CondensedMatrixSolution.py

def getPossibilities(startingPosition, turns):
    """
    calculates the total number of uniques paths from the starting position in N turns for the transition matrix
    """

    # edge case: if turns is 0, total number of unique paths is 1
    if turns == 0:
        return 1

    # edge case: if starting position is 5 and turns is not 0, total number of unique paths is always 0
    if startingPosition == 5:
        return 0

    # map original positions to new positions
    mappedStartingPosition = getMappedStartingPosition(startingPosition)

    # our transition matrix
    transitionMatrix = np.matrix([
        [0, 1, 1, 0],
        [2, 0, 0, 0],
        [2, 0, 0, 1],
        [0, 0, 2, 0]], dtype=object)
    
    # get our new matrix N turns ahead
    newMatrix = np.linalg.matrix_power(transitionMatrix, turns)
    
    # sum the relevant row
    return np.sum(newMatrix[mappedStartingPosition])