Week 4, Assignment #1

This week we had the first assignment due, which I worked in conjunction with 2 other people. We each focused on one or two questions, and then helped explain the solutions for the others to the rest of the group - this seemed to work out well. 

I mostly focused on the 2nd question which was about the menu problem for the restaurant. The following is my work on the solution using the Polya approach:

1. Understanding the Problem:
   Try to come up with a method to design a menu which suits the constraints given, then prove that it works.
2. Devise a Plan:
   Start with the simpler menu example given and then to 3 items, and onwards, finally generalize to 2^n, and then prove.
3. Carry out the Plan:
   First, I started by creating each of the possible menus using the set of items {L, S} given that any 2 consecutive days must only differ by exactly one meal.
   
   I then extended this to a set of 3 meals, {1, 2, 3} and created a menu cycle where they all differed by only one day. And then once again for 4 meals. At this point my lazyness kicked in and I decided I no longer wanted to write any more meal menus. So I began looking for patterns and realized that the menus, when extended for any n+1 could be represented recursively based on its previous element n. That is,
   
   n+1 = n + [\reverse(n) with the (n+1)th element added to each element in \reverse(n)]After some further testing (I actually ended up doing 5 meals...meh) it seemed to me that this was correct enough, so I wrote out the proof and that's what was handed in.
4. Looking Back:
   Looking back, the solution still seems correct in my mind however I'm not absolutely sure if the format in which I showed my proof is correct. So will have to wait for the assignment to be handed back before I can see the result....
   

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