Do your best to solve each one. Always explain your reasoning $-$ what concepts you are drawing on$,$ how you are moving from step to step, why your conclusions

are plausible. Even if you can't completely solve a problem, describe how you would approach it and why.

You may submit your answers as text in the response boxes here. If it is easier to write up answers to some or all of the problems using another platform$/$tool you

can produce a single PDF which you submit below. If the solution to a given problem is in a PDF submission, please note that in the response box for that

problem $($e$.$g$.$ "See PDF$").$

A few code cells are included in case you want to do anything with Python while you work on these problems. Except for problem $3,$ no code needs to be

submitted.

Question $1$ of $5$

Problem $1$ Selling Tickets and Making Change

A university event requires a fifty dirham "donation" per attendee. Assume that every person who comes to the event pays with cash and each one has either a $50$

Dhs bill or a $100Dhs$ bill. The team that collects payment at the door neglected to get any small bills to be able to make change. If at least as many attendees pay

with $50$ as pay with $100$ then the ticket sellers will not have to issue anyone an "IOU" $($I owe you$)$ and send them their change later.

Suppose $2n$ people come to the event and every individual pays for their own "ticket" and that by the end of the evening there were exactly $n$ with $50$ Dhs notes

and exactly $n$ with $100$ Dhs notes. We want to think about different the implications of them arriving in different orders. For example, if all the people with $100s$

arrived first then we would need to issue $50$ IOUs.

What problem that you have seen in your pre$-$class work does this problem bear a resemblance to$?$

How can you use that analogous problem to give a geometric interpretation of sequences of arrivals of people with $50s$ and people with $100s$ that never require an

IOU to be issued.