Actually it IS still possible to solve this one the "logical" way. You just have to do some patching first, because doing 1023 moves in 60 seconds is really challenging.
BUT, don't do it. I know how to solve the Towers of Hanoi since I'm a child, but still doing 1023 moves needs time, and precision. You must not do any false move, because you cannot correct it within the 1023 moves limit, so you need the shortest possible solution. And also you have to be lucky that the browser doesn't crash and as Z already wrote, that your network connection is stable as the applet is not as stable as it could be. Actually it took me some tries, because I tried to do it too fast and made errors, or the browser crashed, or whatever, and of course not before at least 500 moves were made already.
Doing it the "non-logical" way did cost me about 5-10 minutes and is far more reliable.