Scrabble is not writing if it is regular scrabble. The travel edition where the pieces snap in, is likely a problem of writing.
However, there is a different problem with scrabble, which is that you usually write down the score, since it is hard to keep track of. It is generally assur to do things on shabbos which usually involve writing.
I have heard of a heter that since we only play on shabbos, we never write down the score. You should consult with a competent authority regarding this.