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Ruminations about “Magical Numbers” in boardgame design

In the course of designing “nano-games” that have between 17 and 20 pieces for two sides and very small boards that fit on part of a postcard, I again had occasion to wonder if there is some kind of "sweet spot" or "magical number" of pieces and spaces for a boardgame (or any other kind, for that matter) where pieces occupy locations?

I’ve seen several queries from novice designers asking if there’s an ideal board size, and my immediate reaction has been “it depends”. This question could be expanded to “is there a sweet spot in the number of pieces, compared with board size, in a game that involves maneuver/placement and geospatial location.” For games that are not about maneuver/placement and location, such as many modern boardgames and most card games, the question doesn’t apply. For RPGs and other games where you have an avatar, a single “piece,” as in many kinds of video games, the question also doesn’t apply, although we could ask if there is some kind of sweet spot for the number of characters in an adventuring party. But that question becomes more a matter of psychology than of game.

Let’s look first at traditional non-commercial boardgames. Chess and checkers have 64 spaces, and (respectively) 32 (16 per side) and 24 (12 per side) pieces. Technically the checker board is 32 spaces, since you play on only one color. So the ratio of pieces to board is 1:2 in chess and 2:3 in checkers, and always goes down as pieces are captured. In Tic-Tac-Toe the ratio is ultimately 1:1. In the Roman Empire placement and maneuver game Nine Men’s Morris there are 24 intersections for the 18 pieces, a ratio of 3:4. (Tic-Tac-Toe is actually a much reduced variation of Nine Men’s Morris; both are draws when played perfectly.) Backgammon has 30 pieces and 24 locations, differing from these other games in allowing more than one piece per location. In such instances it isn’t surprising that the ratio is 5:4.

Leaving the games of the West, in African Mancala, which is a boardgame even though it’s a board you can easily dig in the sand, there are many more pieces (stones) than there are board spaces. Oware, for example, has 12 spaces and 48 pieces (seeds). There are relatively few board spaces compared with other games I’ve mentioned. Shogi (Japanese chess) has 40 pieces and 81 squares for a ratio close to 1:2. Xiangqi (Chinese chess) has 32 pieces on a board of 90 intersections, for a ratio close to 1:3. In the Asian game Go, played on the intersections of a 19 x 19 board (361 spaces), we have a situation closer to Tic-Tac-Toe where you could leave all the captured pieces on the board and fill the board up. It may be no accident that Tic-Tac-Toe and Go are placement games rather than maneuver games. If you can’t move from one place to another then the only way to occupy a particular place is to fill it up with a piece.

Intuitively, you might expect that fewer areas means the game is simpler to deal with, though the game may still have a great deal of gameplay depth. You might also ask how much are the limited numbers of pieces and spaces simply a reflection of manufacturing limitations in ancient and medieval times?

Let’s look at some traditional commercial games for comparisons. Many traditional commercial games such as Monopoly, the Game of Life, Clue, and Careers have just one piece per player, a kind of avatar. Many of these games are not actually games of maneuver/placement and location; Monopoly has location but the players have no control over where they are so there’s no maneuver. Careers and possibly Game of Life have diverging paths but most of the time there is no choice about where your avatar goes.

How about "classic" area-based wargames? Risk has 42 areas. Britannia has 37, plus 5 sea areas that can only be temporarily occupied and where combat cannot take place. Vinci has 45 (by quick count). Diplomacy has 56 land regions and 19 sea regions (75 total) for 34 pieces. History of the World and Axis and Allies have many more areas (and original Nova Games’ A&A had a lot).

How about piece to area ratios? The average for Britannia depends on the era, but is roughly 55 pieces for the 37 areas, about 3:2. But as with several of these games, one area can hold more than one piece. The ratio in Vinci is something over one piece per area. In Risk it's a lot higher, at times, as massive armies build up. Diplomacy's ratio is much like that of chess, one piece for a little more than two areas (and no more than one piece per area). Unlike chess and checkers, the ratios in Diplomacy don’t change much over time because the overall game economy is slightly positive and then stabilizes.

My motto in game design is "A designer knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away" (Antoine de Saint-Exupery) or in another form, about Japanese gardening, "Your garden is not complete until there is nothing else that you can remove." Consequently, nowadays I look for the smallest number of pieces and smallest board that gives the experience I have in mind.

For example, I have a couple prototypes that are vaguely Stratego-like. One dates back about 30 years, the other is quite recent but grew out of the first. I’ll describe the latter one below, and how I discovered I might need to reduce the number of pieces. The older one is a space battle game played on squares, and when I first designed it I stuck with something near the typical Stratego 40 pieces per side, though on a more roomy board. But when I resurrected it several years ago, when it was played mostly by video game students, I cut the number of pieces down to 19 and eliminated some of the special pieces - they became options that we rarely used. The ratio of pieces to board became 38:113 or about 1 to 3. This made for a much quicker game, as little as 15 minutes, that was just about as interesting and certainly as fun to play as the big version.

But this cutting back is harder to do when you're trying to model a particular historical situation such as a battle or war. Going back to the “nano-games”, the postcard games, it’s quite difficult to provide an historical model with a very small number of pieces and very small board - you can really only focus on a few things. One reason why the traditional games mentioned above can have small numbers of pieces in relatively small boards is that many are abstract games rather than models of any reality. Even Diplomacy gives the appearance of World War I but virtually none of the reality. Risk is very far from reality. In the commercial game world we really didn’t have boardgames intended to be models of reality until Avalon Hill wargames came along in the late 1950s. (We can probably think of isolated examples before then.) Before then, only wargames played by the military tried to reflect some reality.

What about the numbers of pieces and numbers of playing spaces in relation to human comfort and human capacities? This brings to mind the famous journal article “The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information” http://en.wikipedia.org/wiki/The_Magical_Number_Seven,_Plus_or_Minus_Two. It discusses the number of things a human can keep in “working memory” at one time (barring memory tricks in specific instances). I don’t know of other research of this kind, but I’d be willing to predict that if average human capacity has changed it has gone down, rather than up, as we live in a world of more and more distractions and more and more multi-tasking leading to more and more routine mistakes and inability to focus. Human memory capacity more generally seems to continually fall, for lack of practice if nothing else. Where people in non-literate societies had to memorize any information they needed to retain, we went to writing things down, and now to recording them digitally. The joke used to be that old people had a disease called “CRS” (can’t remember “stuff” - or perhaps some other “s” word), but my observations of college and high school students is that they have CRS quite badly even as they usually refuse to write things down in class.

Whatever research may show, I think it’s fair to say that 21st century game players by and large prefer fewer rather than more assets to keep track of, and less rather than more memorization. There are lots of exceptions, of course. Yet I’m put in mind of a very intelligent freshman game player who said she didn’t like wargames because there were too many decisions, too many things to keep track of. For her it became too much like work, no longer entertainment, though she was very capable of doing it. I myself quit playing chess at age 15 because it was too much like work, if you wanted to be any good.

Perhaps one reason why Diplomacy has been so popular and durable is the small number of pieces for each player. Players start with 3 or (in one case) 4 pieces. Even a winning player probably averages no more than 10 pieces over the course of a game (18 is a certain win). And in the much more common draws (2-, 3-, or 4- way for a 7 player game) a player averages even less.

When considering numbers of pieces we should also look at how many different kinds of pieces there are. Most of these games have just one kind of piece, the army in Risk or Vinci or the piece in Nine Men's Morris. Japanese chess has eight different types of pieces, Western chess six types. In between we have Diplomacy with two types of pieces, armies and fleets.

I think the variety of pieces is sometimes almost as important as the number of pieces. Clearly variety appeals to those who play Western and Eastern versions of chess.

One way to avoid the potential problems of using a variety of pieces is to avoid exceptions to the general rules. Going back to Stratego and my prototype, both games have lots of different pieces, but Stratego is almost entirely hierarchical, so that there’s no difference between a “2" and a “7" (using the old numbers where the 1 is the strongest piece) other than its strength. Only the engineers, scouts, bombs, spy, marshall, and flag are “unusual”, though many of them also participate in the hierarchy. And each of them offers just a single exception to the rules. There’s a hierarchy in my prototype as well, but submarines and aircraft make it considerably more complex, as some ships can sink the subs, subs can sink some ship types if the sub is the attacker, and the aircraft are all over the place and may combine in attack (two can move at once, remember). In other words, there are a lot more exceptions.

I saw what we might call “resistance to too many choices” in relation to one of my playtest prototypes (World War II naval) vaguely related to Stratego. A casual player who loves Stratego, and played well, quickly buckled when playing my prototype. There were too many choices, too much freedom of movement. Stratego has lots of pieces, with a piece to board space ratio of 80:92. The number of movement choices is very much limited by the board and setup. In my prototype, though there were fewer pieces (25 per side for a total of 50), there were many more hexagons (13 by 12 for 156, piece to board ratio 1:3); the length of the “border” between the two players was much longer (12 or 13 hexes instead of 6 squares); and all pieces had a choice of moving two in a straight line as well as one space. (Aircraft can move in segments, and two can move in a turn, much more complex than scout movement in Stratego.) Even the winning conditions are more complicated in my game. In Stratego you only have to take the non-moving enemy flag. In my game you need to get a merchant ship to the other side of the map, or wipe out all the four opposing merchant ships, which can move of course.

In both games the uncertain identity of opposing pieces helps avoid over-analysis, thus simplifying the mental task. Seeing the casual player’s reaction was quite a lesson to me, as I had thought that fewer pieces would simplify the game and bring it closer to the mass-market. But it became clear that I’d need to reduce the number of pieces even further, given the other parameters. (Veteran wargamers have no problem with 25 pieces per side, since they’re only moving one or two at a time. Imagine how tough it would be on casual players if all the pieces could move at once, as in traditional hex and counter wargames.)

I’ve described this at some length because the heart of my inquiry has become, what do we do in designing games that makes them less like the classics, or more important, makes them less comfortable for players? It’s not just choice, or chess would not be as popular as it is. It’s a combination of things.

Games with no uncertainty, such as chess and checkers, promote “analysis paralysis” as a player tries to find the very best move: because there IS a very best move, which is not true of most games with uncertainty (and which is why chess and checkers are essentially puzzles, because there’s an always-correct solution). For most purposes you wouldn’t want to design a game today that lacks uncertainty. (Of course, in the real world uncertainty is ever-present; games without uncertainty are usually abstract, and abstract games are hard to sell.)

(And as an aside, if chess did not exist and was published today, I think it would be just another not-very-successful game, not a hit. It promotes severe “analysis paralysis”, which is why we have chess clocks, and there’s a huge advantage to first-mover. Moreover, it ends in a draw about a third of the time. Plus, abstract games are hard to sell. And it only accommodates two players (I know three and four player variants exist, but likely don’t work well by the nature of the game). Doesn’t sound good, does it?)

With the above in mind, about a year ago I tried to figure out what an “ideal” introductory wargame would be like. One of the pretty obvious characteristics was “10 to 15 pieces per side”. And not moving all the pieces at once. (In fact, I realize I arrived at close to the same solution as Shenandoah’s Battle of the Bulge iPad game, that only pieces from two or three areas could move in one turn.) I also think the number of spaces must be limited, thus prohibiting the prospect of a board of hundreds of hexagons. In other words, keep it to the kinds of parameters we see in most of the games mentioned above. If the game can have both strategic and tactical elements it is a better introduction to the possibilities of typical wargames. I’ve ended up with a (two part geomorphic) area board for strategic movement, and a hex board for battles. There’s also an economy, which is what separates war games from battle games, though the economy does not dominate play.

Unfortunately I haven’t been real excited by playtest results, perhaps because the tactical element (which I adapted from a space wargame prototype where it is quite popular) isn’t yet perfected for land battles. So I’ve set it aside for some months.

As I said in the title, these are ruminations, I have no hard-and-fast answers to my queries. So rather than offer a solution, I’ll just end here.

Comments

RPS and Structural Numbers

Quote:
There’s a hierarchy in my prototype as well, but submarines and aircraft make it considerably more complex, as some ships can sink the subs, subs can sink some ship types if the sub is the attacker, and the aircraft are all over the place and may combine in attack (two can move at once, remember). In other words, there are a lot more exceptions.

It's simply a rock paper scisor relationship which is more complex. For example, in advance wars, fighters can only attack planes and helicopters, bombers can only attack ground units, helicopter can attack ground and other helicopters while SPAAG can attack helicopers but might not be enough for high altitube bombers, but SPSAM can do the job. So it's only a matter of remembering a rock paper scisor ralationship between the units. The more units the more complex it becomes.

The other problem is the multiple weaponry. In dai senryaku, the game is much more realistic, consequence is that units has multiple weaponry configuration which changes their RPS relation. For example fighters can also have bombs, or harpoon missiles to target ships. While submarines can also use harpoon missiles instead of torpedoes and they can even use BGM missile that attacks ground target very far away. So at this point, the RPS relationship becomes way more complex than the usual and units are much more versatile.

But is you keep a dozen of units with 1 type of main weaponry, it should not be much of a problem.

--- Structural Number---

About magic numbers, what I normally use is a structural number. Which means a number that will occur multiple times though the course of the game. Normally, it's a number relatively small: between 2 and 8. Not only it creates a structure to the game making it looks like if the elements are working together but also makes it easier for the player to remember. Here are some examples:

In my game Fallen Kingdoms, the structural number is 3. 6 is also used intensively in the game, for example having 6 groups of 3 is common. This number appear the following ways through the game:

3 type of resources
3 type of trophies
3 building types
3 production track with size in multiples of 3 (3-6-9)
3 Card in each of the 6 knowledge category (6x3 = 18 cards)
3 Rumor cards for each value (3x2, 3x3, 3x5, 3x8, 3x13, 3x21 = 18 cards)
3 Invasion paths can be used
3 dice are rolled during invasions

In my Master of Magic board game idea, the structural number was 5 because they were 5 sphere of magic like in magic the gathering. Si 5 appeared as:

5 sphere of magic
5 type of Natural and 5 type of magical resources
5 actions groups could be done each turn

I normally place numbers in relations with each other. For example 3 and 5 or 5 and 8 strengthen the relation using the fibonacci sequence. So for example, players chose 3 action groups out of 5, so the 3-5 made a fibonacci connection between both values.

I thought I never used the number 4 again while I liked so much when I was younger, but in an adventure game of mine I wanted to have a symmetry between the physical and mental stats, so an even number was essential. I ended up using number 4 to have a 4 quadrant stat system where left is physical, right is mental, top is strength, and bottom is dexterity. Besides the stats, the number 4 was used for other sauces:

4 type of creatures
4 type of special powers

So my Magic Numbers are normally used for structural numbers. And I'll have to say that most of the games out there does not have any structural numbers which makes them feel incomplete or imperfect. The elements does not look like if they belonged together because there is not integrity or consistency between the elements of the game.

Anytime you have one on one

Anytime you have one on one unit combat and no chance element, you can try to describe it as rock-paper-scissors if it isn't strictly hierarchical.

I don't see how using the same number throughout the game makes any difference (over not being so consistent) other than for the tidy-mindedness of the designer! That is, if it makes you more comfortable, fine, but I see no pseudo-numerological significance in it.

I recognise this. in my

I recognise this. in my design behaviour.
I had the 6 throughout my entire game. Although, it doesn't always work that way for certain games to be properly balanced. It would be fun though, if a game could entirely be designed around 1 number. But be wary of the limits that these designs hold.

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