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that satisfies a property). Namely, we can generate all strings of length n
, which may satisfy the given property, and then show that no string can appear more than once (or else the property would be true). To do this, we first define a "graph" for $S = \1,2,\ldots,n$; this consists of all directed edges from i
to j
whenever we can string together an i
-word and a j
-word. In your case, we have $S=\1,2,\ldots,n$; note that this is not the same set as your S
! The directed edges will be from i
to 2i
and from 2i
to 2i+1
. Note that, for 1 \leq i \leq n/2
, there are i
such edges, since a word of length n/2
can be built up using the first 847798691e