I have always been intrigued by the Pythagorean triple (20, 21, 29). It interests me for two reasons. Firstly it looks really improbable and I am fascinated by the things that remain counter-intuitive in mathematics despite my repeated verification of them. Secondly apart from the basic (3, 4 ,5) triangle, it is the only small triple in which the first two numbers are consecutive. For ease of reference I shall call these special triples. I had never even been sure how many more of these there were, although I had tracked some down by computer at one point. In the middle of preparing some follow-up work for a lecture, I decided to look at these again to see whether there was anything suitable for the students to do. I uncovered a connection that certainly I had never known before.