Why all the confusion and heartache? Some people afford the AMC way more importance than it deserves. It's a patch of cloth for goodness sake. It's not going to get you a better job, get you laid, or get you to the summit of K2. It's a patch of cloth with some writing on it, indicating you followed the rules of their game, 48 times. That's all it is. If that kind of thing is important to you, then follow their rules.......it's really so very simple. Hiking per a list is good clean fun; if you want to do it, just follow the rules....so easy to do.
But if you do drive to the summit of Washington and then hike to Madison, trust me, it'll be one of the most beautiful hikes you'll ever do in the northeast. Keep in mind, none of these peaks (Washington, Jefferson, Adams, Madison) will "count" toward your 4k patch, so you'll have to tax your imagination and think of some other reason to enjoy this hike.