Zeller's Congruence turns the a date (yyyy/mm/dd) into a number from 0-6, where Saturday is 0, Sunday is 1, and so on. And the first question has us change it to where Sunday is 0, and Monday is 1, and so on. It was rather easy to figure out with the hint. All I had to do is play around with the hint our prof gave us. That and read the wikipedia article as it has the answer in it.
The Doomsday Rule, is where things really get interesting. Given a formula, we can find out what day of the week, any day of the year lands on. Each year has an "anchor day", which for example this year's is a Monday. And there are certain days of the year that are always the anchor day. Such as 4/4, 6/6, 8/8, 10/10, 12/12. And then the rule for uneven months after 4, I work 5-8, at 7-11. Where 5/9, 7/11, 9/5, 11/7 all fall on the anchor days. Which as previously said this year is a Monday, so all those dates are on a Monday. From this it is rather easy to pick a day of the month and calculate which day of the year it falls on. Other dates that fall on an anchor day are Halloween, Boxing day, and a few others.