We have taken steps towards developing a method that enables an interactive humanoid robot to adapt its speed to a walking human that it is moving together with. This is difficult because the human is simultaneously adapting to the robot. From a case study in human-human walking interaction we established a hypothesis about how to read a human's speed preference based on a relationship between humans' walking speed and their relative position in the direction of walking. We conducted two experiments to verify this hypothesis: one with two humans walking together, and one with a human subject walking with a humanoid robot, Robovie-IV. For 11 out of 15 subjects who walked with the robot, the results were consistent with the speed-position relationship of the hypothesis. We also conducted a preferred speed estimation experiment for six of the subjects. All of them were satisfied with one or more of the speeds that our algorithm estimated and four of them answered one of the speeds as the best one if the algorithm was allowed to give three options. In the paper, we also discuss the difficulties and possibilities that we learned from this preliminary trial. Copyright 2007 ACM.