  The floor and ceiling functions appear often in mathematics and manipulating
sums involving floors and ceilings is a subtle game. Fortunately, the
well-known textbook Concrete Mathematics provides a nice introduction with a
number of techniques explained and a number of single or double sums treated as
exercises. For two such double sums we provide their single-sum analogues.
These closed-form identities are given in terms of a dual partition of the
multiset (regarded as a partition) of all b-ary digits of a nonnegative
integer. We also present the double- and single-sum analogues involving the
fractional part function and the shifted fractional part function.
