132 lines
1.9 KiB
TeX
132 lines
1.9 KiB
TeX
\section{Crosses}
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You are given an $n \times n$ grid. Some of its squares are white, some are gray. Your goal is to place $n$ crosses on white cells so that each row and each column contains exactly one cross.
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\vspace{2ex}
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Here is an example of such a grid, including a possible solution.
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\newcommand{\bx}[2]{
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\draw[
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line width = 1.5mm
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]
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(#1 + 0.3, #2 + 0.3) -- (#1 + 0.7, #2 + 0.7)
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(#1 + 0.7, #2 + 0.3) -- (#1 + 0.3, #2 + 0.7);
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}
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\newcommand{\dk}[2]{
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\draw[
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line width = 0mm,
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fill = gray
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]
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(#1, #2) --
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(#1 + 1, #2) --
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(#1 + 1, #2 + 1) --
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(#1, #2 + 1);
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}
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\begin{center}
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\begin{tikzpicture}[
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scale = 0.8
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]
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% Dark squares
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\dk{0}{2}
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\dk{1}{0}
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\dk{1}{1}
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\dk{1}{2}
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\dk{1}{4}
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\dk{2}{2}
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\dk{2}{4}
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\dk{3}{0}
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\dk{3}{1}
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\dk{3}{3}
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\dk{3}{4}
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\dk{4}{3}
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\dk{4}{1}
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% Base grid
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\foreach \x in {0,...,5} {
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\draw[line width = 0.4mm]
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(0, \x) -- (5, \x)
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(\x, 0) -- (\x, 5);
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}
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% X marks
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\bx{0}{4}
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\bx{1}{3}
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\bx{2}{1}
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\bx{3}{2}
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\bx{4}{0}
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\end{tikzpicture}
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\end{center}
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\problem{}
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Find a solution for the following grid.
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\begin{center}
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\begin{tikzpicture}[
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scale = 1
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]
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% Dark squares
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\dk{0}{2}
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\dk{0}{3}
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\dk{0}{6}
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\dk{0}{7}
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\dk{1}{0}
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\dk{1}{1}
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\dk{1}{4}
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\dk{1}{5}
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\dk{1}{6}
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\dk{1}{7}
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\dk{2}{0}
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\dk{2}{1}
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\dk{2}{3}
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\dk{2}{4}
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\dk{2}{5}
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\dk{2}{6}
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\dk{2}{7}
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\dk{3}{1}
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\dk{3}{2}
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\dk{3}{3}
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\dk{3}{4}
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\dk{3}{5}
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\dk{3}{6}
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\dk{4}{0}
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\dk{4}{1}
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\dk{4}{2}
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\dk{4}{3}
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\dk{4}{6}
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\dk{5}{1}
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\dk{5}{4}
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\dk{5}{5}
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\dk{5}{6}
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\dk{6}{0}
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\dk{6}{1}
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\dk{6}{2}
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\dk{6}{3}
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\dk{6}{4}
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\dk{6}{5}
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\dk{7}{0}
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\dk{7}{4}
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\dk{7}{6}
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\dk{7}{7}
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% Base grid
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\foreach \x in {0,...,8} {
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\draw[line width = 0.4mm]
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(0, \x) -- (8, \x)
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(\x, 0) -- (\x, 8);
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}
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\end{tikzpicture}
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\end{center}
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\pagebreak
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\problem{}
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Turn this into a network flow problem that can be solved with the Ford-Fulkerson algorithm.
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\vfill
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\pagebreak
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