It shouldn't be surprising that babies begin to laugh when they are six to twelve weeks old. After all, six weeks is usually when parents notice that colic, which can make babies fussy, is starting to improve.
Sources:
Functional developmental evaluation. Prerequisite to habilitation. Capute AJ - Pediatr Clin North Am - 01-FEB-1973; 20(1): 3-26
Linguistic and auditory milestones during the first two years of life: a language inventory for the practitioner. Capute AJ - Clin Pediatr (Phila) - 01-NOV-1978; 17(11): 847-53
Clinical linguistic and auditory milestone scale: prediction of cognition in infancy. Capute AJ - Dev Med Child Neurol - 01-DEC-1986; 28(6): 762-71
The Denver Developmental Assessment (Denver II)
