Perspective: Learning Math From an Ancient African Board Game

Coming from East Africa to Toronto, Canada, in the late eighties, my family very quickly realized that most cultures have something in common. One of the most memorable ways that we shared our culture with others during my childhood was through games, especially owari.

Owari, more commonly known as mancala, is a two-player board game where both sides routinely place beads from one end of the rectangular game board towards the other in a successive pattern. 

Counted as one of the oldest two-player board games in the world, the origin of mancala dates back to 700 AD. The word itself is derived from the Arabic word “Naqala” (na-ka-la), which means “to move” or “to transfer.” Over centuries, the game has become a cultural and spiritual staple in African cultures. 

One of my very fond memories was playing mancala on the steps of my family home in Nairobi, so imagine my surprise at seeing a version of the game on my friend's shelf in downtown Toronto. It was a sign of globalization, especially in a diverse city like Toronto — and as I’ve grown up, the world has become more aware of African culture, through art and design, literature, food, and board games.

How is Owari played?

In his book African Fractals: Modern Computing & Indigenous Design, Ron Eglash breaks down the rules: 

There are at least four playing pits on both of the longer sides of the rectangle and can sometimes have a large store for each player on both of the shorter sides. Players take turns picking up all the stones from one of their pits and placing one in each of the pits in a counterclockwise direction until they run out of stones. The goal is to have the most stones in your store at the end of the game.

If a player runs into their own store during their turn, they have to deposit a piece in it. If they run into the opponent’s store, they skip it and move onto the next pit.

Players can get an advantage by either capturing their opponent’s stones or placing the last stone during their turn in their store so they can get an extra turn. If the last stone to be deposited is in an empty pit on the player’s own side, the player is able to capture any stones that lie in the pit directly opposite to it. The game ends when all the pits on one side of the board are empty, and the player who still has stones on their side is able to capture and store them. The player with the most stones in their store wins. 

How does this relate to mathematics?

Mathematicians and educators across the world have long been studying the game from a mathematics-based strategy perspective. 

In Ghana, players of owari often use marching groups. A player will move stones from the largest pit counterclockwise one by one. If each successive pit has one less stone than the last, the player will win by repeating the pattern, capturing the most stones. As a self-replicating pattern, marching groups are fun expressions of sophisticated mathematics.

In the 1970s, French-American mathematician Benoît Mandelbrot coined the term “fractal” for what he called the “art of roughness.” Fractals are geometric patterns that follow specific mathematical rules, developing highly intricate, complex, and dimension-warping qualities. These rules describe complex patterns that show self-similarity at different scales, creating complex and irregular structures, versus Euclidean geometry, which creates simple and regular patterns.

Fractal mathematics, such as the Mandelbrot Set, is thought to represent the thumbprint of God or the equation of life. The concept of fractals has helped researchers understand several biological phenomena, unleashing a new era of science. They also have applications in understanding earthquake distribution, and improving cell signaling, computer graphics, and even cryptography, which often uses fractals as a founding principle behind several solutions.  Last year, researchers at Northwestern University found fractal-like structures in the brain, a finding that may be universal across species. When fractal concepts are applied to topological insulation, they have also been seen to reduce energy loss, becoming a key piece of development for quantum computing.

When expressed as mathematical equations, fractals can be a form of organized chaos, providing order to randomness on infinite scales. Take the example of a tree: the growth of a tree is not a linear process, but rather takes place in stages where a root will grow several smaller roots and those smaller roots will grow even smaller roots. 

Eglash writes that the “marching order” strategy in owari as “a self-replicating pattern is at the heart of some sophisticated mathematical concepts.” Owari can get players thinking about fractals even as children, while being a fun way to explore African identity and culture from a precolonial context. 

Africa's lineage of fractals

Fractals are not popular among common games or culture, but they are commonly found in a lot of African art and design. Small bits of culture, like cornrows, to larger specimens like the medieval city of Edo, several aspects of African culture stem from an understanding of fractals.

But fractals have made an even bigger revival in modern times by playing a role in the dominance of artificial intelligence. Large language models that are being used to train AI are using fractal structuring to recreate patterns of language. But this also means that fractals are now becoming a part of AI’s increasingly authoritarian and unequal access.

A group of African innovators, in partnership with Eglash, are working to democratize AI. “We were one of ten recipients of an OpenAI grant to democratize AI,” Eglash wrote over email. Through the grant, Eglash and his collaborators are developing a platform called Ubuntu AI that will be owned and run by African creatives, including crafters, artists, and designers. “It allows them to upload their work and get paid if folks want to use it to train AI. We have about 350 [people] participating so far.”

Ubuntu, a Zulu phrase, roughly translates to “I am because you are.” It is part of a larger Pan-African humanist philosophy representing mutuality and community. A 2024 paper co-authored by Eglash and Micheal Nayebare at the University of Michigan questions the power imbalance in AI as well as other technologies and presents African fractals as a technosocial solution to increasing inequality.

“AI democratization, in other words, needs to operate across multiple scales. Nature allows the multiscale flourishing of biological systems through fractal distributions.”

As the world population increases and our societal structures fail to support our growing populations, scaling political and social institutions across international, national, local, and community scales via fractal patterning can improve democratization. 

And this is what a board game like owari has the potential to teach children. Fractals aren’t just a part of African history but also a part of our collective future.