The Winograd Challenge has been advocated as a test of computer understanding with respect to commonsense reasoning. The challenge is based on Winograd Schemas: sentences that contain correferential ambiguities. Most Winograd Schemas are relatively easy for human subjects, and today the best computer systems for the Winograd Challenge can work close to human performance. In this paper, we examine the assumptions behind the Winograd Challenge, and investigate how far we can push the difficulty level of Winograd Schemas, proposing various strategies to build really challenging schemas.