We propose a simple distributed algorithm for balancing indivisible tokens on graphs. The algorithm is completely deterministic, though it tries to imitate (and enhance) a random algorithm by keeping the accumulated rounding errors as small as possible. Our new algorithm surprisingly closely approximates the idealized process (where the tokens are divisible) on important network topologies. On d-dimensional torus graphs with n nodes it deviates from the idealized process only by an additive constant. In contrast to that, the randomized rounding approach of Friedrich and Sauerwald [STOC'09] can deviate up to Omega(polylog n) and the deterministic algorithm of Rabani, Sinclair and Wanka [FOCS'98] has a deviation of Omega(n1/d). This makes our quasirandom algorithm the first known algorithm for this setting which is optimal both in time and achieved smoothness. We further show that also on the hypercube our algorithm has a smaller deviation from the idealized process than the previous algorithms. To prove these results, we derive several combinatorial and probabilistic results that we believe to be of independent interest.