A schema mapping is a declarative specification of the relationship between instances of a source schema and a target schema. The data exchange (or data translation) problem asks: given an instance over the source schema, materialize an instance (or solution) over the target schema that satisfies the schema mapping. In general, a given source instance may have numerous different solutions. Among all the solutions, universal solutions and core universal solutions have been singled out and extensively studied. A universal solution is a most general one and also represents the entire space of solutions, while a core universal solution is the smallest universal solution and is unique up to isomorphism (hence, we can talk about the core). The problem of designing efficient algorithms for computing the core has attracted considerable attention in recent years. In this paper, we present a method for directly computing the core by SQL queries, when schema mappings are specified by source-totarget tuple-generating dependencies (s-t tgds). Unlike prior methods that, given a source instance, first compute a target instance and then recursively minimize that instance to the core, our method avoids the construction of such intermediate instances. This is done by rewriting the schema mapping into a laconic schema mapping that is specified by first-order s-t tgds with a linear order in the active domain of the source instances. A laconic schema mapping has the property that a “direct translation” of the source instance according to the laconic schema mapping produces the core. Furthermore, a laconic schema mapping can be easily translated into SQL, hence it can be optimized and executed by a database system to produce the core. We also show that our results are optimal: the use of the linear order is inevitable and, in general, schema mappings with constraints over the target schema cannot be rewritten to a laconic schema mapping.