# Multiplications in a chart, problem solving

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.

• is it mendatory to use both of them...? – manshu Dec 19 '15 at 13:30
• No, you can use only 1 if you like. – algebra1 Dec 19 '15 at 13:30
• 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 $$\begin{array}{|c|c|c|c|c|c|} \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & 1 & 1\\ \hline 1 & 1 & 1 & 1 & -1 & -1\\ \hline \end{array}$$ ? – GentlePurpleRain Dec 19 '15 at 14:21
• @GentlePurpleRain How did you post those images in your comment? Thanks – Ruchir Baronia Dec 20 '15 at 7:15
• @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. – GentlePurpleRain Dec 20 '15 at 13:51

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: