On October 23, a large group of students and math professors gathered in a room in the CMC, took handfuls of pennies from a huge penny jar at the front of the room, arranged their pennies into piles, and shuffled them around for an hour in pursuit of mathematical truth.
This was one of the Math Department's weekly Math Colloquia, a series of lectures by visiting professors about any mathematical topic they want to talk about, usually showcasing the less-serious side of the mathematical world.
This time, the lecture was given by Dr. Suzanne Doree, a professor at Augsburg College. It concerned an activity called Bulgarian Solitaire, which Dr. Doree was hesitant to call a game. She thought "a puzzle" or just "a thing" would be a better description.
You'll see why when you read the rules:
~The Rules of Bulgarian Solitaire~
1. Take some coins.
2. Put them in piles.
3. On your turn, you must take one coin from each pile and make a new pile with those coins.
4. Keep repeating Step 3 until the game gets stuck in a loop.
5. You cannot win.
Yes, that's right- for each position there is only one possible move, so you can't make any choices in this game. You will always eventually get stuck. And there's no way to win. When looked at this way, Bulgarian Solitaire is a rather nihilistic and depressing game.
But mathematicians don't PLAY Bulgarian Solitaire for fun. They ask themselves questions about it for fun.
For example, Dr. Doree asked the audience the following questions:
Does every game of Bulgarian Solitaire eventually make it back to its starting point?
Can the game ever get stuck in a one-step loop?
What kinds of positions get the game stuck in a loop?
Are there any positions which are unreachable (unless you start in that position)?
(Answers at the end)
The tradition of asking and then answering questions about Bulgarian Solitaire started when Martin Gardner wrote an article called "Bulgarian Solitaire and Other Seemingly Endless Tasks" in his famous "Mathematical Games" column in Scientific American. As Dr. Doree explained, anything Martin Garner writes about instantly becomes a smash hit in the mathematical community (even if it's a game where you make no choices and you can't win!). Martin Gardiner didn't actually invent Bulgarian Solitaire- it first appeared in a paper called "Cycles of Partitions" by Jørgen Brandt, which Dr. Doree assured everyone is not a fun read.
After asking questions and figuring out solutions, Dr. Doree discussed variants of Bulgarian Solitaire, such as multiplayer games or Mancala-like games where there is more than one choice to make, which she and her students have written papers about.
At the end of the lecture, she encouraged the audience to invent their own variants.
I'd be interested in seeing a Bulgarian Freecell.
Answers for the Curious (Those Not Curious About the Answers are Permitted to Stop Reading):
No.
Yes. Triangular pile arrangements, like the one in the picture, do this. (The vertical columns are the piles).
Triangular "staircase" arrangements, which may or may not have one extra coin "person" sitting on each step. These are called "Escher Escalators" because, as you take turns repeatedly, the coin people move down the steps of the escalator infinitely, reappearing at the "top," like something in an M.C. Escher painting. All loops look like this.
Yes. They're called Eden states (because everything starts there, but once it leaves the Eden state becomes unreachable). For all possible arrangements with three coins, there is only 1 unreachable Eden state. increasing the number of coins to 4 gives you 1 Eden state, increasing it again gives you 2, then 3, then 5, then... 7. No, not 8. Why would you think it would be 8?
