## Problem of the Week

** Thanksgiving Week Challenge!** This week's puzzle comes in two parts. The first part will count as the normal puzzle for the week of 11/27. The second part is a challenging bonus question. You have the week of Thanskgiving and the following week to come up with a soluion.

Due Friday, December 1, 2023 at 3 pm submitted to
this Google form. ** You must be logged into your NAU gmail to submit via this form. **

### Part 1

Aditya is hosting Thanksgiving dinner. They have 48 guests coming. Adiyta will seat everyone at a vrey large round table. On the table between each pair of settings is a wine glass with a cloth napkin arranged inside. As each person is seated (by Adiyta), they take a napkin randomly from the left or the right. Adiyta does not see which one they take.
In what order should the seats be assigned to maximize the expected number of guests who don't get napkins?

### Part 2

The set up is the same as above, but this time Adiyta is not seating the guests. The seats are occupied in random order. If the number of guests grow to be very large, what fraction of them (asymptotically) will end up without a napkin?

**Rules for Problem of the Week**
The contest is open to all undergraduates at Northern Arizona University. Solutions are due at 3pm every Friday.

Submit your solutions to this Google form as a pdf of either typed or neatly handwritten and scanned work. If you don't have access to a scanner use a phone app like CamScanner or Adobe Scan or scan at the library.

**The answers should be clearly and logically explained. ** The goal is to write a convincing mathematical argument that supports your solution, not to write down the answer and draw a box around it.

Send your questions to Rachel Neville at
Rachel.Neville@nau.edu.
If your instructor gives you credit for submissions to problem of the week, please include their name
and the class in the form (e.g. Swift, MAT 239) the first time you submit a solution.
(After that I have the information in my spreadsheet.)

Problems will be graded on a scale of 1 to 4. A model solution is posted each week.
A ladder listing the points earned is posted in the lobby of the Adel Math Building
(across from the MAP room). Your name will be printed on the ladder, but no names will be published on the web.
Let me know if you want to remain anonymous on the posted ladder and what name you would prefer to use.