Abstract: In this 30-minute combinatorics talk, we will discuss a relationship between the box-ball systems and Robinson-Schensted-Knuth tableaux. The Robinson-Schensted-Knuth (RSK) insertion algorithm is a bijection between the symmetric group and pairs (P,Q) of standard Young tableaux (arrays of positive integers with increasing rows and columns) of the same shape. The tableaux P and Q are called the insertion tableau and the recording tableau of the corresponding permutation. A box-ball system (BBS) is a discrete dynamical system which can be thought of as a collection of time states each containing a permutation. We can move forward and backward in this system by rearranging the permutation using certain rules. A box-ball system will reach a steady state after a finite number of steps. From any steady state, we can construct a tableau (not necessarily standard) called the soliton decomposition of the box-ball system. The minimum number of time steps needed to go from a permutation w to a steady state in a box-ball system is called the steady-state time of w.  We prove the following: if the soliton decomposition of a permutation is a standard Young tableau or if its shape coincides with its RSK shape, then its soliton decomposition and its RSK insertion tableau are equal. We conjecture that the steady-state time of a permutation is determined by its recording tableau Q; we prove this conjecture for a family of recording tableaux and are currently working on proving the conjecture in general.