The Golden Child will complete the 11th puzzle on his own. You don't need to try to make him do anything (and it won't work if you try). Just give it some more time, and he will take care of it for you. You could just watch him for awhile longer - it's entertaining to see him solve that puzzle, especially when you have a lot of villagers.

I have found that the anticipated event happens sooner if you can leave the game running (rather than leaving it off most of the time and checking periodically). When I leave the game running, I've never seen it take more than a few hours. Some people have reported waiting weeks (if they check on the game somewhat infrequently). Whatever your play style, it will happen eventually.