I recently read the book "Word Freaks" which is about competition scrabble players and their world. One mentioned was an Orthodox Jew who had to have someone else arrainge the tiles and record his score.
There is also a long discussion of various rulings he got from different religious authorities; some argued that using the board with the raised partitions between letter-spaces is acceptable -- you are not "writing" as long as the Scrabble letters are separated, and you are not "breaking" when you disassemble the board -- while others argued that the flat board is better because it has no permanence -- and therefore is not like "writing".
Interestingly, he was able to keep score by moving tokens around on a pre-written scoresheet - a technique that can probably be applied to D&D.