Ducci's Four-Number Game

Math Puzzle: Le jeu à quatre chiffres de Ducci

Ducci's Four Numbers Problem is a well-known math game or puzzle, allowing for a rich exploration of several mathematical ideas. One starts with four numbers placed at the four vertices of a square (or any quadrilateral, if so desired). The distance (absolute value of the difference) between the two numbers connected by a side is then placed at the midpoint of that side. The procedure goes on. With whole numbers, the puzzle can be used with school students for the development of number sense and basic operations. With the right questions, the problem quickly becomes an intriguing one involving abstract algebra at advanced levels.

The current version includes a probability distribution simulation. We define the order of four numbers to be the number of steps taken for all the four numbers to reach all zero, excluding the initial step/square. For example, the order of (94, 81, 56,10) is 14, an extremely rare quadruple. What is the probability of getting four numbers with an order of 5? Further, you could rotate a set of four numbers or swap them around and explore what happens to the order. Have fun!