Bashar_AL-Rfooh wrote: ↑Sat Feb 06, 2021 5:20 am
Hi I am really confused about reflections and rotations if some one can verify for me how many unique and repeated pattern are there in 3 * 3 matrix I will be thankful

Let me see if it helps to explain a bit more before giving out hint numbers. Consider the following 3x3 matrices:

$

\begin{pmatrix} 1&2&3 \\ 4&5&6 \\ 7&8&9 \end{pmatrix}

\begin{pmatrix} 7&4&1 \\ 8&5&2 \\ 9&6&3 \end{pmatrix}

\begin{pmatrix} 9&8&7 \\ 6&5&4 \\ 3&2&1 \end{pmatrix}

\begin{pmatrix} 3&6&9 \\ 2&5&8 \\ 1&4&7 \end{pmatrix}

\\

\begin{pmatrix} 3&2&1 \\ 6&5&4 \\ 9&8&7 \end{pmatrix}

\begin{pmatrix} 9&6&3 \\ 8&5&2 \\ 7&4&1 \end{pmatrix}

\begin{pmatrix} 7&8&9 \\ 4&5&6 \\ 1&2&3 \end{pmatrix}

\begin{pmatrix} 1&4&7 \\ 2&5&8 \\ 3&6&9 \end{pmatrix}

$

"Up to rotations and reflections", those are all the same matrix. Start with the upper left matrix, then pick it up and rotate it 90 degrees clockwise at a time, and you'll get the other 3 on that row. The lower left matrix is just the upper left one, flipped across the vertical axis of the middle column. That's a reflection. You can then take that reflected matrix, and rotate it 90 degrees clockwise at a time to get the other 3 on the second row. So there are 8 ways to move this matrix around, and still have fundamentally the same matrix.