  In this article we develop counterexamples to the Hasse principle using only
techniques from undergraduate number theory and algebra. By keeping the
technical prerequisites to a minimum, we hope to provide a path for
nonspecialists to this interesting area of number theory. The counterexamples
considered here extend the classical counterexample of Lind and Reichardt. As
discussed in an appendix, this type of counterexample is important in the
theory of elliptic curves: today they are interpreted as nontrivial elements in
the Tate--Shafarevich group.
