Sandpiles. Just the name suggests recreational science. The science in question is a mix of combinatorics and probability. I believe that the model came from statistical physics, although that does not necessarily mean much in terms of relevance to reality, although I am not sure I know what "reality" means anyway. No matter. What matters is that it's supersimple to define - even your kindergarten child could understand it - and yet it leads to strikingly beautiful, complex pictures that look like infinitely elaborate mosaics in mosques.

There is a board game called "chaos" that's based on sandpiles. The board is a grid, two players each have n grains of sand, they take turns placing a grain on the board, making little piles with the grains, and whenever a pile reaches height 4, it topples: each of the 4 grains gets moved to one of the 4 neighbors (in a predetermined order, in the case of the board game). Repeat toppling as needed. Any grain that falls off the board gets returned to its owner, and the winner is the player who first gets rid of all of their grains. That's the board game. Level: 8 years old and up.

In reality (by which I now mean, in the mathematical model), the grid is infinite, we start with n grains at the origin, keep toppling, and look at the final configuration as n goes to infinity, where every vertex holds a pile of grains of height 0,1, 2 or 3. The challenge is to analyze the convergence rate and the final configuration. Level: Math PhD and up.

Programming this might be a good first project: straightforward to implement, yet gives pretty pictures.

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