Dots and Boxes Variations

Variations That Change the Starting Board

One of the simplest ways to modify Dots and Boxes is to begin with some lines already drawn.

This changes the opening without requiring a completely different game system. Instead of spending the first part of the game deciding how to build the position, players begin with some of the structure already established.

The standard blank board, Swedish board, and Icelandic board are all documented starting configurations in analyses of Dots and Boxes. The Swedish board begins with the outside lines already drawn, while the Icelandic board starts with only the left and bottom borders drawn. These names are used in mathematical and game-theory discussions of the game.

Standard blank board

In the familiar version, every playable edge starts empty.

Players build the position from scratch, so the opening contains the largest number of unrestricted choices. Safe-move management becomes increasingly important as the board fills.

This is the version most people encounter when they first learn the game.

Swedish board

A Swedish board starts with the entire outside border already drawn.

That seemingly small change has a direct effect on the position. Perimeter edges are no longer available as opening choices because they have already been supplied.

The result is a different starting geometry, not merely a standard game with fewer moves. The pre-drawn boundary changes which edges are available and how the interior develops.

Icelandic board

The Icelandic version goes partway between the blank board and the Swedish board. The left and bottom borders are pre-drawn, producing an asymmetric starting position.

Unlike the Swedish board, the starting position is not surrounded by a complete boundary. The asymmetry is itself part of the game because the two sides do not begin from identical edge conditions.

These three starting arrangements are useful to compare because they demonstrate how much the opening can change without replacing the basic box-claiming framework.

Change the Board Shape Instead of the Rules

Another family of variations keeps the basic idea of drawing between nearby dots but changes the lattice on which the game is played.

The familiar version uses a square grid, so the natural regions being claimed are squares. Other lattices create different cells and therefore different completion conditions.

Researchers have studied Dots-and-Triangles and Dots-and-Polygons as explicit variations built from the same general lattice-game idea.

Dots and Triangles

Dots and Triangles replaces square boxes with triangular regions.

Instead of completing four sides around a square, players work with triangular cells. The underlying idea remains recognizable: players draw edges and claim a region when their move completes it.

The change in geometry is strategically significant. A triangle has three sides rather than four, so the relationship between an unfinished region and its remaining edges is different from ordinary Dots and Boxes.

Dickson and Perrier studied Dots and Triangles as a distinct lattice variation alongside Dots and Polygons, examining how strategies from Dots and Boxes relate to these new structures.

Dots and Polygons

Dots and Polygons takes the geometric idea further by allowing players to claim polygonal regions rather than only 1 × 1 squares.

The game still uses a lattice of dots and connecting line segments, but the claimed regions are no longer restricted to the small square cells of the standard game. Dickson and Perrier studied this variation alongside Dots and Triangles and analyzed how familiar Dots and Boxes ideas carry over to the new structures.

This makes Dots and Polygons particularly interesting as a mathematical extension. The board still looks related to Dots and Boxes, but the regions being claimed can have considerably more varied shapes.

Triangular and Hexagonal Grid Variants

The underlying dot arrangement can also be changed.

Instead of placing dots in a square lattice, players can arrange them on triangular or hexagonal-style grids. This changes which dots are neighbors and which shapes can be formed from the available edges.

The important distinction is that this is not simply a larger or smaller Dots and Boxes board. The adjacency structure itself has changed.

On a square grid, every interior dot has four immediate horizontal or vertical neighbors. A different lattice produces a different network of possible edges and different enclosed regions.

As a result, familiar terms such as “box” may no longer describe what players are actually trying to claim. Unusual grids, including triangular and hexagonal grids, are documented extensions of the game.

Narrow Dots and Boxes

The standard game does not have to use a large rectangular field.

A particularly useful mathematical family is 1 × n Dots and Boxes, where the board is only one box high but can extend horizontally across many boxes.

A 1 × n board contains two rows of n + 1 dots and produces n boxes. Researchers have studied narrow boards because their restricted geometry makes certain positions easier to analyze exactly.

Narrow boards are interesting because reducing the board’s width does not merely make the game shorter. It removes many of the spatial relationships available on a normal rectangular board.

That makes the resulting game useful for studying how board geometry affects strategy.

Changing the Number of Players

Dots and Boxes is normally presented as a two-player game, but the game is also documented as being playable by two players or sometimes more.

The important change is what happens to turn order when more than two people share the board.

Suppose three players are using the normal extra-turn rule. Player A completes a box and therefore takes another turn. If A completes another box, that extra turn continues. Once A makes a move without completing a box, the turn passes to Player B, who then has the same opportunity to continue through any boxes they complete.

That means the familiar two-player question of “who gets control next?” becomes a multi-player sequence involving the entire turn order. A scoring run can therefore affect two other players rather than simply handing the position back to one opponent.

This is why two-player strategy should not automatically be treated as optimal multiplayer strategy. The underlying box-completion mechanic remains recognizable, but the consequences of giving another player the next opening are different.

Changing the Extra-Turn Rule

The extra turn after completing a region is one of the defining mechanics of standard Dots and Boxes.

Removing that rule produces a fundamentally different game.

In a no-bonus-turn variation, a player still claims a completed box but the turn immediately passes to the next player instead of continuing. Educational adaptations have used modified versions of the game in which the usual turn sequence is changed to serve additional learning objectives.

This single change removes one of the most important features of standard Dots and Boxes: a scoring move no longer automatically creates another opportunity to score.

That means a long sequence of forced captures does not behave in the same way. The chain-control ideas that dominate standard endgames therefore cannot simply be transferred to a no-bonus version.

Adding Randomness

Another way to alter Dots and Boxes is to introduce an external random element.

Educational adaptations have combined the grid with dice or numbered squares, requiring players to use the result of a roll to determine which move or square-related action is available.

This creates a different balance between calculation and uncertainty.

Standard Dots and Boxes gives players complete information about the board and available legal moves. Once randomness determines part of the move, the player must account for outcomes that cannot be controlled directly.

These adaptations can be useful when the purpose of the game extends beyond pure Dots and Boxes strategy, such as combining gameplay with arithmetic practice.

Strings and Coins

Strings and Coins is closely related to Dots and Boxes but changes the way the board is represented.

Instead of drawing lines between dots to enclose boxes, the game can be represented as a graph in which coins are vertices and strings are edges. Players cut strings, and when a move leaves a coin with no remaining strings, that coin is claimed and the player receives another move.

The important connection is structural: both games can be understood through edges, claimed regions or vertices, and extra turns generated by completing the relevant objective.

Strings and Coins can also be played on arbitrary graphs, which takes the idea beyond the rectangular grid used by ordinary Dots and Boxes.

For someone learning Dots and Boxes, it is better understood as a related graph-game formulation rather than simply another board size.

What Actually Changes From One Variant to Another?

Not all variations affect the same part of the game.

The distinction matters because calling all of these “Dots and Boxes variations” can hide how different they actually are.

A larger board changes the scale of the standard game. A Swedish board changes the starting position. Dots and Triangles changes the claimed shape. A no-bonus version changes a core turn rule. Strings and Coins changes the representation of the underlying game.

Those are different kinds of variation.

Which Variations Stay Closest to the Original Game?

If you want the familiar Dots and Boxes experience with a slightly different opening, the Swedish and Icelandic boards are the closest changes.

They retain the basic square-grid structure and the familiar box-claiming framework while modifying which edges are already present at the beginning.

Changing the grid geometry is a larger step. Dots and Triangles and Dots and Polygons preserve the idea of drawing edges and claiming regions, but the regions themselves and their mathematical behavior change.

Removing the extra turn or adding randomness changes the incentives even more because those modifications affect the way a move influences what happens next.

This gives a useful progression:

The further you move along that progression, the less appropriate it becomes to assume that standard Dots and Boxes strategy will work unchanged.

Dots and Boxes Variations FAQs

Exploring Beyond the Standard Board

Dots and Boxes is more flexible than its familiar square grid suggests. You can change the starting borders without replacing the game, move to a different lattice and create new region shapes, modify the turn structure, or translate the underlying idea into a graph-based game such as Strings and Coins.

The important distinction is knowing what the variation changes. That tells you how much of the familiar game still applies and when you need to approach the new version as a different mathematical game.

For the standard game itself, start with Dots and Boxes. You can also read the Dots and Boxes rules before trying a variation, or explore How to Win Dots and Boxes for strategy on the standard board.