#ATTENDANCE QUIZ FOR LECTURE 14 of Dr. Z.'s Math454(02) Rutgers University

# Please Edit this .txt page with Answers

#Email  ShaloshBEkhad@gmail.com 
#Subject: p14
#with an attachment called
#p14FirstLast.txt
#(e.g. p14DoronZeilberger.txt)

#Right after finishing watching the lecture but no later than Oct. 24, 2020, 8:00pm


THE NUMBER OF ATTENDANCE QUESTIONS WERE:

PLEASE LIST ALL THE QUESTIONS FOLLOWED, AFTER EACH, BY THE ANSWER

Q1. Where and when was Herbert S. Wilf born? Look up his paper about "random generation of combinatorial objects" [in Advances in Mathematics] and at least skim it. 
A1. 
    1931, Philadelphia

Q2. Let m = Age of Donald Trump; n = Age of Joe Biden What is the probability that if you take a random lattice walk from [0, 0] to [n, m] you will always stay in the region x >= y
A2. 
    m = 74
    n = 77 
    To stay in region x >= y means we need to have a Good Walk, the total probability wuold be NuGPaths(m, n) / NuPaths(m, n) = 2/39


Q3. What is a Wilf-Zeilberger Pair
A3. 
    In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. 

Q4. What nationality was Catalan? What is the constant named after him (definition of value?)
A4. 
    His nationality is Belgium
    Catalan–Mersenne number

Q5. Do the same for [1, 1, -1, -1, 1] and get a set of 10 elem seq [01:09:09] and verify manually that is the whole of Paths(3, 2)
A5. 
    {op(CycShi([1, -1, 1, -1, 1])), op(CycShi([1, 1, -1, -1, 1]))} = {[-1, -1, 1, 1, 1], [-1, 1, -1, 1, 1], [-1, 1, 1, -1, 1], [-1, 1, 1, 1, -1], [1, -1, -1, 1, 1], [1, -1, 1, -1, 1], [1, -1, 1, 1, -1], [1, 1, -1, -1, 1], [1, 1, -1, 1, -1], [1, 1, 1, -1, -1]}

Q6. You are in a circular track in the desert. At random places, there are gas containers with random amount of gas. 
a1, a2, ... , ak
Such that the a1+a2+...+ak gallons are exactly what you need to drive in this track. 

Ex. [[0, 0.1], [0.2, 0.5], [0.7, 0.4]] 
Prove that there exists a location on the circular track such that if you start there you can drive the whole track. 
Device a method for doing it in general. 
Extra Credit: write a maple function that does it 

A6. 
    pass. Tried, too difficult :(