Est. 1998 — 27 years of woodworking knowledge
Tuesday Puzzle
Bill Earl
A group of 20 people are seated at a round table at a restaurant. 10 of them order beef and 10 of them order chicken. The waiter forgets who ordered what and just delivers the plates in random order.
Is there a way to rotate the table that guarantees at at least 10 people will get what they ordered?
Re: Tuesday Puzzle
Alex Y
I don't know how, but if there were not ...
Hint
Bill Earl
Alex might be on to it, but I'm not sure.
It seems intuitive that there should be a way, but can you prove it?
Can you prove that there is not?
Re: Hint
Alex Y
Yes, my answer was confusing. I initially thought that the proof would be proof by contradiction, hence my answer. But it is a more straight-forward, if circular, argument that proves the answer.
An ancillary question:
There are some seating arrangements in which is it possible to put down the plates so that exactly ten people will have the right meals, no matter how the table is turned. Is that possible no matter what the seating arrangement?
Re: This will sound smart, but..
David Weaver
.. I think you've got a good chance of getting everyone satisfied if you send half of the people away and give everyone left both dishes (well, everyone means everyone in the half that is left).
Or if you give everyone the same dish.
But with orienting chairs, I'm having trouble figuring out why the served entrees couldn't just be entirely wrong, or entirely right...or basically any pair 0 right, 2 right, 4 right, ....
Re: This will sound smart, but..
Bill Earl
You are correct. There are some arrangements that will be entirely wrong. The question is whether there is guaranteed to be a way to satisfy at least half of the people simply by rotating the table.
Re: Tuesday Puzzle
Alex Y
Yes, there will be a rotated position in which at least ten people have the right meal.
Proof: Rotate the table through all 20 positions. During that rotation, each of the 20 guests will be sitting in front of the meal of their choice ten times. Thus, when the table goes full-circle, there will have been 200 correct meal-guest pairings. Since there are only 20 table positions, that means the average number of guests sitting in front of their desired meal is ten -- pretty had to do if there are less than ten at all table positions!
Actually
Lee Schierer McKean, PA
If you rotate the table 20 times, all the food will get cold and no one will be satisfied. 
Lee