  When can $n$ given numbers be combined using arithmetic operators from a
given subset of $\{+, -, \times, \div\}$ to obtain a given target number? We
study three variations of this problem of Arithmetic Expression Construction:
when the expression (1) is unconstrained; (2) has a specified pattern of
parentheses and operators (and only the numbers need to be assigned to blanks);
or (3) must match a specified ordering of the numbers (but the operators and
parenthesization are free). For each of these variants, and many of the subsets
of $\{+,-,\times,\div\}$, we prove the problem NP-complete, sometimes in the
weak sense and sometimes in the strong sense. Most of these proofs make use of
a "rational function framework" which proves equivalence of these problems for
values in rational functions with values in positive integers.
