We study some combinatoric properties of the Morse sequence, linked wi
th its ergodic properties of local rank one and local funny rank one;
we show that the maximum part of the Morse sequence that may be covere
d by disjoint translates of one word is exactly of density 2/3, even a
llowing for some errors in the tiling; when we replace words by patter
ns (words with holes), 2/3 can be replaced by at least 5/6.