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:


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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.