Summary: It has become gradually accepted that proof and proving are essential at all grades of mathematical learning. Among the various aspects of proof and proving, this study addresses proofs and refutations described by {\it I. Lakatos} [Proofs and refutations. The logic of mathematical discovery. Cambridge: Cambridge University Press (1976; ME 2015d.00503)], in particular a part of increasing content by deductive guessing, to introduce an authentic process into mathematics classrooms. This paper analyzes proof problems with diagrams as an appropriate task for this mathematical action, and illustrates, with a sequence of three lessons in the eighth grade, that a certain type of such problems can provide students with an opportunity to engage in this action.