Rosie, look at the Unused Whitney Permits 2010
The last Saturday in 2010 when all dayhike permits were used up was Sept 18. And those numbers are after the day ended, and all comers were given what was available. (I imagine people drop in on Saturday and ask for a dayhike permit just to hike up the trail a ways.)
If you show up for the dayhike permit lottery at 2 PM the day before, when they declare the no-shows and pull numbers from a hat, there will be an abundance of permits.
I wouldn't even worry about calling in to reserve permits for that late date. You have to pay $15 per slot for the reservations, but going in on Friday, they are free.