In a search for fast (≥60MeV) Σ-hyperons emitted from multi-nucleon K--captures at rest, 65 events were found. Using our events, as well as those published before by other groups, we have been able to set an upper limit of ∼ 6% to the reaction K nn→ Σ+n, relative to the total fast (2N)Σ -production. This was achieved by a detailed study of the distribution of the visible energy associated with K -stars giving rise to fast identified Σ + and Σ - hyperons. Also, from the visible energy distribution, we have been able to obtain an unbiased estimate for the ratio of the K-np→ Σ-p and K(su-) pp ar Σ(su+)n reactions. From the Auger electrons and short prongs associated with the 2N-events and with pion-emitting stars we can show that, very probably, the total 2N-yield increases significantly with the atomic number of the nucleus in which the K(su-)-capture takes place. A short discussion of the results, in view of several proposed models for the multi-nucleon capture process is also presented.