Last weekend, my little cousin taught me how to play Congkak (Malay traditional board game) or conventionally known as mancalas. It occurred to me that mancalas are straight out of numbers. So I gave it a thought to derive an equation of the mechanics of the game. As a matter of fact, it has been my hobby to try out new puzzles or games and to derive an equation or two out of them. Some were fruitful some were out of my league.
Anyhow, this time it worked swimmingly that I put an extra effort by presenting it into a research-paper-kind of thing – of which I’ve attached a link redirecting to the paper at the end of this read.
This paper is the author’s personal side project and is of no affiliation to any institution or courses. Congkak is a Malay traditional board game. It is played by which the player drops each marble into every successive hole progressing from the hole the player has chosen to start. The objective of this paper is to derive an equation determining the final nth hole with a known starting point (hole) and the number of marbles at hand. The resultant equation is impressively a mere linear equation with a dependence on a restricted domain of the function.
Modelling the Problem
For convenience, we do not attempt to express the whole aspect of the Congkak game into equations. We are more interested to work on the determination of the final hole with the knowledge of the number of marbles belonging to any arbitrary initial hole. With that, this model is of one player and the game is nothing but consists of an uninterrupted, continuous turn. To start, we assign numerical values to each hole, as shown by Figure 1.
This model creates a loop for every 15 moves. This is one of the major factors influencing the formulation of the equation. Also, we do not take into account the role of the 8th hole as the store in this model. Nevertheless, in practicality, the equation is intuitively able to determine either the starting hole or the number of marbles for the arrival to the 8th hole, in the use of the actual game.
Formulating the Equation
I leave all the long, painstaking mathematical work to be later referred in the actual paper I’ve written – of which can be accessed via the link attached.
The equation is both intuitively and impressively a linear function. However, it is not as sensible as there exists an extra variable whose values change depending on different intervals. In short, the Equation (3) is the practical equation determining the final position (nth hole) with a known quantity of marbles belonging to any arbitrary initial position (hole).
Appendix A: Extension
In the appendix, we show Equation (3) fails over a certain interval when we work on n as the function of the sum of the initial position and the number of marbles.
Again, the details of the appendix can be referred from the paper.
In short, the new equation, Equation (A.4), is a quadratic function that does the same – determining the final position (nth hole) from the knowledge of the number of marbles belonging to any arbitrary initial position (hole).
To conclude, the derivation of Equation (A.4) shows that we managed to obtain one single function that exclusively churns out the final position (nth hole) from only two variables, the h-naught and p, without any other unnecessary variable involved. However, the elegance achieved comes with a price, that the immediate assumptions applied to the model yield a certain degree of discrepancy through the range of the function, exclusively between the vertex of the quadratic function and the larger-end of the second previously-derived interval; apart from (ℎ0 + 𝑝) > 30.