i'm making my first chess board. needed a full size portable so i made one that separates into 4 4x4 sections that connect via magnets and moulded edges. fwiw - i got pretty good results getting the squares to align by gluing the long strips alternating between light and dark. then once that set cutting more strips from the alternating color panel. flip every other one and glue up the strips into the final panel.
i would glue the 1/4" board to a backer.