  There are many examples of 3-folds of general type with $\chi(\mathcal
{O})=1$ found by Iano-Fletcher and Reid about twenty years ago. Iano-Fletcher
has ever proved $P_{12}(X)\ge 1$ and $P_{24}(X)\ge 2$ for all minimal 3-folds
$X$ of general type with $\chi(\mathcal {O}_X)=1$. In this paper, we improve on
Fletcher's method. Our main result is that $\phi_m$ is birational onto its
image for all $m\ge 54$. To prove this we establish a general theory on
birationality for general 3-folds which improve known results. Also we will
show $P_m\geq 1$ for all $m\geq 14$ and $P_{18}\geq 3$.
