  An augmented generalized happy function $S_{[c,b]}$ maps a positive integer
to the sum of the squares of its base $b$ digits plus $c$. In this paper, we
study various properties of the fixed points of $S_{[c,b]}$; count the number
of fixed points of $\S_{[c,b]}$, for $b \geq 2$ and $0<c<3b-3$; and prove that,
for each $b \geq 2$, there exist arbitrarily many consecutive values of $c$ for
which $S_{[c,b]}$ has no fixed point.
