rules of Play

  • Begin by drawing dots on a clean sheet of paper to form a grid. The smallest grid size you can create is a 2×2 grid having nine dots. Players also need to know that the grid shapes are not limited to four-sided forms. You can play an unusual variant with a triangular layout too.
  • Next, take turns to connect two un-joined dots either vertically or horizontally.
  • To lay a claim on any box, a player must draw the fourth line that completes the box. Once the box is claimed, the player has to write their initials to mark the box.
  • After winning a box, that player has the chance to draw another line.
  • The game ends when there are no more lines to draw. The initials are counted, and the player with the highest number is the winner.

Dots-and-Boxes is essentially a game of dominance, like in war. Each player aims to win. To do that, each would try to gain control early enough in the game. 

Levels of Play in Dots and Boxes

Out of the several strategies involved in this game, there are five known levels of play.

1. Random Play

As long as a move is legal, the player can make a move at this level. There is no spatial analysis or predictions. There are no levels of look-ahead. Just random moves that connect two dots on the grid.

2. Completing a Box

At this level, spatial analysis is restricted to winning a box at a time and has just one temporal look-ahead level. Positions are tested to predict if a move can create a square. Although a simple way of play, it is highly effective, with a win of 99.63% of games (sample size= 200 000).

3. Avoiding Drawing the Third Side

This level corresponds to two temporal look-ahead levels. Spatial analysis is still confined to a one-box window at a time. Players avoid drawing the third side of a box because it allows the opponent to complete a box. This strategy is also effective, with a win of 99.69%.

4. Conceding Boxes Optimally

At this point, temporal look-ahead and spatial analysis are indefinite. It is more than completing one box at a time. A player has to determine which boxes to give to the opponent to secure their chances of winning.

5. Ignoring a Box

A player can also politely turn down a box to avoid giving away chains or loops to the opponent. Such strong plays are referred to as double-dealing strategies. Here there is no limit to temporal and spatial analysis. This is the highest tactical level. Experienced players commonly use double-dealing strategies.

Is Dots-and-Boxes an Impartial Game?

A game is said to be impartial if it meets these two criteria;

1. The set of moves available to the players depends on the game’s position and not who is playing at the moment.

2. Any particular move has equal value for both players.

The only difference between the two players in an impartial game is that one person plays first. The game continues until no possible moves are left. The player who drew the last possible line is the winner, and the opponent is the loser.

Impartial games are games played with perfect information. 

However, what is ‘perfect’ information?

A game is said to have perfect information if both players make moves in sequence and have complete knowledge of all the game’s previous events.

Both players know and can see what is going on. If any aspect of play is hidden from any player, such a game has imperfect information. Any game that doesn’t qualify as impartial is termed a partisan game.

In a typical impartial game, the player who makes the last move wins. This is the normal play convention in combinatorial game theory. Examples of regular impartial games include Nim, sprouts, draughts, hex, tic-tac-toe, and Y. Chess and Go are not impartial games as each player has pieces of different colors. Games that rely on chance like poker, dice, and dominos are partisan games.

However, the dots and boxes game is different. This impartial game does not follow the normal play convention. The player with the maximum number of boxes wins. This aspect makes Dots-and-Boxes an unusual impartial game.

Is Dots-and-Boxes a Solved Game?

Most impartial combinatorial games are solved using the Sprague-Grundy Theorem, which states that every finite impartial game is equivalent to a nim heap. If you can predict the outcome of a game (a win, loss, or tie) from any position of play, that game is said to be solved as long as both participants make optimal moves.

Combinatorial two-player games can be strongly solved like Tic-tac-toe and Nim. Some have weak solutions, such as Maharajah. Others are termed ultra-weak, such as Hex and Chomp.

Reports from research carried out by mathematicians show that Dots-and-Boxes can be solved using several algorithms.

For instance, if both players play a grid of 4×5 dots perfectly, the outcome will be a draw. This was solved in 2012 and was the largest solved game at that time. In 2014, a 5×5 grid was solved, resulting in a 13-12 win for player 1.

Dots-and-Boxes being a solved game does not make it less interesting for the players. This is because its solution is a bit too complex to be memorized. Strongly solved games with complex solutions are no less interesting than weakly solved games with simple, easily-memorized solutions.

Is Dots-and-Boxes Fair?

Dots-and-Boxes involve fair play. The game is configured so that the same moves are available to both players and nothing is hidden. Both players have several strategies to use to up their chances of winning. This doesn’t make it less fair. All moves are in the open, including the past ones played from the game’s onset.

Is Dots-and-Boxes a Rigged Game?

Playing Dots-and-Boxes requires constant vigilance. You have to avoid drawing the third line early in the game and hope that your opponent does. You have to set up your chances of winning. Knowing your best possible move at all times and how to get out of a double-cross is crucial.

But does this mean the game is rigged? No. If anything, both players have fair chances of winning. The value of each specific move is equal for both players. Nothing is hidden. You only have to play optimally to win.

Final Thoughts

Dots-and-Boxes is a fair game. The rules are clear, simple, and not rigged. Nothing is hidden on the board. It is somewhat different from other impartial games under the normal play convention. Rather than aiming to make the last move, strive to claim as many boxes as possible. The highest number count wins.

It’s all about the boxes. Join the dots, make the boxes, and claim them. Every move you make must be optimal and serve the purpose of securing your win. Don’t expect the opposition to make a mistake because they may be two steps ahead.