AbstractA key property of CANDECOMP/PARAFAC is the essential uniqueness it displays under certain conditions. It has been known for a long time that, when these conditions are not met, partial uniqueness may remain. Whereas considerable progress has been made in the study of conditions for uniqueness, the study of partial uniqueness has lagged behind. The only well known cases are those of overfactoring, when more components are extracted than are required for perfect fit, and those cases where the data do not have enough system variation, resulting in proportional components for one or more modes. The present paper deals with partial uniqueness in cases where the smallest number of components is extracted that yield perfect fit. For the case of K×K×2 arrays of rank K, randomly sampled from a continuous distribution, it is shown that partial uniqueness, with some components unique and others differing between solutions, arises with probability zero. Also, a closed‐form CANDECOMP/PARAFAC solution is derived for 5×3×3 arrays when these happen to have rank 5. In such cases, any two different solutions share four of the five components. This phenomenon will be traced back to a sixth degree polynomial having six real roots, any five of which can be picked to construct a solution. Copyright © 2004 John Wiley & Sons, Ltd.