I got the strategy for this guide from a youtube video of a draw match of Reversi.

https://www.youtube.com/watch?v=txkEiawzS0c

I realize there are also videos of the achievement from the game, but I am dumb and didn't think to look that up before making this guide. Oh well...

The first thing to do is click the Mirror Tiles app in the Rubbish Bin.

When the window opens, you will be presented with an 8x8 grid. For the purposes of this guide (and probably every grid-based game ever), the horizontal tiles will be labeled A-H, and the vertical tiles will be labeled 1-8. So in case you are still not following, the top left tile is A1, and the bottom right tile is H8.

Click the following tiles in order to get the achievement: