Hex vs Square

A tweet got me thinking about hex and square tiles in board games. It is several months old at this point, but that means I’ve given the subject a thorough going-over.

Board-game fans: Hex tile vs Square tile?

— James Hostetler (@metkis) September 23, 2015

You might be interested in this blog post I read titled 20 Fun Facts about Hex Grids.
Or Red Blob Games’ really cool demonstration of some stuff about hexagons.

Neighbours

Obvious difference is a hex tile has 6 neighbours. Squares have only 4, unless you’re counting diagonals, and then you’ve got 8.

Diagonals

Which brings up diagonals. With squares, you’ve got these neighbours that might or might not count as neighbours, depending on how you want to interpret that. In a hex grid, you haven’t got that ambiguity.

In hex chess variants, sometimes they come up with something for the bishops to do, but it feels kind of forced, to me. Diagonals are a natural idea for squares but not for hexes.

For purposes of orthogonal unit differentiation, it’s simple to have pieces that move diagonally, cardinally, or both. With a hex grid, you have to work a little harder to find different movement patterns.

Distances

One thing that’s odd with squares is the difference between diagonal distance and cardinal distance.

The length of the diagonal of a 1×1 square is the square root of 2, or about 1.4. But if you instead just count the number of tiles it takes to traverse the distance, you end up with a different ratio. If you allow diagonal movement, you can travel 40% faster than you would intuitively expect. If you don’t allow diagonal movement, that takes two steps and you’re traveling 30% slower than you might expect.

If you think about distances as the non-tiled distance, a circle contains all points that are equal distance from the centre of that circle. But for square tile-based movement, you’ll get a diamond shape. And for hexagonal tile-based movement, you get a hexagon.



If you find a formula for the number of tiles X tiles away from a centre-point, (doubling back is not permitted) it does work out to 6X for hexagons and 4X for squares. If you were looking for the circumference of a circle, the formula is 2PI*X. 2PI is roughly 6.28318. So 6X comes a lot closer to the non-tiled version than does 4X.

And if you look at the shape of those two, the one based on square tiles makes a diamond shape. If you consider the circumference as travel distance without diagonal movement it would take twice as many moves, and then the ratio is even further off.

So I think if consistency of distance is important in your game, you might prefer hex tiles.

Clustering

One way to measure a connected graph is the clustering coefficient. This is defined by the number of triangles made of three points divided by the number of “triplets.” Triplets in this case are any three points where at least one point is connected to the other two.

According to this value, there’s a dramatic difference between a square grid and a hex grid. A hex grid has a global clustering coefficient of 0.4, a square grid has a global clustering coefficient of 0.

On a square grid, (assuming no diagonals) the smallest loop you can make is a square, with 4 points. So no triangles.

So if you want slightly more or less interconnectedness, consider that in light of this.

Let’s consider two games and how the balance of the game is changed according to these differences.

Carcassonne

I’ll ignore for the moment the river, but each edge of a tile can be either open field, road, or city. The simple formula is 3*3*3*3=81, but you have to take into account some of those are equivalent when rotated. I didn’t think of a more elegant way to do it, but a brute-force approach says there are 24 possibilities when you account for rotations.

If it’s got 6 sides, then 3*3*3*3*3*3=729, but by my brute-force accounting for rotations, that’s down to 130. That is over 5 times as many possible tiles.

In Carcassonne, when you play a tile, you must match it to one or more of the existing tiles, road to road, city to city, etc.

So what percentage of cards will meet the requirements of one edge? Of two? Of three?

Again, I didn’t do anything too fancy, just brute-forced an algorithm that loops through all the possibilities and counts how many could fit.

What I found is if you do a straight comparison of 1 edge hex to 1 edge square, 2 edges hex to 2 edges square, etc. the hex tiles find a match a higher percentage of the time. Which makes sense. You have six ways you can rotate the tile instead of four. But of course, with square tiles, we can have at most four edge requirements, while with hex tiles, we might have up to six. So it starts out easier to make a match, but it becomes much more difficult once you’ve added more sides.

Note there is variation between, for example, field + road and field + field. I didn’t consider every possible combination, so these numbers are imprecise, but they should be roughly representative. The variation within a specified number of sides seems to be smaller than the difference between different numbers of sides and between hex and square tiles, so I think it’s still useful without being more precise.

So if you compare them as matching one edge, matching two edges, matching three edges, then hex tiles are easier to play. But if you compare 3 edges on a hex to 2 edges on a square, because both are 50% of the edges, then hex tiles are more difficult to match.

I don’t have a solid way to prove this right now, but I feel like this would encourage a more disperse layout. A tightly-packed layout requires you match more edges. But I think tightly-packed Carcassonne is the more interesting Carcassonne.

Hive

In Hive you win a game by surrounding the other player’s queen bee. Obviously having 6 neighbours or 4 changes the difficulty of that. Note in the photo the the white queen is surrounded on 3 sides. If a square was surrounded on 3 sides, that’s one move from losing, but as a hex, it’s only half-way.

There’s also a restriction that you can’t move a piece if it’s the only thing keeping two parts of the game connected. So for example here, the white queen and white ant can’t move, but the white beetle or grasshopper could. I think here the possibility of triangles makes a big difference. If a tile and two neighbours form a triangle, then any one of the three can be moved without disconnecting the other two. With square-based tiles, the smallest loop with that property requires four tiles.

Or another way to think about it, if a hex tile has two adjacent neighbours, they are also neighbours of one another. For a square tile, this doesn’t happen, and it requires a further tile to bridge them together. So if you tried to play Hive with square tiles, I think you’d have a lot more pieces that become trapped because they are the only piece connecting other pieces together.

More than two possibilities?

Now these two are the most popular, but if the right circumstance comes up, I think some other tiles could make sense.

Triangular tiles would be pretty straightforward, just alternate triangles pointing up and down. Only three neighbours. Those neighbours are not neighbours, so not very tight clustering. (In fact the smallest possible loop is 6 tiles arranged in a hexagon.) Distances can be twice as far depending on whether you’re lined up with one of three directions.

Five-sided tiles?

There was an announcement that a new tiling pentagon was discovered. There were already 14 different classes of tiling pentagon though there are some complexities here.

None of them are regular (equal-length sides) like our squares and hexes are. So mechanics that rely on being able to rotate the tile in any direction aren’t going to work.

Even though they have five sides, many of the patterns have more than five neighbours. (And some cases they might have different numbers of neighbours.)

Some rely on the tile being mirrored, so whether that’s acceptable for your purposes may vary. If you are printing tiles for a board game, for example, you’d need to print both sides. And it might be an additional complication for players.

Free-form maps?

And of course there are games with maps that don’t use on a single repeating tile, but are more free-form. For example Risk, or Metropolys have boards where at any given location the number of neighbours at that location can vary. That puts more emphasis on the landscape of the board. A tactic that works in one location needs to be adjusted for other locations. So when considering the question “squares or hexes” consider whether you even need things to be tileable.

Here’s the map from Metropolis with the number of neighbours for each written down.

That’s 4 ones, 5 twos, 21 threes, 20 fours, and 5 fives, if you’re curious. Risk (not pictured) has 4 twos, 13 threes, 13 fours, 7 fives, 5 sixes. So, overall more connected than Metropolys. I’ll leave that as an exercise for the reader to consider how changing that would change the balance of the game.