This one is hurting my brain!

Well, not entirely, but it is right if you've roundeded it. But the real question is... what is 1.414 more commonly expressed as?

Oh, and the solution for 10 is actually lower. Get your head around that one!

That is definitely correct, and it should be straightforward from where Ganesh left it. Now, it should be pointed out, as mathsyperson said, that the solution you get for y=10 in this way is lower than for y=2.

But if you think about it, for x and y > 1, x < y implies that x^x^x^... is always less than y^y^y^... , so this implies that 10 < 2, which is nonsense. I'm still thinking about this, but it should mean that there is a cutoff value for y above  which there is no x solution.

So my final puzzle related to this is: find the largest value of y such that

Last edited by fgarb (2006-03-12 16:38:16)

That is what I get as well. If anyone has any idea why e ends up a solution to this problem, I'd love to hear it! That annoying constant seems to have a way of popping up everywhere.

Ah. Fair enough. Sorry.

Wouldn't the cut-off point be y=e?
It makes sense that the cut-off point is e, because that's when the gradient of x starts becoming negative. But maybe I've missed something.

Edit: I've done some research in Excel and you're right.

The 10th root of 10 is implied to be the solution of x for y=10, but using that value makes y converge to 1.371288574.

Yes sabujakash, above the magic number 1.44466786... the series diverges.