Throw back, found these pictures and had to share them :).
Over the course of about 40 hours I did this room, first I painted everything white, then I used painters tape to make the grid. Then I painted white over the painters tape to seal it well so the other colors wouldn't bleed through under the tape.
Then I used a notebook to make a template of the squares to randomly place colors where they wouldn't be same color next to each other. Painted all those and then finally eventually was able to remove the tape. Still needs the crown molding though! I've since repainted the room to re-purpose as a guest room and it still doesn't have the crown molding :). Now it needs to be repainted anyway because it's my son's room and he needs boy colors in there.
Things I'd change if I had to do it all over again, I'd probably paint three walls a solid color and use only three colors for the squares. Four walls of squares is time consuming AND a little busy. But my daughter loved it for a lot of years.