If you have a solution you would like to post here, email it to me and I’ll add it.

- B2: Anonymous submission: Putnam20142B2
- B3: Jesse Freeman: FreemanPutnam2014B3

I use homemade sweet treats as prizes for going above and beyond the course work. This semester in my Stat 101 class, the first person in each section to explain the solution to the “Birthday Problem” in front of the class without notes got a Triple Chocolate Espresso Brownie. I’ll give you the premise of the question and and then the recipe for these great, easy brownies.

The “Birthday Problem” is a fascinating probability problem that can be answered with a few basic probability rules that you learn in introductory statistics. The problem statement is *how many people do you need to have in a room such that there is at least a 50% chance of at least two people in the room having the same birthday?* To answer this, we need to simplify our world a bit. Let’s ignore Leap Day and assume there is an equal chance of being born on each of the 365 days of the year (which is probably not true these days with modern maternity medical practices).

First off, we notice that it is easier to calculate the chance that you don’t have at least two people with the same birthday in the room. The first person in the room could have any of the 365 days for their birthday (365/365) and then the second person in the room has to have a birthday on one of the 364 days left (364/365) and then the third person in the room has a birthday on one of the 363 days that aren’t called for yet (363/365). If there are n people in a room, the probability of getting at least two people with the same birthday is

This probability surpasses 0.5 when n = 23. One of my students eloquently explained this in class and got a homemade Triple Chocolate Espresso Brownie. The brownie recipe that I used, adapted from Cooks Illustrated, is listed below.

**Triple Chocolate Espresso Brownies **

5 oz. semisweet chocolate, chopped

2 oz. unsweetened chocolate, chopped

8 T. unsalted butter, cut into quarters

3 T cocoa powder

1.5 T instant espresso powder

3 large eggs

1 1/4 C. sugar

2 t. vanilla extract

1/2 t. salt

1 C. all-purpose flour

1. Heat oven to 350 with oven rack in lower-middle position. Spray 8-inch square pan with cooking spray. Take one 12-inch piece of foil, fold it and lay it horizontally and press it into the corners. Take a second 12-inch piece of foil, fold it, and lay it vertically on the pan (making an cross with the other piece) and press it into the corners. Spray foil with cooking spray.

2 . In medium saucepan, melt chocolates and butter on low, stirring frequently until mixture is smooth. Take off heat. Whisk in cocoa and espresso until smooth. Set aside to cool.

3. Whisk together eggs, sugar, vanilla, and salt in medium bowl until combined, about 15 seconds. Whisk warm chocolate mixture into egg mixture; then fold in flour until just combined. Pour mixture into prepared pan, spread into corners. Bake until slightly puffed and toothpick inserted in center comes out with a small amount of sticky crumbs clinging to it, 35-40 minutes.

4. Cool pan on wire rack to room temperature, about 2 hours, and then remove brownies from pan using foil handles. When ready to serve, cut into 1-inch squares. Enjoy!

]]>I didn’t quite know what to expect from a small college tucked away in the Berkshires, but it didn’t take long for me to fall in love with this place and be proud to call it home. There are many reasons for this, although it’s impossible to pinpoint a single defining one. Perhaps it’s the students, who are intelligent, motivated, and teach me just as much as I teach them. Or perhaps it’s the faculty in the math department, who are inspiring teachers and brilliant individuals, but are also fun to be around and always have some interesting stories to tell over lunch (come join us one day, you’ll see!). Maybe it’s the fall colors (check out the view from my office!) and watching the mountains change from season to season.

Either way, happy Mountain Day!

]]>- We have a weekly problem solving dinner at 5:30pm in Dennett Private Dining Room at Mission on Wednesdays. There’s no prep work; feel free to drop in any time (and if you’re not on the meal plan we’ll provide a swipe). The way it works is we print out a math competition from somewhere in the world, and then brainstorm and attack the problems together.
- We’ll also meet for lunch on Wednesdays (in Mission at noon, room TBD) to do Project Euler problems.
- There are several math competitions each year. Some time in October or November we’ll defend the Green Chicken when Middlebury travels here. There’s also the Virginia Tech math competition (we’ll do this remotely Saturday October 25th), and the Putnam exam (which will probably be Sat Dec 6th).
- We also frequently field teams for the Mathematical Contest in Modeling.
- I’m also teaching a class on math puzzles and problem solving, Math 331: The little Questions. Feel free to check out the homepage for resources, as well as lectures (each class is recorded and uploaded to YouTube).

Many people love math puzzles or riddles. They’re often fun, frequently illustrate a beautiful concept or perspective, and unlike real world research problems they typically have an elegant answer. Below is one of my favorites. It can be solved by brute force but only at great cost (and a high probability of forgetting a case); however, if you look at it the right way it’s just one line (and this proper perspective illustrates a powerful technique which is of use in research mathematics).

**Problem: Imagine you have 10 distinct cookies but 5 distinguishable, hungry graduate students (so they’re not engineering students working away in a lab!). How many ways can you divide all the cookies among the people? In other words, we only care about how many cookies each person gets, not which ones.**

I run a math riddles page (see http://mathriddles.williams.edu/ ), and this was the first problem I added; the problems and resources posted there for students and teachers are used in schools throughout the world, and if you’re interested in helping with the site drop me a line at sjm1@williams.edu. It turns out that this problem is equivalent to Waring’s Problem with exponent 1; I’ve used this idea with SMALL students in research problems related to Fibonacci numbers (this is discussed in another post, To Bead or Not To Bead). Enjoy!

]]>One of the things I’ve missed most about the US and about Williams especially is the involvement that undergraduates have with the broader math community. Giving a senior colloquium, doing original research, or presenting at a conference are all essentially unheard of at the undergraduate level in the UK. For that reason, I decided to organize a conference at which both Williams and Exeter college students could present mathematics.

The event was a huge success. Thanks to a generous contribution from the John and Louise Finnerty fund, we were able to purchase food, coffee, and bus tickets for conference participants. About half of the WEPO students attended. This included every math major in the WEPO program, but even more Williams students majoring in other disciplines! Four Oxford students attended, including a few very enthusiastic first year maths students and a second year physicist. Everyone who attended stayed for at least 4 of the 5 talks. These talks were given by

Jesse Freeman

Jesse Liu (2nd year physicist, Exeter college)

Isaac Loh

Christopher Huffaker

Elliot Chester

I would like to extend a special thank you to John and Louise Finnerty, Professor Katie Kent, Exeter College Rector Frances Cairncross, Exeter College Sub-Rector Jeri Johnson, Professor Steven Miller, and the Williams math department for their invaluable support in organizing this event.

Here are some pictures from the event, the Exeter College Quarrel Room, refreshments, and the presentations:

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ROSENBURG PRIZE for excellent senior: **Ilya Amburg **and** Vu Le.**

GOLDBERG AWARD for best colloquium:

**Nina Horowitz** “The Mathematics Behind Playing Hard to Get“

**Joseph Iafrate** “Random Walk, Random Strikeout: Baseball as a Markov Chain”

WYSKIEL AWARD in Teaching: **Jeff Brewington**

MORGAN PRIZE in Applied Math: **Carson Eisenach**

MORGAN PRIZE in Teaching: **David Stevens**

ROBERT M. KOZELKA AWARD in Statistics: **Faraz Rahman**

OLGA R. BEAVER PRIZE for department service: **David Stevens**

BENEDICT PRIZE for outstanding sophomore:

1^{st} prize: **Peter McDonald**

2^{nd} prize: **Eva Fourakis and Elizabeth Frank**

WITTE PROBLEM-SOLVING PRIZE: **Samuel Donow, ****Jared Hallett, Benjamin Kaufman**

COLLOQUIUM ATTENDANCE: **Michael Gold ’14, John Bihn ’16**

Prof. Johnson thanked visitors Michael Biro, Holley Friedlander, and Ed Hanson, and recognized SMASAB, the student math advisory board:

Craig Corsi, Philippe Demontigny, Jared Hallet, Caroline Miller, Faraz Rahman, Jiripat Samranvedhya, David Stevens, Kirk Swanson, Sam Tripp, Carrie Chu, Jesse Freeman, Joe Kinney, Anna Spiers, Phonkrit Tanavisarut, Jaclyn Porfilio.

Following tradition, seniors gave advice, for example:

Michael Gold: “Go to office hours.”

Ilya Amburg: “Do a thesis.”

Sam Austin: “Work with friends, if you have any.”

Alex Albright: “If you think you understand something, try explaining it to someone.”

Caroline Miller: “If you want to stump a job interviewer, start talking about research.”

David Stevens: “Be promiscuous, mathematically speaking, because you never know what you’ll like until you fool around.”

Heidi Chen: “Learn LaTeX.”

Matt Micheli: “Try not to overuse your beverage wrench.”

Gabor Gurbacs: “Work on an unsolved problem in math and never give up.”

The banquet was supported in part by the John and Louise Finnerty Class of 1971 Fund.

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The Golden Ratio and the Fibonacci Sequence by **Megumi Asada**

Benford’s Law and the Fibonacci Numbers by **John Bihn**

Isoperimetric Regions in the Plane and Beyond by **Wyatt Boyer**

Limit of Sequence in non-Hausdorff Space by **Yuanchu Dang**

Generalizations of the Vector Cross Product by **Edward Hanson**

The Convex Body Isoperimetric Conjecture by **Frank Morgan**

A Proof of the Arrow’s Impossibility Theorem by **Jeffrey Wang**