Well for green machine washers: at a farm where I used to live, we had a gas powered wringer machine--but we took the gas motor off, added a starter engine and ran it all on solar collectors. The drying was also solar / wind, on a clothes line.
McGyver, I suspect some of the problem with the competitions is not the green element, but the fact that people are trying to invent such things all the time and if it worked, it would soon be on the market. Probably there are examples of some such competitions that actually created a result by proposing new threads of thought--but probably best suited for "appropriate technology" solutions for rural or undeveloped areas.