Now I'm stumped too. You're right, if we use x(t)=t+1-sqrt(1+e^(-2t)) and y(t)=2cosh(t)-sqrt(e^(2t)+1)) then (x(t)-x(s))^2 = 4y(t)y(s) produces a really horrible equation. Wolfram Alpha can't even find the next point. Maybe this isn't the best method?
Might I ask what the equation you got that related s and t was? If a circle intersects at t, then its center lies at x=t+1-sqrt(1+e^(-2t)) (and from that I derived the equation r = 2cosh(t)-sqrt(e^(2t)+1)). Is that the equation you used?
I don't know if this helps, but I found that the radius of a circle r can be expressed explicitly in terms of where the circle touches e^(-x), at x. r = 2cosh(x)-sqrt(e^(2x)+1).
Thanks for your responses. Vargo, I was wondering the same thing about idea 2; it would be very interesting if they did lie on such a curve. Would that imply their centers also lie on the curve?
I thought of a problem a few days ago and I have no idea as to its solution. I posted this on Reddit and xkcd forums earlier but not much has been solved apart from the area of one circle. Suppose you have a boundary formed by the curve y=e^(-x), and the lines x=0 and y=0. In this boundary you...