  Kuniba, Okado, Takagi and Yamada have found that the time-evolution of the
Takahashi-Satsuma box-ball system can be linearized by considering rigged
configurations associated with states of the box-ball system. We introduce a
simple way to understand the rigged configuration of $\mathfrak{sl}_2$-type,
and give an elementary proof of the linearization property. Our approach can be
applied to a box-ball system with finite carrier, which is related to a
discrete modified KdV equation, and also to the combinatorial $R$-matrix of
$A_1^{(1)}$-type. We also discuss combinatorial statistics and related
fermionic formulas associated with the states of the box-ball systems. A
fermionic-type formula we obtain for the finite carrier case seems to be new.
