# Putnam Seminar

## Cody Johnson

Solving problems is all about asking the right questions and finding the right thing to explore. Therefore, I advise asking the following questions while solving any problem:

• What problems that I have seen before does this problem remind me of?
This is the absolute best question to ask. Think of every single relevant problem, recall the solutions of them, and try to find a similar idea in the problem you have in front of you.
• What techniques does this problem feel like?
If you see "prove for all integers $n$," then think of induction. If you see a problem involving making a sequence of moves, try to find an invariant. If you see a problem that says "prove there exists" but doesn't have any structure you can see, think of the pigeonhole principle. If it asks to prove that the number of one things is equal to the number of the other, then find a bijection. If it is a game, then look at what positions are winning and losing for small values.
• What tricks does this remind me of?
As you solve more and more problems, you will see tricks that appear in several problems.

# Welcome

Welcome to my blog for the Carnegie Mellon Putnam Seminar! I'm Cody Johnson, a junior at CMU, and I'm a grader for this course. I designed this web page to interact with the students of this course, hopefully to the avail of making CMU even better at Putnam! On the blog tab, I will post topics related to the lectures that either Professor Loh didn't have enough time to cover, or additional points that I find important for the competition. On the discussion tab, I will discuss solutions to homework problems that I found either a large number of students missed, made the same mistake on, or didn't attempt. I'll post both a discussion of the problem and its motivation and also a concise solution like I would write on the actual Putnam. On the resources page, you can find extra materials related to Putnam. Feel free to contact me at ctj@math.cmu.edu for any reason.

Important note: I have office hours! Schedule them by email for any reason, whether it be to discuss potential grading errors, math problems and ideas, or even just to say hi and meet me.