It’s Time To Party

Hello everyone! This is Andrew, Sunday 1/12/2025, a rainy afternoon here in Seattle (… wait, Seattle?), all excited for our class to start tomorrow.

Here are some problems to start our semester off with! I’m just throwing these at you—don’t try to look up how to do these! It’s fine if you don’t know how to solve all of them—for that matter, I don’t EXPECT you to know how to be able to solve all of them—I just want you to think deeply and work hard and see if you can make any progress figuring any of them out! That’s our goal. Not to memorize formulas, but to explore. Later, perhaps, we’ll learn some formulas, but for now, let’s see what we can figure out ourselves.

Oh! And if you already know how to figure out any of these problems, don’t give away any spoilers to your classmates! It’s about the process, not the result. We’re trying to learn math here not a as a bunch of individuals, but as a team. (We’re soccer, not cross-country) (sorry runners) (I’m a runner!)

1. You’re planning a party!!! You need to pick a time. The party has to be after school between 2pm and 4pm. If you can only start at whole or half hours (like 2:00, 2:30, 3:00, 3:30), how many possible start times are there?
2. Except, actually, you’re printing out fancy invitations on your friend’s letterpress (look up what a letterpress is if you don’t already know), and because it’s a vintage repurposed letterpress, she doesn’t have all the glyphs (i.e., all the little metal stamps of type). Instead she just has three copies of the digit 4, one copy of the digit 2, one three, two sevens, a zero, and an eight. So if you want the party to still start sometime between 2 and 4, how many different choices for starting times are there?? Can you list all of them?
3. What if you no longer require that the party has to start at an even hour or half-hour? How many possible start times are there?
4. At the party, in Jackee Bruno’s honor, everyone decides to high-five everyone else. There are 20 people at the party. In order for everyone to high five everyone else at least once, what’s the minimum number of high fives that have to happen?
5. Everyone’s hungry! It’s time to order pizzas! Patrick, in his increasing Italianization, now has a side gig running a pizzeria (“Patrick’s Perfect Pies”), so you decide to order from him. He has eight different toppings for his pizzas (one size only). You can have any combination of all eight of the toppings. How many possible pizzas can you order?
6. Patrick’s Perfect Pies is expanding. Now he sells pizzas in three sizes: large, extra-large, and eXtremely-large (L, XL, and EL for short). With these three sizes, and eight choices of toppings, how many possible pizzas could you order?
7. It turns out that these perfect pies are imperfect: they’re not free. In particular, Patrick charges $20 for a plain L pie, $25 for a plain XL pie, and $30 for an EL pie. The L pizzas are 16" in diameter; the XL pizzas are 18" in diameter, and the EL pizzas are 20" in diameter. With each of these pies, how much do you pay per square inch of pizza? Which is the best deal??
8. Each topping is $2. After collecting money from everyone at the party, you end up with $200. With that $200, what’s the maximal square inches of pizza (per person) that everyone could eat? That, of course, would require some pretty boring plain cheese pizzas… what if you want to be more creative? How many different possible pizzas can you buy with the $200??