  We consider the problem of allocating indivisible goods to agents with
additive valuation functions. Kurokawa, Procaccia and Wang {[JACM, 2018]}
present instances for which every allocation gives some agent less than her
maximin share. We present such examples with larger gaps. For three agents and
nine items, we design an instance in which at least one agent does not get more
than a $\frac{39}{40}$ fraction of her maximin share. {Moreover, we show that
there is no negative example in which the difference between the number of
items and the number of agents is smaller than six, and that the gap (of
$\frac{1}{40}$) of our example is worst possible among all instances with nine
items.}
  For $n \ge 4$ agents, we show examples in which at least one agent does not
get more than a $1 - \frac{1}{n^4}$ fraction of her maximin share. {In the
instances designed by Kurokawa, Procaccia and Wang, the gap is exponentially
small in $n$.}
  Our proof techniques extend to allocation of chores (items of negative
value), though the quantitative bounds for chores are different from those for
goods. For three agents and nine chores, we design an instance in which the MMS
gap is $\frac{1}{43}$.
