"Greetings, Professor Falken. Shall we play a game?
How about a nice game of chess?"
-- WOPR (WarGames)
What is the maximum number of queens that can be placed on an n x n chessboard such that no two queens attack one another? (A queen can move any number of squares vertically, horizontally or diagonally.)
The answer is n queens.
However, the number of different ways the n queens can be placed on an n x n chessboard depends on n.
Algebraic notation notates the standard 8 x 8 board in the following way (files are labeled a to h and ranks are numbered 1 to 8):
a8 b8 c8 d8 e8 f8 g8 h8
a7 b7 c7 d7 e7 f7 g7 h7
a6 b6 c6 d6 e6 f6 g6 h6
a5 b5 c5 d5 e5 f5 g5 h5
a4 b4 c4 d4 e4 f4 g4 h4
a3 b3 c3 d3 e3 f3 g3 h3
a2 b2 c2 d2 e2 f2 g2 h2
a1 b1 c1 d1 e1 f1 g1 h1
...which is very suggestive of a representation for a solution (or partial solution) to the n-queens problem:
[2 5 7 1 3 8 6 4]
In the above example, 2 corresponds to a queen on a2, 5 corresponds to a queen on b5, etc... Note that this vector is a solution to n-queens for n=8. It's also a partial solution for n=9; [2 5 7 1 3 8 6 4 9] is a complete solution for n=9.
For project 1, write the following functions in Clojure:
(qextends? partial-sol rank)
Given partial solution vector partial-sol of length k, and candidate queen placement on file k+1 on row rank, return true iff the new placement is valid (i.e. no queens attack one another). For example (qextends? [1] 2) should return false, since a queen on a1 can clearly attack a queen on b2. (qextends? [1 3] 5) should return true.
(qextend n partial-sol-list)
Given a list partial-sol-list of all partial solutions of length k (e.g. [[1 3] [1 4] ...]), return a list of all partial solutions of length k + 1. n is the size of the board (i.e. an n x n board).
(sol-count n)
Returns the total number of n-queens solutions on an n x n board.
(sol-density n)
Return the density of solutions on an n x n board where density is number of solutions / number of possible placements. Only consider placements that have exactly one queen per file (since this is the only type of placement the board representation can represent.)
(We won't test sol-count or sol-density on n < 1, so don't worry about the number of "solutions" for a board of size 0.)
Finally, plot a graph of solution count as a function of board size (n) from n = 1 to 10. (You may use whatever plotting software you like.)
We have provided lastname_project1.clj and public_test.clj as a skeleton file and a basic test file respectively.
Name your code lastname_project1.clj, your plot lastname_project1.png and push it to the submit server (submit.cs.umd.edu).
Note that we will be grading your projects by script, therefore it is very important to use the exact function names and arities described above. (You may change the name of the formal parameters if you wish, as this will not affect the grading scripts.)
We will provide a short set of public tests. We strongly recommend you write your own additional unit tests.
Don't use the ns macro - if you don't know what that is, don't worry about it.
All of the release tests will be suffixed with "-test". Please don't name any of your own functions in lastname_project1.clj this way to avoid naming collisions.
The grading will break down (roughly) into
- 90% - Correct performance
- 5% - Plot
- 5% - Documentation and coding style
Well-formatted Clojure code looks like this (see the Source Code Layout & Organization section):
https://github.com/bbatsov/clojure-style-guide#source-code-layout--organization
We're not going to be too strict about these rules nor should you treat them as absolutes. Just try to use some common sense. (If you can't easily follow the control flow of your own code, that's a bad sign.)