Abstract. Kaplansky's zero divisor conjecture envisions that for a torsion-free group G and an integral domain R, the group ring ROEG does not contain non-trivial zero divisors. We define the length of an element˛2 ROEG as the minimal non-negative integer k for which there are ring elements r 1 ; : : : ; r k 2 R and group elements g 1 ; : : : ; g k 2 G such that˛D r 1 g 1 C C r k g k . We investigate the conjecture when R is the field of rational numbers. By a reduction to the finite field with two elements, we show that if˛ˇD 0 for non-trivial elements in the group ring of a torsion-free group over the rationals, then the lengths of˛andˇcannot be among certain combinations. More precisely, we show for various pairs of integers .i; j / that if one of the lengths is at most i, then the other length must exceed j . Using combinatorial arguments we show this for the pairs .3; 6/ and .4; 4/. With a computer-assisted approach we strengthen this to show the statement holds for the pairs .3; 16/ and .4; 7/. As part of our method, we describe a combinatorial structure, which we call matched rectangles, and show that for these a canonical labeling can be computed in quadratic time. Each matched rectangle gives rise to a presentation of a group. These associated groups are universal in the sense that there is no counter-example to the conjecture among them if and only if the conjecture is true over the rationals.