## Mathematical Qualities of the Nsolo Zambian/African Traditional Game

The Nsolo Zambian or African traditional game is played in most parts of the continent of Africa. Glimpses of it can be seen occasionally in African documentaries. These are scenes of men squatting and standing under a shady tree with dusty hands playing the game. There are various versions of the game. The easiest and most commercially lucrative is known as the Mankala. This is the small board game sold in toy stores. It is made out of small wooden boards with two rows of holes. When playing the game, marbles are moved and spread from hole to hole according to the rules of the game.

The Nsolo Zambian traditional version is more elaborate. In Zambia in Southern Africa, the game consists of digging a number of holes in rows and columns in a rectangular shape on a flat ground in the dirt or soil.  In some cases, some of the holes are made on a flat cement concrete slab. There are always four rows of holes but the columns can range from as small as 4x 12, 4x18, 4x40 up to as large as 4x 100 or more. The opposing players or teams of players sit on the opposite of the rows of holes. Each side controls only the two rows closest to them.

How to Play Nsolo

Two stones are placed in each of the outside rows at the beginning of the game. The two sides flip a coin to determine who starts. The player can then pick up any 2 stones from any of his holes and drop one in each hole. Where ever the player lands with the last stone, they will then pick up all the stones in that hole and continue to distribute them again starting from the next hole. They only stop is when they land with 1 stone in an empty hole. The player then says aloud: "Chenti!" This signifies that the player has finished their play and this prompts the opposing player(s) to take their turn. The stones are always and without exception moved from left to right. Normally, after 5 or 6 initial relatively routine harmless plays, they can now begin to score on each other. This is when serious mathematical strategy comes to play. Players have to count stones and holes and try to anticipate 2 to 3 or more moves ahead while the opposing player(s) will do and try to thwart, pre-empt or evade their opponents’  next possible moves.

Nsolo Rules and Terms

The winner of the game has more than one stone to play with while the loser is the team or side that runs out of stones to play with. The Nsolo game has a built-in culture of both teams discussing and arguing strategy very openly before each crucial play. Laughter, friendly teasing, bunter, bluffing an opponent, baiting, and feigning victimhood to trick or lure  opponent(s) into over confidence, are all part of the friendly game of Nsolo. After each game is over, the players will laugh, comment,  and reminisce about perhaps the one move that lost or won the just ended game.

The opposing player(s) have to agree whether the game is touch and go or not. If they agree that it is touch and go, it means that if the player begins play and half way they discover it is a bad move they cannot retract the play.

Chenti - a term said loudly to indicate that the player has finished their turn.

Sula - when you score on the opponent, you penalize them by confiscating or putting out of commission of play all stones from 1 or 2 holes depending on whether the score is marley or changena. This is perhaps one of the most important actions  that players have to take and must take sometime to think carefully before they make their move.

Marley - when you score or strike the opponent such that you can confiscate all stones only from one of any of  their strategic holes.

Changena - when you score or strike the opponent such that you can confiscate all stones from any 2 of their most strategic holes.

Para this is when intentionally or by accident, a player forces the opponent(s) to have only single stones in all their holes. This is generally regarded as dangerous or suicidal play. But it is not uncommon for players also to use it as a strategic move.

Kuwalata a strategy in which during the course of play, the player spreads the stones in as many holes as possible. This reduces the player or team's vulnerability while at the same time bolsters the player or team's strategic position.

Mphiri - since you can never play single stones until you have run out of a minimum of twos, mphiri refers to the unfortunate position of being stuck with a 2 stone hole-play towards the end of the game. It is normally disadvantageous for the player or team. The opposing player or team normally will capitalize on this condition of the opponent having or being stuck with a  mphiri. The disadvantaged team will try their best to quickly get rid of their mphiri.

Mutu (head) - is a metaphor used to refer to the the front stone of the game that is used to strike the opponent as in excitedly saying during play: "Lets sula this hole to strike the head!"

Mchila (tail) - is a metaphor used to refer to the back or tail end of the game which is often the most powerful especially toward the end of the game. A player might say or suggest excitedly: "Lets sula this hole to cut off the tail to weaken them!"

Nsolo General Strategies

1. You have to kuwalata the stones (see above) as much as possible during the game. Avoid as much as possible bunching too many stones in one hole
2. The more you kuwalata during play along both your 2 rows, the less changena  opportunity the opponent will have when they score.
3. You can shield you vulnerable stone(s) by escaping at an opportune time to the safety of the outside row.
4. Know when to stall, lay low, and wait to ambush your opponent as they are compelled to move toward you in a dangerous play.
5. Know how to sula (see above) so that you will maximize changena when you score again in the future.
6. Avoid to para (see above) as much as you can unless you are deliberately using it as a strategic move.
7. Know when and how to slow down an opponent.
8. Know when to use sacrifice play in order to gain and advantage in subsequent play.

Mathematical Analysis

The Nsolo uses many mathematical concepts.  Some of the more complex may be beyond this article or are still to be discovered.

1. The Nsolo game uses simple subtraction, addition, and multiplication of stones
2. The game uses natural numbers as contrasted to whole numbers, as the concept of zero does not seem to exist. All the holes are always counted even though they may be empty. Since the game never goes backwards, negative numbers may be either inconceivable or the concept is merely ignored as a fundamental rule of the game.
3. It is unclear historically whether the ten base numbering system or what type of base system has ever been used as there are no fixed number of holes for the game or standard number of stones as a requirement for the game except for the 4 rows.
4. The counting, piling up, and adding stones strategically seems to mimic the use of exponentials. For example, a hole which has 3 stones suddenly becomes more powerful with the strategic addition of 1 more stone to make it 4 in order to score on an opponent.
5. The game has many elements of probability as well as conditional probability. This is because there are often numerous possible moves that depend on so many factors at once from the team playing and the team waiting their next turn. The choice of each one play depends on some luck combined with the nature of the skills for sula (if they had previously scored) the opponent made and may be the nature of their kuwalata.
6. The game seems to be linear as you never skip a hole and at the same time cyclical in structure.
7. The game may also have elements of infinity as the two rows that each opponent is designated to play in, go around in a continuous circle.
8. It is uncertain whether the game can be subjected to one or a series of moves based on known mathematical logical calculations and principles that would guarantee a win for  the player. It seems as though there are too many unpredictable often disruptive factors and moves to make such a scheme infinitely complex.  For example, if the game had a standard number of stones, number of holes, and certain standard plays, it would be possible to string a series of plays together that would guarantee a particular player the win.
9. It is often said that Chess may be the most complex game. But it is worth contemplating  as to whether the 4 rows x 100 Nsolo Zambian game, for example, generates so many complex play combinations, on both defense and offence simultaneously,  that it may be more difficult or challenging for even a talented, gifted player or mathematician to master the game easily.  I am not even sure the computer may play the game as many of the moves players make may not have linear fixed rules and too many of the moves depend on conditional probability.