Board game rules

I would like talk about board games again – about game rules specifically. I downloaded some rulebooks and counted words: see the chart above. It can be seen that, in general, games for older players have longer rulebooks.

There are some exceptions, though. For example, Saint Petersburg and 7 Wonders both have very long rulebooks (for 10+ games). This means that new players need advice during the game, and in the case of 7 Wonders are often steered away from the more complex strategies at first. For Dominion, where there is a choice of “action cards” each game, one generally starts with an easier set if a novice is playing. For games like Pandemic, Power Grid, or Forbidden Island, an experienced player will take on the responsibility for the “machinery” that is common to all players.

Conversely, Taj Mahal is an example of a game that is a lot more difficult than the size of the rulebook would suggest. But even simple games can be enjoyable for adults: I would quite happily play Sushi Go! or Kingdomino if offered the chance (in part because the artwork is so much fun). Ticket to Ride is, of course, famous as a “gateway game” for introducing people to modern board games.

As a side issue, many people disapprove of the financial morality Monopoly teaches. More modern games will generally (a) not make money the object of the game and (2) not refer to “dollars.” For example, 7 Wonders refers to “coins,” Dominion has “treasure” cards (which may be “copper,” “silver,” or “gold”), Puerto Rico refers to “doubloons,” Saint Petersburg has “rubles,” and Power Grid refers to the unit of currency as an “elektro.”

I also looked at other keywords (listed below in decreasing frequency). The words “card,” “draw”, “board,” “tile,” “move,” “domino,” “dice,” and “buy” indicate major game mechanisms. For Forbidden Island, Monopoly, Pandemic, Puerto Rico, and Sushi Go! (among others), the text of the rules highlights the “theme” of the game. For example, Forbidden Island has a mechanism similar to Pandemic, but a theme involving a collaborative search for treasure on an island that is gradually sinking beneath the waves (hence the importance of “flood,” “island,” and “water”). Sushi Go! is inspired by Japanese sushi restaurants (hence the importance of “nigiri” and “wasabi”).

  • 7 Wonders: CARD, city, build, age, point, victory, coin
  • Carcassonne: TILE, score, meeple, place, city, point, road
  • Chess (basic rules): MOVE, square, king, piece, pawn, white, black
  • Citadels: district, CARD, character, gold, turn, city, gain
  • Dominion: CARD, action, discard, pile, deck, turn, phase
  • Forbidden Island: CARD, treasure, TILE, flood, island, discard, water
  • Kingdomino: DOMINO, king, kingdom, point, place, turn, line
  • Love Letter: CARD, round, hand, deck, choose, turn, discard
  • Monopoly: property, bank, DICE, house, mortgage, CARD, BUY
  • Pandemic: CARD, city, infect, disease, cube, discard, cure
  • Power Grid: plant, power, city, resource, step, phase, market
  • Puerto Rico: build, goods, colonist, ship, place, occupy, CARD
  • Saint Petersburg: CARD, phase, score, ruble, market, worker, exchange
  • Sushi Go!: CARD, score, point, nigiri, hand, round, wasabi
  • Taj Mahal: CARD, token, visit, place, score, province, point
  • Ticket to Ride: CARD, route, ticket, train, destination, score, DRAW
  • Tigris & Euphrates: TILE, leader, place, kingdom, temple, BOARD, color

Board game taxonomies

I am a huge fan of board games, as I am sure my readers have realised. So today, I would like to introduce a simple taxonomy.

Abstract combat games

For the moment, I will focus attention on 2-player abstract combat games like Chess (rather than games which are primarily race games, like Backgammon). I include games based on blocking as well as games based on capture, but not games played with dice, such as Daldøs or Chaturaji.

The first question is: are the pieces of one player all the same at the start of play? For games in the top half of the chart, like Checkers (Draughts), the answer is Yes; for games in the bottom half (like Chess), the answer is No.

A slightly different question is: is the collection of initial pieces of one player identical to that of the other player? For most games (in the left half), the answer is Yes; for games in the right half, like Fox & Geese or the obscure Nosferatu (where one side has “pawns” and a “king”), the answer is No.

A third question is: do the initial pieces remain the same during play? For games in the outer columns, like Go, the answer is Yes; for games in the centre half (games with piece promotion, like Checkers and Chess), the answer is No.

For games not illustrated, one can add Alquerque or Surakarta or Five Field Kono or Gomoku to Mū Tōrere and Go; Chaturanga to Chess; Tiger & Goats (Bagh-Chal) or Asalto or Hnefatafl to Fox & Geese; and the rather odd Owlman to Nosferatu.

Race Games

I now turn to race games, played on a track with dice and/or cards as a random element, where players race to reach the end. The game of Chaturaji has sometimes been seen as an intermediate between race games and the abstract combat games we have just looked at.

The first question is: how many pieces does a player have? For games in the top half of the second chart (like Backgammon or Ludo), the answer is Several; for games in the bottom half (like Snakes & Ladders), the answer is Only one (in which case the sole piece is really a token or meeple). This last category is really boring, unless it is supplemented with other game elements, such as the question cards of Trivial Pursuit.

A rather different question is: are pieces captured during play? For most games (in the left half), the answer is No; for running-fight games in the right half (like Daldøs or Fang den Hut), the answer is Yes.

A third question is: how many players are there? For games in the outer columns, like Backgammon, the answer is Two; for games in the centre half (like Snakes & Ladders or Ludo), the answer is Several.

For games not illustrated, one can add the Royal Game of Ur or Ludus duodecim scriptorum to Backgammon and Senet. As in the previous chart, the top left corner is the most popular.


Board games!

I love board games. I really, really, love them. The chart below shows my least favourite game (Monopoly) and some personal favourites:

The chart shows the rating on boardgamegeek.com, the age range, the number of players, the approximate time to play the game (games with a predictable duration are better in social terms – particularly if you want to finish a game before dinner), and three other quality characteristics:

  • Artistic quality: I am rating this subjectively, but it’s always nice to have something beatiful to look at while you are waiting for your turn.
  • Educational value: Ticket to Ride and Pandemic teach geography, 7 Wonders teaches something about history (especially with the Leaders expansion), and Kingdomino teaches multiplication (I give partial credit to Forbidden Island for being a collaborative game, and negative credit to Monopoly for teaching that money is the most important thing in life).
  • Winning: ideally, all players have a chance of winning right up to the end (I give extra credit to Kingdomino for having a mechanism that helps players that “missed out” on a good tile, and to Pandemic and Forbidden Island for being collaborative).

It can be seen that Monopoly scores badly in every possible way: it can take forever; the artwork is poor; it teaches bad moral lessons; and players are actually eliminated from the game during play. That is why I dislike it.


Snakes and Ladders


Snakes and Ladders board, dated 1966, from the Auckland Museum (credit)

Snakes and Ladders is an ancient board game originating in India. It is totally random, and hence not very interesting. If players start on square #1, then after one turn, they have equal probabilities of being on squares #2, #3, #4, #5, #6, and #7. This image shows the probability distribution:

After two turns, the probability distribution is as follows (the most likely total of two dice rolls is 7, taking a player to square #8 and up a ladder to #26:

After 8 turns, players would be scattered all over the board. There is a 1% chance that any given player has won:

After 19 turns, there is a 24.7% chance that any given player has won:

This probability grows to 50.4% after 35 turns. But no matter how long you play, it remains possible (though increasingly unlikely) that nobody has won yet. Yet another reason why children tend to rapidly tire of the game.

For an alternative view of the probability analysis, see this animation:


The Game of Mu Torere

The New Zealand game of mū tōrere is illustrated above with a beautiful handmade wooden board. The game seems to have been developed by the Māori people in response to the European game of draughts (checkers). Play is quite different from draughts, however. The game starts as shown above, with Black to move first. Legal moves involve moving a piece to an adjacent empty space:

  • along the periphery (kewai), or
  • from the centre (pūtahi) to the periphery, or
  • from the periphery to the centre, provided the moved piece is adjacent to an opponent’s piece.

Game play continues forever until a draw is called (by mutual consent) or a player loses by being unable to move. Neither player can force a win, in general, so a loss is always the result of a mistake. For each player there is one “big trap” and four “small traps.” This is the “big trap” (Black wins in 5 moves):

  
The board on the left is the “big trap” for White – Black can force a win by moving as shown, which leaves only one move for White.

  
Again, Black moves as shown, which leaves only one move for White.

  
Now, when Black moves as shown, White cannot move, which means that White loses.

Here is one of the four “small traps” for White. The obvious move by Black results in White losing (but avoiding this does not require looking quite so far ahead as with the “big trap”):

Here (click to zoom) is the complete network of 86 game states for mū tōrere (40 board positions which can occur in both a “Black to move” and a “White to move” form, plus 6 other “lost” board positions). Light-coloured circles indicate White to move, and dark-coloured circles Black to move, with the start position in blue at the top right. Red and pink circles are a guaranteed win for Black, while green circles are a guaranteed win for White. Arrows indicate moves, with coloured arrows being forced moves. The diagram (produced in R) does not fully indicate the symmetry of the network. Many of the cycles are clearly visible, however:


Revised SOLAR RACING board game!

In honour of the upcoming World Solar Challenge, I’ve revised my SOLAR RACING board game (see picture above). Game play is more interesting now, as game play uses a deck of 54 cards, rather than just dice. The cards are illustrated with 16 photographs of central Australia and of previous races. If you’re interested, game rules can be downloaded from the game page (see the link in the “Downloads” section).


Chemical Compounds: the board game!

I have previously mentioned my strong interest in science / technology / engineering / mathematics education and in networks and in board games. This has prompted me to start designing educational games, such as the World Solar Challenge game. Joining the collection is my new Chemical Compounds game, which looks like this:

The online game store (faciliated by the wonderful people at The Game Crafter) has a free download link for the rules, should anyone wish to take a look. I also have a few other educational games there.


World Solar Challenge: the board game!

Readers of this blog will know that I am passionate about science / technology / engineering / mathematics education, and that I am passionate about board games, and that I am passionate about solar car racing (with the ESC and the Sasol Solar Challenge coming up soon). Wouldn’t it be great if those three things could be combined?

Well, now they can! To assist solar car teams with education/outreach efforts, I’ve put together a simple board game based on the World Solar Challenge, and aimed mostly at kids. It looks like this:

The online game store (faciliated by the wonderful people at The Game Crafter) has a free download link for the rules, should anyone wish to take a look. I also have a few other educational games there.


Games: the Good, the Bad, the Ugly

I recently redrew a classic graph by Oliver Roeder from fivethirtyeight.com, showing the ratings of various board and card games at boardgamegeek.com. These ratings run from 1 (“Defies description of a game. You won’t catch me dead playing this. Clearly broken.”) to 10 (“Outstanding. Always want to play, expect this will never change.”). I have used the same dataset (downloaded by Rasmus Greve in 2014, so slightly old now), but removed games rated by less than 100 people, leaving a total of 5121 games. The average rating for these games is 6.42 (or 6.92 for the average weighted by number of ratings).

I’ve labelled three kinds of outlier in the graph above, and listed the corresponding games below. The Frequently Rated Games on the right are rated often because they are played often, and so they are generally very good games (the graph shows a weak correlation, reflecting this popularity–quality link). These games include Carcassonne (a superb family game, because very young children can join in if they are given hints about the best move), Dominion (my favourite card game), and Pandemic (one of the best collaborative games). Overlapping with this category are the Highly Rated Games at the top, some of which are aimed at hard-core gamers, while others (like Puerto Rico) are more widely popular. It should be noted, however, that game expansions tend to get deceptively high ratings, since they are generally only played by fans of the original game.

At the bottom are a number of Poorly Rated Games, which (sadly!) includes many of the games I grew up with. These flawed games include those which are too simple (Tic-Tac-Toe, Battleship); which are too heavily based on chance (Snakes and Ladders, Risk); which eliminate players before the end of the game (Risk, Monopoly); which take an unpredictable amount of time (Risk, Monopoly); or which have other problems. This category includes seven of the eight games on this list of flawed games by Ben Guarino, and all six in this post by Luke McKinney (which recommends Power Grid, Settlers of Catan, Ricochet Robots, Alien Frontiers, Ticket to Ride, and King of Tokyo as substitutes for Monopoly, Risk, Battleship, Connect Four, The Game of Life, and Snakes and Ladders, respectively).

Frequently Rated Games

Highly Rated Games

Poorly Rated Games

Mathematics in action: Risky Random Walks


A game of Risk (photo: A.R.N. Rødner)

The board game Risk, though far from being my favourite game (and rated only 5.59/10 on Board Game Geek), nevertheless has some interesting strategic aspects and some interesting mathematical ones.


Combat units in Risk (photo: “Tambako The Jaguar”)

A key feature of the game is a combat between a group of N attacking units and a group of M defending units. The combat involves several steps, in each of which the attacker rolls 3 dice (or N if N < 3) and the defender rolls 2 dice (or 1 if M = 1). The highest value rolled by the attacker is compared against the highest rolled by the defender, and ditto for the second highest values, as shown in the picture below. For each comparison, if the attacker has a higher value, the defender loses a unit, while if the values are tied, or the defender has a higher value, the attacker loses a unit.


Comparing attacker (left) and defender (right) dice in Risk (photo: “Val42”)

Working through the 65 possibilities, the attacker will be down 2 units 29.3% of the time, both sides will have equal losses 33.6% of the time, and the attacker will be up 2 units (relative to the defender) 37.2% of the time. On average, the attacker will be up very slightly (0.1582 of a unit). A fairly simple computation (square each of the outcome-mean differences −2.1582, −0.1582, and 1.8418; multiply by the corresponding probabilities 0.293, 0.336, and 0.372 and sum; then take the square root) shows that the standard deviation of the outcomes is 1.6223.

When this basic combat step is repeated multiple times, the result is a random walk. For example, with 10 steps, the mean attacker advantage is 1.582 units, and (by the standard formula for random walks discussed in a previous post) the standard deviation is 1.6223 times the square root of the number of steps, i.e. 5.1302.

The histogram below shows the probability of the various outcomes after 10 steps, ranging from the attacker being 20 units down (0.0005% of the time) to the attacker being 20 units up (0.005% of the time). Superimposed on the plot are a bell curve with the appropriate mean and standard deviation, together with five actual ten-step random walks. While the outcome does indeed favour the attacker, there is considerable random variability here – which makes the game rather unpredictable.