In how many ways can you construct a 6x6 chart with only 1 and -1 such that in every row and column the product is always positive.
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5
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$\begingroup$ is it mendatory to use both of them...? $\endgroup$– manshuDec 19, 2015 at 13:30
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$\begingroup$ No, you can use only 1 if you like. $\endgroup$– algebra1Dec 19, 2015 at 13:30
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$\begingroup$ Are reflections and rotations counted as separate ways, or would $$\begin{array}{|c|c|c|c|c|c|} \hline -1 & -1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline \end{array} $$ be considered the same as $$\begin{array}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & -1 & -1\\ \hline \end{array} $$ ? $\endgroup$– GentlePurpleRainDec 19, 2015 at 14:21
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$\begingroup$ @GentlePurpleRain How did you post those images in your comment? Thanks $\endgroup$– Ruchir BaroniaDec 20, 2015 at 7:15
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$\begingroup$ @RuchirBaronia They aren't images; they're mathematical "equations" using MathJax markup. You should be able to right-click on the grid and view the underlying MathJax code (I'm on my phone right now, so I can't verify). Choose the "Tex" option. $\endgroup$– GentlePurpleRainDec 20, 2015 at 13:51
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1 Answer
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The number of ways is
$2^{25}$
The idea is that
you can put whatever you want in the upper left 5 by 5 square, but then there is a unique way to complete this to a valid 6 by 6 square.
This is explained in the below picture:
answered Dec 19, 2015 at 14:23