Three Men's Morris

The board for Three Men's Morris

For my birthday recently my daughter Ruth sent me the rules of the game Three Men's Morris. In order for me to play the game, she also sent six small shells that she had picked up off a beach in Uist, three of a cool grey colour cast and three of a warmer colour. The game is played with three counters per person on a board with nine points. The objective is to be the first to get your three counters in a straight line, something that is called a "mill". Players take turns placing pieces until all six are on the board. Once all counters are placed, players move one per turn to an adjacent, empty intersection following the lines on the board, aiming to be the first to align their three pieces in a row; a move that wins the game.

It's an ancient game and its moves reminded me of the dynamics of the square as set out by Rudolf Arnheim. He believed that everything we see is part of a visual field of forces and these forces are attractors that force us to look in certain ways. Using drawings he demonstrated how the formal elements that underpin visual thinking are integrated into fields of visual force dynamics. His diagrams pointed to the fact that perception and thought act reciprocally and after reading his text and looking at his drawings, I did find that I had become much more aware of the energy fields that emerge from a geometric form's inner dynamics.

The field dynamics of a square according to Rudolf Arnheim

In the diagram above, the circles represent centres of attraction and how they reach out into the spaces around them and the lines visual forces and their direction. 

Playing the game Three Men's Morris, in some ways re-enacts Arnheim's idea. You gaze at a board that is marked out in the same way as Arnheim's square's emergent energy fields. Each move in effect destabilises the dynamic and as you play you are constantly looking for patterns that seek cohesion, in the case of Three Men's Morris, joining three points together. In effect by playing the game you reveal the underlying structural skeleton of a square. The counters, or in my case small shells, appear to move along the structural axis of the overall form. As they do so they create a 'flow', that moves through the square's focal points, leading your eyes' movements through one part of the square to the next. This 'flow' is a beautiful example of Arnheim's idea that perception and thought act reciprocally, as you think of the next move, you are at exactly the same time 'seeing' the pattern that is both there and implied. 

Perception and thought are tied together, but at some point within the timeframe of almost instantaneous action/reaction decisions, I suspect Bayesian probability theory could be applied by a statistician, in order to determine the probability of a particular move. However there are two types of approaches to Bayesian probability theory, objectivists interpret probability as an extension of logic, probability quantifying an answer that presumes we have a starting point or baseline that we tend to call common sense. However subjectivists, approach probability as something that corresponds to a personal belief. Therefore rationality is subject to substantial variation due to differing beliefs. The objective and subjective variants of Bayesian probability differ mainly in their interpretation and construction of the prior probability, i.e. where are we coming from in terms of an approach to reality? This is I think a vital question in a time of post truth. 

Because of my daughter's gift, I've been stimulated to think about board games as a way of presenting ideas. I have dipped my toe into the territory before and I did make a game once and had it made up with ceramic pieces made in the form of small boats as counters. 

Snakes and ladders type game 2016

I discovered this monkey game recently, which is a variant of The Game of the Goose, which was the first board game to be published commercially.

The new game of the monkey: By E. Wallis: 1820

I was fascinated by the new game of the monkey, as it seems to open doors for a wide range of stories, in particular those stories whereby there are no fixed formats. For instance in space 14 there is a monkey I think trying to seduce a woman with a saucepan for a hat and in space 23 there is a Little Red Riding Hood monkey. Both suggest a lascivious monkey, but without a particular narrative. This opens out the moments of action in the game into unknown territory. I have been using the figure of a monkey a lot recently, and was in particular thinking of the monkey's position as substitute for humans in scientific and medical experiments. My recent time in Gothenburg in particular re-alerting me to the way monkeys are used to test out the implications of new medical procedures before they are used on humans. Although initially working my ideas through as images and making the occasional ceramic, it is the game format that I'm hoping to develop. 

Monkey studies

A board game can be a substitute for a life's journey and another of my go to concepts in terms of how we visualise a life's journey is the Wyrd. This old nordic vision is of the intricate web of fate. Imagine that your umbilical cord at birth has an invisible twin. This one is never cut and as your life unravels it unravels too, winding itself around first of all your immediate family and the landscape of your youth. It gradually wraps itself around all the experiences of life, until a fabric is woven that is your life cloth. It represents the interconnectedness of everything including the past, present and future. Individual lives extend out threads that weave a tapestry of their own destiny. The Norns spin, measure, and cut the threads of life for humans and gods alike.

The final cloth is the result of an amazingly complex decision tree. As I looked at playing Three Men's Morris, the decision tree for that is I thought a symbol, that could stand for the one that underpins the Wyrd.

A decision tree for Three Men's Morris visually maps all possible moves and outcomes, showing each decision point where a player can place or move a piece. It's a game tree that starts with an empty board and branches out with every possible move, leading to a win, loss, or draw for the players. Building this tree involves mapping out the game's two phases: the placement phase (the possibilities surrounding decisions made as all pieces are placed on the board) and the movement phase (pieces are moved to adjacent, empty spots). There are four aspects to the decision tree that need to be taken into consideration. The Nodes, which represent the places from where a player must make a choice. The Branches, that represent the possible moves a player can make from a given node. The Options, that indicate which player's turn it is and what their options are and the Outcomes, that show the result of the sequence of moves (win, loss, or draw).

A simple compact decision tree of a game of three men's morris is an interesting structure in itself.

Three Men’s Morris placement phase has 9 empty positions at the start, when player 1 places a piece, 8 open positions remain. When player 2 places a piece, 7 open positions remain, then when player 1 places a second piece, 6 open positions remain. This continues until player 2 places their third counter, which leaves 3 open positions. There are therefore 9 × 8 × 7 × 6 × 5 × 4 × 3 = 181,440 possible placement sequences, all of which will have taken place before the game actually gets started. The decision tree that shows every placement sequence up to completion of the placement phase (the 6 initial placements), is set out below.  

Three Men’s Morris placement phase decision tree

As the decision tree grows it reminds me of when I was getting fabrics made using a Jacquard loom. Although the loom I was getting my work made with no longer used a card file and was now driven by computer, I was very aware of the complexity that underpinned the process. We tend to forget how complex many of our computer programs must now be. For instance when BBC computers were introduced into the college back in the late 1970s, we were taught how to write code to place a dot into a position on a screen. Then we could join that dot to another by defining co-ordinates. Therefore just to draw one line we had to write several lines of code. Many years have passed since then, but deep down within the complexity of some 3D animation software there will still be the code for joining two points on a screen, but millions of operations have been conducted since then and it is unlikely that I will ever get even an inkling of what they entail. However that doesn't stop me from thinking of that complexity as being not unlike the woven tapestry of the Wyrd. Several people have observed in the past that complex technology when seen by people from non technological societies will seem like magic. Indeed Arthur C. Clarke in his essay, "Hazards of Prophecy: The Failure of Imagination", stated as the third of his laws for future prophecy, "Any sufficiently advanced technology is indistinguishable from magic". My present awareness of new technology is already making me worry that we are now entering a new magical era, whereby AI magicians will conjure new realities out of nothing.

Punched cards used to operate a Jacquard loom

The complex matrix of threads that are interwoven with punch card directions

As I play the game of Three Men's Morris, which is also my granddaughter's favourite game, I'm reminded of the wonderful complexity of life, of how from that initial moment of our conception, when a sperm fertilises an egg and they combine to form a zygote, that single cell will begin to multiply and a decision tree begins that will eventually be of such complexity that it is impossible to hold in the mind, except as an idea and in my case I still find the old nordic vision of the intricate web of fate as being as good as any other that humankind has produced. 

See also: 

Weaving and grids

The square and the rectangle

The first ever drawing was a hashtag

Category theory and drawing