The Four-Numbers Game

The Four-Numbers Game starts with a square and four non-negative integers placed in no particular order on each vertex of the square. The first step of the Four-Numbers Game is to draw a new square using the midpoints of the initial square as its vertices. The number placed at the midpoint is the absolute value of the difference of the two numbers at the adjacent vertices and the new square is formed by joining the midpoints of the first square. The process continues with every step of the game where a new numbered square is formed in another following the same rule. The game ends when the player arrives at a numbered square with a zero at every vertex. The purpose of our research is to discover a method to increase the length of a game and generate one of any length of our choosing. First step of the research was to establish the properties of the Four-Numbers Game. Second step was to be able to predict when a game ends by simply observing a certain form in the numbers such as the significance of a number appearing twice in a step. The last step was to notice a pattern when attempting our own methods of increasing the length of the game. Our reaserched concluded when we discovered that the additive property and tribonacci numbers allows the player to generate a game of any given magnitude.