I thought it might be of interest for people to see progress to date. I will mention here a selection of proposed solutions. If you would prefer your solution not to be listed here, please let me know when you write to me. If you are shy about the consequent publicity, you may suggest a pseudonym!
28 August: Pie Golly gives an elegant "pie-friendly" solution using only 3 pie wedges with an eaten area about 1/12 of the original pie, but this experienced problem solver is not yet content and I am sure improvements are on the way.
31 August: Danny also has essentially the same solution with about 1/12 eaten. Meanwhile Pie Golly produced an arcane and complicated improvement, nontrivial to verify. Mr Tweedy attained the same improved bound with a simple self-explanatory picture. This is very similar to Simple Simon's solution - but SS had a head start on the problem.
I have not received any upper bound less than 1, though I am expecting progress from Pie Golly. Simple Simon has some constructions which may be obstructions to any easy deductions.
6 September: Mr Tweedy, Pie Golly, Danny and Simple Simon are in acchord with constructions corresponding to an eaten area of approximately 3/104. Mr Tweedy sent a picture which illustrated one such construction beautifully. Pie Golly has since made astonishing progress in closing the gap between upper and lower bounds.
You are invited to send me ( ) solutions before September 13, 2008. These could be lower bounds, upper bounds or both, according to taste.
The original stimulus for this problem was a variant where half the pie had been eaten and the problem was to find the smallest plate which could hold the remainder. Solutions were posted which claimed to be optimal but clearly weren't.
The general problem would be to show how the size of the optimal plate varies as the eaten proportion rises from "whatever the solution to my problem is" to 1, where the limit of plate size reaches 1/2. Does this function decrease smoothly or are there stationary points or jumps?