  We find the winning strategy for a class of truncation games played on words.
As a consequence of the present author's recent results on some of these games
we obtain new formulas for Bernoulli numbers and polynomials of the second kind
and a new combinatorial model for the number of connected permutations of given
rank. For connected permutations, the decomposition used to find the winning
strategy is shown to be bijectively equivalent to King's decomposition, used to
recursively generate a transposition Gray code of the connected permutations.
