The investigation of determining solutions for the Diophantine equation 𝑥𝑥4 + 𝑦𝑦4 = 𝑧𝑧3 over the Gaussian integer ring for the specific case of 𝑥𝑥 ≠ 𝑦𝑦 is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

The investigation of determining solutions for the Diophantine equation over the Gaussian integer ring for the specific case of is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known.

Congkak is a traditional counting game played in Southeast Asia including Malaysia, Singapore, Brunei and Indonesia. To start a game, a board that has 16 holes together with 98 marbles are required. Each player controls a set of seven holes and own a store. The winner of the game is the player who captured more marbles into the store at the end of the game. Note that the first move advantage exists in chess; we investigate if the first-move advantage holds in congkak also. We model the route for each player in congkak using a directed graph and adopt these graph representations in our programs, to compute the winning percentage of each player. We focus on games between novices, hence a randomised strategy is used in our algorithm. We present the first experimental results for 100,000 games between novices in congkak. We also suggest some questions for future research in this area.