Hexagon Boards vs. Square Boards Explained
Why board game designers often prefer hexagons
The two most popular board games have to be Settlers of Catan and Monopoly. In many ways, they’re opposites. One is described by board game aficionados as “the perfect board game.” The other is described as, uh, “terrible.” One has inspired legions of fans, spinoff games, and even a novel. The other has inspired legions of angry table flips.
One uses a hexagon board. The other uses a square board.
You often hear that hexagon boards are simply “better” for strategy games. The Civilization video games switched from a square map to a hexagonal map starting with Civilization V. Hexagonal chess was popular in Eastern Europe for decades. Even Monopoly produced a hexagonal version of their game, U-Build Monopoly, in 2010.
But you don’t hear much about why hexagons are preferred. So let’s talk about the differences between square boards and hexagon boards. We’ll start with a super simple example.
The Disadvantage of Square Boards, Made Simple
Imagine you’re a hero trapped in a dungeon. Your name is Quackerton.
The dungeon is dark, so Quackerton can only see one tile in each direction. Tiles you haven’t seen are colored gray.
Quackerton can move one square at a time, in any direction.
Let’s say Quackerton moves 1 tile south:
This reveals 3 new tiles (colored green).
But what if instead, Quackerton had moved 1 tile northeast, diagonally?
5 tiles are revealed instead of 3. Clearly, moving diagonally is more rewarding than moving horizontally or vertically.
Now let’s put Quackerton in a dungeon tiled with hexagons:
Here’s what it looks like if Quackerton moves one tile south or one tile northeast:
No matter which way Quackerton moves, three new tiles are revealed.
On a square board, moving diagonally is fundamentally different than moving up, down, left, or right. But on a hexagon board, all movements are balanced. That’s the main advantage hexagon boards have over square boards.
Using Graph Theory on Board Games
So why does this happen?
On a square board, there are two types of adjacent tiles:
“Side tiles” that border you on one side.
“Corner tiles” that border you on one corner.
Instead of visualizing each tile as a tile, we could visualize each tile as a dot. If it’s possible for Quackerton to move from one tile to another, we’ll draw a line connecting those two dots. Do this for all 8 tiles around Quackerton, and we get a graph that looks like this:
Next to each dot, I wrote the number of lines connected to that dot. In graph theory terms, the dots are called vertices, the lines are called edges, and the number next to each vertex is called the degree of that vertex.
Corner tiles (blue) have degrees of 3, but edge tiles (red) have degrees of 5.
Let’s say Quackerton moves to the blue dot in the top-right corner. On the board, this space would be adjacent to 8 other tiles (every tile on a square board is adjacent to 8 other tiles). But 3 of those tiles are already visible to Quackerton. So when Quackerton moves to the blue dot, 5 new tiles will become visible. Similarly, if Quackerton moves to a red dot, 3 new tiles become visible.
Here’s the same graph for the 6 tiles adjacent to Quackerton on a hexagonal board:
In a hexagon tiling, there’s no distinction between “side” tiles and “corner” tiles. There’s only one type of adjacent tile. Any direction Quackerton moves in will reveal 3 new tiles. This is why movement on a hexagon tiling is balanced, while movement on a square tiling isn’t. In square tilings, there are 2 types of adjacent tiles to worry about. In hexagon tilings, there’s only 1 type.
Warped Distance
You might’ve heard that square boards are disadvantageous because of the distances between the different tiles. If moving one space horizontally or vertically moves you “1 inch,” then using the Pythagorean theorem, we can show moving diagonally moves you “sqrt(2) inches.”
This is true, but it’s technically not distances that cause diagonal tiles to behave differently. Because what if I stretched the square board like this?
Moving to a red tile clearly moves you farther, but moving to a blue tile would still reveal 5 new tiles. This is because I didn’t change which tiles are adjacent to each other. The distances between tiles are different for square boards, but that’s not what makes square boards imbalanced. It’s the varying numbers of adjacent tiles (the degrees) that matter.
Making a game with a square board is not bad. Square boards are (arguably) more intuitive than hexagon boards. As a designer, you just have to be aware of the two types of movement on a square board, and design accordingly. Civilization switched from squares to hexagons because exploration is a huge part of that game, and hexagons make more sense for that concept. Chess uses a square board, but has different pieces that move across the board in different ways—some pieces can only move diagonally (bishops), some can only move horizontally or vertically (rooks), and so on.
Triangle Boards Make Me Want to Scream
On top of hexagons and squares, there’s actually a third way to tile a board: triangles.
Let’s put Quackerton in a dungeon one more time (sorry little duck). This time, the dungeon is tiled with triangles.
Quackerton can see the 12 tiles adjacent to them. Let’s see what happens if Quackerton moves to tile A, tile B, or tile C:
All three moves reveal a different number of tiles!
Hexagon boards have 1 type of adjacent tile. Square boards have 2 types: “Side” tiles and “corner” tiles. Triangle boards have 3 types (which I did my best to name, but honestly it’s a mouthful):
“Side” tiles: tiles that share a side with your tile.
“Same-orientation corner” tiles: Tiles that share a corner with your tile, where the triangle has the same orientation as your tile.
“Inverted-orientation corner” tiles: Tiles that share a corner with your tile, where the triangle has the opposite orientation as your tile.
If we made a graph of the tiles like we did before, we get this monstrosity:
From this graph, we see that moving to an “edge” tile reveals 3 spaces, moving to a “S.O. corner” tile reveals 6 tiles, and moving to a “I.O. corner” tile reveals 7 tiles.
Triangle boards aren’t intuitive like square boards, and they aren’t balanced like hexagon boards. It’s the evil Hannah Montana of board gaming, giving you the worst of both worlds.
Board games with triangular boards are not common, but there are a few out there. The 2006 board game Blokus Trigon is one example.1 Chess also has a triangular variant, invented by George R. Dekle.2 Reading the rules for triangular chess gives me a headache. This is how a bishop moves in triangular chess:
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Since there are three types of movements you could make on a triangular board, it’s even more challenging to design a board game that uses a triangular board. You have to somehow make sure all three types of movements are balanced while still making a game that’s intuitive enough to play.
So that’s why-
You alright Quackerton?
Are you mad I kept putting you in those dungeons?
…What’s that?
…You just hate the mathematical properties of triangular boards? …Yeah, fair. Let me get you a sandwich.
Conclusion
Hexagon boards are preferred over square boards because they’re more balanced, but does this mean square boards are bad? Of course not. There is no right or wrong answer ( …okay, maybe triangular boards are a wrong answer).
The most prestigious award for board games is Germany’s annual Spiel des Jahres award. Looking through the nominees over the years, it’s not hard to find games with square boards (2017 winner Kingdomino, 2018 winner Azul) or games with hexagon boards (2012 winner Kingdom Builder, 2022 nominee Cascadia), or games with unique boards that don’t fit any tiling (2006 winner Thund und Taxis or whatever the hell is going on in 2005 winner Niagara).
So which board is preferable? Whichever one works best for the kind of game you want to make. While designers should be aware of the differences between square, hexagon, and triangle boards, at the end of the day a board game should be fun. Fun can happen on any shape.
I’m no marketing expert, but doesn’t “Blokus Trigon” sound like a made up name? That name’s a pile of consonants. It would be like naming an esports team “Fredit Brion” or something.
Dekle is best-known as the lead prosecutor for the murder trial of Ted Bundy. I’m not making that up. The guy who gave Ted Bundy the death penalty is also the inventor of triangular chess.