In attempting to solve a problem in the fewest possible moves, one mus
t determine moves that are individually necessary and jointly sufficie
nt. This amounts to constructing a demonstration or proof of the solut
ion. The development of proof construction was explored through two pr
oblems-a hidden figure task and a variant of the game ''Mastermind'' (
J. Piaget (1987). Possibility and necessity: Vol. 2. The role of neces
sity in cognitive development. Minneapolis: Univ. of Minnesota Press).
In Experiment 1, children 11 to 12 years of age distinguished necessa
ry from nonnecessary moves and recognized when sufficient evidence had
been established to determine a solution to a greater extent than chi
ldren 7 to 8 years of age. In a second experiment, the original hidden
-figure task was modified with the aim of reducing its information-pro
cessing demands. The performance of 8- to 9-year-olds on the modified
problems was comparable to, and in certain respects surpassed, the per
formance of the oldest children on the original problem in Experiment
1. Seven-year-olds did not perform well on the modified problems even
with training. Results are discussed in terms of theory and research o
n children's understanding of necessity and sufficiency. (C) 1997 Acad
emic Press.