Jayasree S and R. Ramanujam
Consider two children absorbed in a board game. They are focussed, thinking, strategizing and trying to execute plans. All this is worthwhile and children are encouraged to play board games. Does such play also help their mathematization abilities? By itself, perhaps not, with some guidance and help from a teacher, perhaps yes. In this article we would like to suggest that such an exploration is worthwhile for teacher and student alike; it is certainly joyful.
We recently had the opportunity to work with a group of 9th graders in a matriculation school in Chennai on their annual science project. This group of students wanted to dig into ‘ancient mathematics’ and specifically into ‘ancient mathematical games’. While we read about many such games and played them, here we share some of our learning via pallankuzhi, the pits and pebbles game.
The game description
The version of pallankuzhi we played is a two-player game, with a board containing seven pits on each side, and six pebbles in each pit initially. Each player owns the seven pits on her side The first player starts the game by choosing one of the pits on her side and redistributing (sowing) the pebbles there in, one pebble per pit, starting from the adjacent pit and traversing the board in an anti-clockwise direction. Once all the pebbles are ‘sown’, she draws on the pebbles from the pit adjacent to where she dropped the last pebble and continues the ‘sowing’ till such time that the pit from which the pebbles are to be drawn is empty. In this case, the player captures (harvests) all the pebbles from the next pit – i.e., if the pit next to the last drop is empty; the player captures all the pebbles in the next pit (empty or not) and the turn ends. The turn now passes to the next player, who chooses a pit on her side and continues as above. The game continues till all the pits owned by one player are empty on her turn. The goal of the game is to maximize the number of pebbles captured. In general, the game is played with cowrie shells or pebbles or locally available seeds.
Pallankuzhi – a part of culture and tradition
Pallankuzhi being part of the cultural heritage of Tamil Nadu, most of us had played it at some point in our lives and had a game board available, though no longer in regular use. The students, in conversation with their parents and grandparents, realized that this game was part of the elders’ growing-up as well and that they knew a trick or two that would yield a rich ‘harvest’ of pebbles. Some students talked of people who, taking into consideration the current distribution of the board, could use some quick counting strategies to come up with moves leading to a capture. This ‘look-ahead’ in the game could involve being able to visualize multiple rounds of sowing and therefore a certain mathematical acumen.
In the South Indian states, the game is largely played by children and womenfolk. There are traditions and taboos surrounding the game, like the game being played on certain ‘auspicious days’; that the game should be played in the daylight hours only, etc. There are mentions of the game being a recreational activity for agricultural labourers during hot afternoons and of the game being played with currency coins instead of pebbles and seeds soon after a wedding in some communities (Balambal, 2005). The presence of rock engraved game boards in temples points to the role of the game in community life. Balambal (2005) and Murray (1952) list different versions of the game as played in different parts of India and abroad. These include solitaire versions, triangular boards and boards with five, six or seven pits.
Pallankuzhi variants from across the world
The family of ‘pit and pebble games’, usually termed Mancala, all involving sowing and harvesting of identical pebbles, with many distinct rules of play, is quite widely spread geographically and temporally. The types of game boards also vary – rock-carved boards to pits dug in sand and ones made from material as varied as wood to ivory inlaid with gold. Archaeological findings from sites such as the Karnak Temple in Egypt, Pyramids of Meroe, Palymra in Syria, Theseum in Athens indicate the presence of the game from the beginnings of human civilization. You may want to look at some references listed at the end of this article for details.
The game has many connotations. In many communities, the game has been associated with rituals and festivities around invoking rain, celebrating a harvest, birth and death, etc. In parts of Africa it is a “noisy social occasion with cheers and jeers, laughter and occasionally anger and brawling”. Though a two-player game it often takes on the face of a team game, with the boundaries between players and spectators blurring. Among some African tribes the chieftain is chosen by playing a game of Mancala. In most African countries Mancala is a game only for men, unlike in India. Townshend (1979) hypothesizes that a woman defeating a man at a public game, or ridiculing the loser, would be a dent to male power. The game is considered a means to educate young men on the social values embodied by the elders and not learning to play would be considered anti-social.
A peek into the classroom – whiffs of mathematization
We now turn to the classroom and hope to highlight the opportunities for mathematization afforded by the game. A group of five students from class 9 took up this exploration.
Even as students started to play the game, there were differences in rules of play that each student came up with in consultation with knowledgeable elders at home. One of them said that the initial configuration has to be six seeds per pit and another said it has to be five, and that one could harvest the seeds whenever the count of seeds in a pit (on one’s own side) reaches three. We decided to go with the six seeds per pit version, hoping to see if we could observe some patterns that could lead to a strategy to win. Soon we were struggling to keep track of the number of seeds in each pit and counting on to see what configuration of the board would eventually result after a turn. We then decided to play a three seeds per pit version of the game, following Polya’s heuristic of ‘simplifying the problem’. We also had to clearly define the problem, by spelling out the rules of the game – like what constitutes a turn, and when does it end, what are the rules for capture, which pit do we draw from once the seeds in hand have been redistributed and so on.
When we found even the three seeds per pit version hard to keep track of, we decided to simplify even further and strip down to a seemingly trivial configuration of one seed per pit.
With this some patterns started becoming obvious and students came up with statements like “In this game, we could capture two seeds from the pit adjacent to the one we started from.” Soon the students wanted to see if there is a pattern to be observed in the two seeds per pit and three seeds per pit versions, more specifically if they could make predictions about the opening move. With two seeds per pit, the position of the pit from which one could harvest given a starting pit was obvious to them – the pits were moving in tandem – one step at a time. But verbalizing the relationship proved to be a challenge.
The suggestion to number the pits gave them a better way of talking about these relationships. Now they could make statements such as: “In a one seed per pit game, if the first player starts with an odd-numbered pit, the second player has the option of starting from the even numbered pits on her side.” This also gave them a way of listing out the sequence of pits from which seeds would be drawn to be redistributed. Playing with these sequences of numbers helped them predict the course of the game without actually playing it. This constitutes a form of mathematization in itself.
For example, the sequence 1, 4, 7, 10, 13, 2, 6, 10 – is the sequence of pits from which seeds would be drawn for redistribution starting with pit 1 when there are two seeds per pit, (pits are numbered 1-14 clockwise). The two seeds in pit 1 get redistributed to pits 2 and 3 and the seeds in pit 4 are drawn and dropped in pits 5 and 6 and then the seeds are drawn from pit 7 and so on. Coming to pit 13, the seeds get dropped in pits 14 and 1, but now there are three seeds to be picked up in pit 2, and so the next draw would be from pit 6. Continuing thus, the game reaches a stage where one needs to draw from pit 10. But since pit 10 was drawn from in the previous round, it is empty and so the player gets to capture the seeds in pit 11. Writing out these sequences of numbers gave them a way of verbalizing the patterns they observed and made the patterns more obvious as well.
Moving beyond the first move, the second player has a number of options to choose from – she could play any of the non-empty pits on her side. So how does one choose between the options? We knew that we should maximize the number of seeds captured. So then we calculated the seeds that could be captured with each available option and looked for ways to represent this information. A tree-diagram came in handy.
A notation was evolved to represent the configuration of the board at the end of each turn as well. While we are nowhere near finding a winning strategy (obviously!) the exercise has opened up a number of possibilities and points of departure for initiating many more school projects across disciplines.
Points of departure
Some students might want to go on a historical journey, digging into the origins of the game. Marking the areas of prevalence of the game on a map could be an exercise in geography. Comparing the games as played in two different regions and looking for similarities and differences might give an indication of aspects of the game that could be varied. Letting students come up with their own variations of the game could be an interesting and useful exercise. This could lead to multiple questions – For example, the six-seeds-per-pit game that we started out with and the one-seed-per-pit game that we eventually ended up exploring – are they different games or is one a simplified version of another? Varying which dimensions of the game would constitute a ‘simplification’ and which dimensions would result in an altogether different game?
Patterns observed for a three-seed-per-pit game or a two-pits-per-side game may be appropriately extended to a six-seeds-per-pit or eight-pits-per-side games. Having fewer seeds per pit or fewer pits per side are ways of simplifying the game. However in our game, if we had chosen to
• redistribute the seeds from the pit of the last drop instead of the adjacent pit, or
• if we had insisted that a ‘turn’ would end when the seeds drawn from the first pit chosen are redistributed
these games would require a different analysis. So this leads us to the question – change in what aspects of the game call for a fresh analysis?
Though questions related to a winning strategy for a given set of rules and configuration of the board are difficult to answer, students can still investigate simpler questions like:
• how many different configurations are possible, after say three moves, given an initial configuration, or
• does a particular configuration give an advantage to one of the players, or
• are there impossible configurations, and so on.
(The configurations have to be chosen carefully for the questions to be answerable).
The game apart, even investigating the number of ways a given number of seeds can be placed in a given board can be a good exercise in combinatorics (see References at the end).
We suggested at the outset that board games of strategy may provide an arena for guided mathematization. Apart from the attractiveness of pallankuzhi being culturally rooted, we note that it has a low threshold (unlike, say, chess) and everyone playing the game can formulate some strategies. Can it lead to complex mathematics? We do not know but this seems worth exploring. (For instance, is there a Sprague-Grundy type theorem for pallankuzhi?) The fact that pallankuzhi offers multiple starting points, branching pathways for exploration and scope for a local formal language for articulation of strategies strengthens our belief in its scope for mathematical exploration. Understanding how teachers’ guidance can shape such exploration requires further work and sharing of experience among teachers.
Acknowledgements: We thank the matriculation school authorities and the students who participated in this endeavour. We also thank Vinusha Venkatesh for help with documentation and photographs and Shikha Takker and Aaloka Kanhere for their inputs. This is also our tribute to John Conway, a great mathematician who succumbed recently to Covid-19, and who taught the world how complex mathematics resides in simple games.
- Balambal, V. (2005). Folk Games of Tamil Nadu, The C P Ramaswami Aiyar Foundation
- Bikić, V., & Vuković, J. (2010). Board Games Reconsidered: Mancala in the Balkans. Issues in Ethnology and Anthropology, 5(1), 183-209. https://doi.org/https://doi.org/10.21301/eap.v5i1.10
- De la Cruz, R. E., Cage, C.E., Ming-Gon John Lian. (2000). “Let’s Play Mancala and Sungka!” Teaching Exceptional Children 32, no. 3 (January/ February): 38-42
- De Voogt, A. J. (2010). Mancala Players at Palmyra. Antiquity, 84, 1055-1066
- De Voogt, A. J. (2012). Mancala Pyramids of Meroe. Antiquity, 86: 1155-1166
- De Voogt, A. J., Rougetet. L, Epstein. N. (2018). Using Mancala in the Mathematics Classroom, The Mathematics Teacher Vol. 112, No. 1 (September 2018), pp. 14-21
- Gananath, S. N (2017). Sita’s Solitaire, Teacher Plus, August 2017
- Haggerty, J. (1964). KALAH – an ancient game of mathematical skill. The Arithmetic Teacher, 11(5), 326-330. Retrieved February 25, 2020, from www.jstor.org/stable/41184972
- Murray, H.J.R. (1952). A History of Board-Games other than Chess. Oxford at the Clarendon Press
- Townshend, P. (1979). African Mankala in Anthropological Perspective, Current Anthropology, 20(4), pp. 794-796 https://timesofindia.indiatimes.com/city/chennai/Old-temples-a-cache-of-ancient-board-games/articleshow/54135999.cms
Jayasree S is a teacher, currently registered for a PhD at HomiBhabha Centre for Science Education, (HBCSE) Mumbai. She can be reached at firstname.lastname@example.org.
R Ramanujam is a theotetical computer scientist at The Institute of Mathematical Sciences, Chennai. He is also interested in science and mathematics education and popularization. He can be reached at email@example.com.