Description
1. N-Queens [25 points]
In this section, you will develop a solver for the n-queens problem, wherein n queens are to be
placed on an n x n chessboard so that no pair of queens can attack each other. Recall that in chess, a
queen can attack any piece that lies in the same row, column, or diagonal as itself.
A brief treatment of this problem for the case where n = 8 is given in Section 3.2 of the textbook,
which you may wish to consult for additional information.
1. [5 points] Rather than performing a search over all possible placements of queens on the
board, it is sufficient to consider only those configurations for which each row contains
exactly one queen. Without taking any of the chess-specific constraints between queens into
account, implement the pair of functions num_placements_all(n) and
num_placements_one_per_row(n) that return the number of possible placements of n queens
on an n x n board without or with this additional restriction. Think carefully about why this
restriction is valid, and note the extent to which it reduces the size of the search space. You
should assume that all queens are indistinguishable for the purposes of your calculations.
2. [5 points] With the answer to the previous question in mind, a sensible representation for a
board configuration is a list of numbers between 0 and n – 1, where the ith number designates
the column of the queen in row i for 0 ≤ i < n. A complete configuration is then specified by a list containing n numbers, and a partial configuration is specified by a list containing fewer than n numbers. Write a function n_queens_valid(board) that accepts such a list and returns True if no queen can attack another, or False otherwise. Note that the board size need not be included as an additional argument to decide whether a particular list is valid. >>> n_queens_valid([0, 0])
False
>>> n_queens_valid([0, 1])
False
>>> n_queens_valid([0, 2])
True
>>> n_queens_valid([0, 3, 1])
True
3. [15 points] Write a function n_queens_solutions(n) that yields all valid placements of n
queens on an n x n board, using the representation discussed above. The output may be
generated in any order you see fit. Your solution should be implemented as a depth-first
search, where queens are successively placed in empty rows until all rows have been filled.
Hint: You may find it helpful to define a helper function n_queens_helper(n, board) that
yields all valid placements which extend the partial solution denoted by board.
Though our discussion of search in class has primarily covered algorithms that return just a
single solution, the extension to a generator which yields all solutions is relatively simple.
Rather than using a return statement when a solution is encountered, yield that solution
instead, and then continue the search.
>>> solutions = n_queens_solutions(4)
>>> next(solutions)
[1, 3, 0, 2]
>>> next(solutions)
[2, 0, 3, 1]
2. Lights Out [40 points]
The Lights Out puzzle consists of an m x n grid of lights, each of which has two states: on and off.
The goal of the puzzle is to turn all the lights off, with the caveat that whenever a light is toggled, its
neighbors above, below, to the left, and to the right will be toggled as well. If a light along the edge
of the board is toggled, then fewer than four other lights will be affected, as the missing neighbors
will be ignored.
In this section, you will investigate the behavior of Lights Out puzzles of various sizes by
implementing a LightsOutPuzzle class. Once you have completed the problems in this section, you
can test your code in an interactive setting using the provided GUI. See the end of the section for
more details.
1. [2 points] A natural representation for this puzzle is a two-dimensional list of Boolean
values, where True corresponds to the on state and False corresponds to the off state. In the
LightsOutPuzzle class, write an initialization method __init__(self, board) that stores
an input board of this form for future use. Also write a method get_board(self) that returns
this internal representation. You additionally may wish to store the dimensions of the board
as separate internal variables, though this is not required.
>>> list(n_queens_solutions(6))
[[1, 3, 5, 0, 2, 4], [2, 5, 1, 4, 0, 3],
[3, 0, 4, 1, 5, 2], [4, 2, 0, 5, 3, 1]]
>>> len(list(n_queens_solutions(8)))
92
>>> b = [[True, False], [False, True]]
>>> p = LightsOutPuzzle(b)
>>> p.get_board()
[[True, False], [False, True]]
>>> b = [[True, True], [True, True]]
>>> p = LightsOutPuzzle(b)
>>> p.get_board()
[[True, True], [True, True]]
2. [3 points] Write a top-level function create_puzzle(rows, cols) that returns a new
LightsOutPuzzle of the specified dimensions with all lights initialized to the off state.
>>> p = create_puzzle(2, 2)
>>> p.get_board()
[[False, False], [False, False]]
>>> p = create_puzzle(2, 3)
>>> p.get_board()
[[False, False, False],
[False, False, False]]
3. [5 points] In the LightsOutPuzzle class, write a method perform_move(self, row, col)
that toggles the light located at the given row and column, as well as the appropriate
neighbors.
>>> p = create_puzzle(3, 3)
>>> p.perform_move(1, 1)
>>> p.get_board()
[[False, True, False],
[True, True, True ],
[False, True, False]]
>>> p = create_puzzle(3, 3)
>>> p.perform_move(0, 0)
>>> p.get_board()
[[True, True, False],
[True, False, False],
[False, False, False]]
4. [5 points] In the LightsOutPuzzle class, write a method scramble(self) which scrambles
the puzzle by calling perform_move(self, row, col) with probability 1/2 on each location
on the board. This guarantees that the resulting configuration will be solvable, which may not
be true if lights are flipped individually. Hint: After importing the random module with the
statement import random, the expression random.random() < 0.5 generates the values True and False with equal probability. 5. [2 points] In the LightsOutPuzzle class, write a method is_solved(self) that returns whether all lights on the board are off. >>> b = [[True, False], [False, True]]
>>> p = LightsOutPuzzle(b)
>>> p.is_solved()
False
>>> b = [[False, False], [False, False]]
>>> p = LightsOutPuzzle(b)
>>> p.is_solved()
True
6. [3 points] In the LightsOutPuzzle class, write a method copy(self) that returns a new
LightsOutPuzzle object initialized with a deep copy of the current board. Changes made to
the original puzzle should not be reflected in the copy, and vice versa.
>>> p = create_puzzle(3, 3)
>>> p2 = p.copy()
>>> p.get_board() == p2.get_board()
True
>>> p = create_puzzle(3, 3)
>>> p2 = p.copy()
>>> p.perform_move(1, 1)
>>> p.get_board() == p2.get_board()
False
7. [5 points] In the LightsOutPuzzle class, write a method successors(self) that yields all
successors of the puzzle as (move, new-puzzle) tuples, where moves themselves are (row,
column) tuples. The second element of each successor should be a new LightsOutPuzzle
object whose board is the result of applying the corresponding move to the current board. The
successors may be generated in whichever order is most convenient.
>>> p = create_puzzle(2, 2)
>>> for move, new_p in p.successors():
… print(move, new_p.get_board())
…
(0, 0) [[True, True], [True, False]]
(0, 1) [[True, True], [False, True]]
(1, 0) [[True, False], [True, True]]
(1, 1) [[False, True], [True, True]]
8. [15 points] In the LightsOutPuzzle class, write a method find_solution(self) that
returns an optimal solution to the current board as a list of moves, represented as (row,
column) tuples. If more than one optimal solution exists, any of them may be returned. Your
solver should be implemented using a breadth-first graph search, which means that puzzle
states should not be added to the frontier if they have already been visited, or are currently in
the frontier. If the current board is not solvable, the value None should be returned instead.
You are highly encouraged to reuse the methods defined in the previous exercises while
developing your solution.
Hint: For efficient testing of duplicate states, consider using tuples representing the boards
of the LightsOutPuzzle objects being explored rather than their internal list-based
representations. You will then be able to use the built-in set data type to check for the
presence or absence of a particular state in near-constant time.
>>> for i in range(2, 6):
… p = create_puzzle(i, i + 1)
… print(len(list(p.successors())))
…
6
12
20
30
>>> p = create_puzzle(2, 3)
>>> for row in range(2):
… for col in range(3):
… p.perform_move(row, col)
…
>>> p.find_solution()
[(0, 0), (0, 2)]
>>> b = [[False, False, False],
… [False, False, False]]
>>> b[0][0] = True
>>> p = LightsOutPuzzle(b)
>>> p.find_solution() is None
True
Once you have implemented the functions and methods described in this section, you can play with
an interactive version of the Lights Out puzzle using the provided GUI by running the following
command:
python homework2_lights_out_gui.py rows cols
The arguments rows and cols are positive integers designating the size of the puzzle.
In the GUI, you can click on a light to perform a move at that location, and use the side menu to
scramble or solve the puzzle. The GUI is merely a wrapper around your implementations of the
relevant functions, and may therefore serve as a useful visual tool for debugging.
3. Linear Disk Movement [30 points]
In this section, you will investigate the movement of disks on a linear grid.
The starting configuration of this puzzle is a row of L cells, with disks located on cells 0 through n –
1. The goal is to move the disks to the end of the row using a constrained set of actions. At each step,
a disk can only be moved to an adjacent empty cell, or to an empty cell two spaces away, provided
another disk is located on the intervening cell. Given these restrictions, it can be seen that in many
cases, no movements will be possible for the majority of the disks. For example, from the starting
position, the only two options are to move the last disk from cell n – 1 to cell n, or to move the
second-to-last disk from cell n – 2 to cell n.
1. [15 points] Write a function solve_identical_disks(length, n) that returns an optimal
solution to the above problem as a list of moves, where length is the number of cells in the
row and n is the number of disks. Each move in the solution should be a two-element tuple of
the form (from, to) indicating a disk movement from the first cell to the second. As suggested
by its name, this function should treat all disks as being identical.
Your solver for this problem should be implemented using a breadth-first graph search. The
exact solution produced is not important, as long as it is of minimal length.
Unlike in the previous two sections, no requirement is made with regards to the manner in
which puzzle configurations are represented. Before you begin, think carefully about which
data structures might be best suited for the problem, as this choice may affect the efficiency of
your search.
>>> solve_identical_disks(4, 2)
[(0, 2), (1, 3)]
>>> solve_identical_disks(5, 2)
[(0, 2), (1, 3), (2, 4)]
>>> solve_identical_disks(4, 3)
[(1, 3), (0, 1)]
>>> solve_identical_disks(5, 3)
[(1, 3), (0, 1), (2, 4), (1, 2)]
2. [15 points] Write a function solve_distinct_disks(length, n) that returns an optimal
solution to the same problem with a small modification: in addition to moving the disks to the
end of the row, their final order must be the reverse of their initial order. More concretely, if
we abbreviate length as L, then a desired solution moves the first disk from cell 0 to cell L – 1,
the second disk from cell 1 to cell L – 2, . . . , and the last disk from cell n – 1 to cell L – n.
Your solver for this problem should again be implemented using a breadth-first graph search.
As before, the exact solution produced is not important, as long as it is of minimal length.
>>> solve_distinct_disks(4, 2)
[(0, 2), (2, 3), (1, 2)]
>>> solve_distinct_disks(5, 2)
[(0, 2), (1, 3), (2, 4)]
4. Feedback [5 points]
1. [1 point] Approximately how long did you spend on this assignment?
2. [2 points] Which aspects of this assignment did you find most challenging? Were there any
significant stumbling blocks?
3. [2 points] Which aspects of this assignment did you like? Is there anything you would have
changed?
>>> solve_distinct_disks(4, 3)
[(1, 3), (0, 1), (2, 0), (3, 2), (1, 3),
(0, 1)]
>>> solve_distinct_disks(5, 3)
[(1, 3), (2, 1), (0, 2), (2, 4), (1, 2)]