  We investigate the decomposition problem of balls into finitely many
congruent pieces in dimension $d=2k$. In addition, we prove that the $d$
dimensional unit ball $B_d$ can be divided into finitely many congruent pieces
if $d=4$ or $d\ge 6$. We show that the minimal number of required pieces is
less than $20d$ if $d \ge 10$.
