Why 15 and 290 are cool (making numbers out of sums of squares)

Given a particular recipe for making numbers out of sums of squares, such as or , when can the recipe make all positive integers? And if you have a recipe that you have successfully tested on all small numbers, can you be confident it will work for all numbers? Some of these questions have recently (last 20 years) been answered by John Conway, Manjul Bhargava, and others. But what do 15 and 290 have to do with it? Well, watch the videos: