BULGARIAN SOLITAIRE

This is a nice and simple example of a finite dynamical system, just like the 3D Game of Life. It is finite because only a finite number of states are possible.

In addition, it is not exactly a game either, because there is a precise rule of what we have to do and no choice is given to us, that is, despite the name, it is not a game of solitaire.

At first, we distribute a certain number of blocks into piles of arbitrary sizes (for example, if we have 10 blocks, we can divide them into two piles of 5, into ten piles of 1, into three piles: one with 7, one with 2 and one with 1, etc.). The rule is simple: you must take a block from each stack to build a new one, that is, if the zero state is two stacks with 5 blocks, in the next step, we will have two stacks with 4 and a stack with 2. Then we repeat the rule again and again...

The interesting thing is to try to understand what happens when we repeat the rule many times. As there are only a finite number of states, at some point none will repeat and you will be stuck in a periodic cycle. This cycle can have period 1 (in this case it will be a fixed point) or have period greater than one.

The photo sequence below is an example of an iterated sequence with 10 blocks starting from any position (the zero state), from which the rule is applied several times. Note that the last configuration is repeated, being called fixed point.

This is a simple game to play. Get chips, or soap, or anything you can stack, and play around! Start by fixing a number of blocks and test different starting positions. What happens? What if I change the number of blocks? Test, write down, make your conjectures and theories!

If you want to see a little more about this game, check out the video below! Seize the opportunity to check out the other cool videos on Matemateca's channel on YouTube!

                 

USEFUL LINKS

This Wikipedia page gives a brief overview of how Bulgarian Solitaire works, mentioning the triangular numbers and what they have to do with the configuration possibilities of each possible game. In addition, he mentions at the end the so-called Random Bulgarian Solitaire (or Stochastic), a version of the game more focused on Statistics.                                                              

This Tipping Point Math's channel video introduces the game, showing some examples of initial settings and what happens in each round. It also talks about triangular numbers and adds a discussion of the Cradle Model, which can be used to better visualize and understand what happens throughout the game.                                                                                                                            

This Vesselin Drensky's article focuses on the mathematics involved in this game, giving a more formal look at the proofs that can be made from it. Vesselin describes some theorems and uses mathematical tricks to support himself, such as Young's Diagram, and ends by making generalizations and mentioning other similar games, such as Austrian solitaire.

As said, this game is an example of a dynamic system, a mathematical concept. This Wikipedia page brings an explanation of what dynamic systems are, their history, definition, sub-areas, terminologies, among other things. It also brings some examples of dynamic behaviors, including the concept of fixed point and showing its more formal definition.

Still on dynamical systems, in this Leonardo Tambellini's article an intersection between dynamical systems and game theory is covered (if you want to know more about it, take a look at our page that mentions them). The author studies, starting from Bulgarian Patience and Carolina Patience, the dynamics in finite sets and the number of distinct periodic orbits.