1. Modeling the Game of Brain Cube Leslie Muzulu, Kristal Jameson, and Kathy Radloff; St. Catherine University Game Description Brain Cube is a puzzle game that consists of a board and a game piece that moves around the board. The three-dimensional game piece is made of two attached cubes that move as one. There are three types of states in which the game piece may be situated on the board: standing upright, occupying one square; lying on its side “horizontally,” occupying two adjacent squares in the same row; and lying on its side “vertically,” occupying two adjacent squares in the same column of the board. If the piece is lying on its side, two types of moves are permissible in any direction the size of the board permits: it can roll on its side such that after the move it will still be on its side, or it can be tipped up so that it is standing on end. If the piece is standing on end, it may be tipped down in any of the four directions the board permits (i.e., the piece will not partially or wholly fall off due to the move), such that the piece is lying on its side after the move is executed. The goal of the game is to move the game piece from the given starting position to the given end position. Research Questions 1.For which game setups does a solution exist? 2.Regarding those setups for which a solution exists, can we find a most efficient solution? 3.Still regarding those setups for which a solution exists, is there a unique most efficient solution? Future Questions 1.If moves are chosen at random and the game piece cannot fall off the board, what is the expected number of moves to reach the winning game state? 2. Still choosing moves at random, what is the probability of reaching the winning game state if the game piece is allowed to fall off? What is the probability of the game piece falling off the board before reaching the winning game state? Major Results 1.For a given game setup, a solution exists precisely when the vertices representing the starting and ending states lie in the same connected component of the graph. 2.Any given board can be modeled by a graph as described for the simple 3x3 board. The question of finding a most efficient solution then translates to finding a shortest path between the starting and ending vertices. One can use Dijkstra’s Algorithm to find such a shortest path on the graph. 3.Whether there is a unique most efficient solution is sometimes easy to see, and at other times not. We do not have a general algorithm that will answer this question. Our Models We devised notation to represent every possible game state. States where the game piece is standing on end are represented by u 1, u 2, … u mn. States where the game piece was lying on its side occupying two adjacent squares in a column or row are denoted by v or h, respectively. Examples for a 3x3 board are shown below. This notation can be used to create a graphical model of the relationships between the game states for any given board size. Each vertex represents a game state, and two vertices are adjacent if and only if their corresponding game states are within one move of each other during game play. The graph of the game states for a 3x3 board is below. The graph above tells us, for example, that from the game state h 2 on a 3x3 board we can move directly to h 1, h 3, or u 6. It also shows that every game state can be reached from any of the others except for u 5. Adjacency matrices were utilized to confirm the fewest number of moves possible to cross a rectangular board. Additionally, transition matrices showed the probability of reaching a given game state after a given number of randomly chosen moves. u1u1 u2u2 u3u3 u4u4 u5u5 u6u6 u7u7 u8u8 u9u9 h1h1 h2h2 h3h3 h4h4 h5h5 h6h6 v1v1 v2v2 v3v3 v4v4 v5v5 v6v6 A Special Class Here we assume that our board is rectangular, that we are starting in the upright position on the upper left corner square, and finishing the game in the upright position on the square in the bottom right corner of the grid. 1.This situation admits a solution provided both the number of rows and columns of our grid is at least 3. 2.For an m x n board with both m and n at least 4, the number of moves required in a most efficient solution is: 3.There is a unique most efficient solution only for relatively small rectangular boards. Once the dimensions are 4 x 4 or larger, uniqueness fails.