  By a tight tour in a $k$-uniform hypergraph $H$ we mean any sequence of its
vertices $(w_0,w_1,\ldots,w_{s-1})$ such that for all $i=0,\ldots,s-1$ the set
$e_i=\{w_i,w_{i+1}\ldots,w_{i+k-1}\}$ is an edge of $H$ (where operations on
indices are computed modulo $s$) and the sets $e_i$ for $i=0,\ldots,s-1$ are
pairwise different. A tight tour in $H$ is a tight Euler tour if it contains
all edges of $H$. We prove that the problem of deciding if a given $3$-uniform
hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved
in time $2^{o(m)}$ (where $m$ is the number of edges in the input hypergraph),
unless the ETH fails. We also present an exact exponential algorithm for the
problem, whose time complexity matches this lower bound, and the space
complexity is polynomial. In fact, this algorithm solves a more general problem
of computing the number of tight Euler tours in a given uniform hypergraph.
