Two dominant models of mitotic spindle assembly are search-and-capture (SAC) and acentrosomal microtubule assembly (AMA). SAC predicts that kinetochores are captured randomly by dynamically unstable centrosomal microtubules (MTs), while AMA posits that kinetochore-associated MT bundles get integrated with centrosomal asters at random times. Both models predict a slow spindle assembly plagued by errors. Recent data shows that randomness in kinetochore capture is very low, that spindle assembles in relatively precise stages, and that small and dynamic MT ‘clouds’ near the kinetochores are critical in assisting centrosomal MTs in the spindle assembly. We used 3D tracking of centrosomes and kinetochores in mitotic animal cells to inform a computational model, according to which: 1) initially, when the centrosomes are proximal, the centrosomal MTs rapidly and indiscriminately ‘skewer’ the kinetochore MT clouds, effectively establishing lateral MT-kinetochore connections. When the pole-pole distance reaches a threshold, the polarity sorting of the kinetochore MTs and integration of the kinetochore and centrosomal MTs lead to establishing a vast majority of amphitelic attachments within a narrow time window.