Thanks Charlie, I can now see my error – my assumption that all solutions must

lie on y = x is wrong. In fact, the curves y = b^{x} and x = b^{y} can also intersect at

points off that line and, because of the symmetry about y = x, there would then

be an odd number of solutions, with one on the line and others in symmetric

pairs about y = x.

This is what is happening when 0 < b < e^{-e}, as Broll discovered.

At (e^{-e}, e^{-e}) the curve y = b^{x} crosses y = x with gradient -1 and for b < e^{-e}

there will be three solutions. For example:

when b = 0.05,x ~= 0.35022(y ~= 0.35022) x ~= 0.13736(y ~= 0.66266) x ~= 0.66266(y ~= 0.13736)

So:when 0 < b < e^{-e}there are 3 solutions whene^{-e} <= b <=1there is 1 solution when1 < b < e^{1/e}there are 2 solutions when b = e^{1/e}there is 1 solution whene^{1/e} < bthere are no solutions