Your best bet for getting a permit is to be flexible about start dates. I visit the Canyon 1-2 times per year and have not yet failed to get a permit for an itinerary near my request.

What to do? First, I don't bother faxing in my request; it is a frustrating process at best. I download the permit application form and fill it out with an itinerary that avoids any of the "corridor" camp sites such as Indian Gardens. At the bottom of the form is a space in which you can enter a range of start dates and other potential modifications of your itinerary. Fill this out using as much flexibility as possible. I usually enter at least a two week start date window and also alternative campsites such as Hermit Rapids rather than Hermit Camp or Granite Rapids rather than Monument Creek. I then mail the form to the backcountry office aiming at getting it there a day or so before the first of the appropriate month. The permit request cannot be made more than four months before the planned trip. So, if you wanted a start date for this coming April you should have mailed or FAXed your form in by the end of November.

I just sent in a permit request for this coming April. I listed acceptable start dates anywhere from April 15 to 30. My desired itinerary was: Boucher Creek for night 1, Monument Creek for night 2 and Horn Creek for night 3. I plan on exiting via the Bright Angel Trail. I listed a number of alternative camps such as Hermit Creek, Hermit Rapids and Granite Rapids. I don't know yet whether I've got a permit but if I don't I'll try again at the end of this month for May dates.

And, as WD mentioned, you can always just show up, get in line at the Backcountry Office and hope for a no-show or a cancellation.