The purpose of this article is to describe a couple of challenging mathematical problems that involve paper folding. These problem solving tasks can be used to foster geometric and algebraic thinking among students. The context of paper folding makes some of the abstract mathematical ideas involved relatively concrete. When implemented appropriately these activities have the potential to address many of the mathematical proficiencies, as delineated by Australian Curriculum and Assessment Reporting Authority (ACARA, 2014). [This paper was first published in the "Australian Senior Mathematics Journal," in 2014 (EJ1093386).]