Quite large (400k) mysql database design for multiple users
Use (word_id, user_id) table to store solved combinations, that way you won't store the NULLs. How many words will an average user solve? 5-20 maybe? 400k * 20 = 8M rows, thats nothing (the two column table is quite compact).

Do not even try to alter the table on each new user, that would stop working really quickly.

I suggest InnoDB for anything new unless there is a specific reason against it.