University of Ottawa Putnam webpage

This webpage is currently being will someday be reorganized.

Problem sets for 2018.


General Information

The putnam exam is an annual mathematics exam originating from Harvard, aimed at undergraduate students in North America. It is written on the first Saturday in December, in two three hour papers. It's all long-answer, no multiple choice.

If you like playing with fun math problems, then this might just be your thing.

If you are interested in being part of our putnam team at University of Ottawa, then send Mike Newman an email (mnewman at you-know-where).

If you're trying to track me down, you find me here: schedule, office and email.

In the meantime, here is a problem (that I shamelessly stole from the Putnam website itself).

Players 1, 2, 3, …, n are seated around a table and each has a single penny. Player 1 passes a penny to Player 2, who then passes two pennies to Player 3. Player 3 then passes one penny to Player 4, who passes two pennies to Player 5, and so on, players alternately passing one penny or two to the next player who still has some pennies. A player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers n for which some player ends up with all n pennies.

Here's an old one that I shamelesssly stole from somewhere that I can no longer remember.

Find all rational numbers a and b such that ab=ba.

Here's another that I will attribute to Claude Shannon (if you know the reference or just know better than please tell me).

A message is to be transmitted over some imperfect medium, so it may be that some of it is corrupted. Consider the message to be a (large) number. It is to be written in base b and transmitted one b-igit at a time. Each b-igit has a probability p of being destroyed enroute.

If we choose b huge then there will only be one b-igit; it will have a pretty good chance of making it through unscathed. But there is a small chance we will lose everything. On the other hand if we choose b small then each failed b-igit will only lose a small amount of information but there may very well be many failures.

What value of b should be chosen so as to minimise the expected amount of lost information?

Note that the amount of information in a b-igit is b. You need a bit of probability theory to set this one up properly.

plan

The general idea is to work on solving problems, share our insights, solutions, and stumbling blocks. I'll suggest some things to work on for a given week, but you can also bring your own problems (solved or not!). Presenting solutions is highly encouraged. It's a time-honoured truth that nothing crystallizes your own thinking like explaining it to someone else. It really is a win-win.

resources

A few links for now, mostly to other peoples' collections of problems. Thank you to all of those people.