The model behind the toy
This site's toy chain is a tiny version of the HP model, a lattice protein model biophysicists actually use to study folding. It keeps only one property of each amino acid — hydrophobic (H) or polar (P) — and confines the chain to a grid, one residue per cell, never doubling back on itself. Stripping away everything else about real chemistry is the point: it leaves a version of the folding problem small enough to search completely, while keeping the one force that matters most for how proteins collapse into shape.
The energy score is a count: for every pair of H residues that end up next to each other on the grid without being next to each other in the chain, the score drops by one. That's the model's stand-in for buried hydrophobic surface — the more H residues shield each other from the (implied) surrounding water, the lower the energy, and the closer the fold is to the one water would actually settle into. "Reveal the optimal fold" doesn't take a shortcut to find that minimum; it tries every possible fold and keeps the best, which is also why the length slider on the home page tops out at 22 — beyond that, the number of folds to check grows too fast to search in real time.
Even at this toy scale, that search is already the hard part of the real protein-folding problem: for a real chain, hundreds of residues long, twenty amino acid types, folding in three dimensions rather than two, finding the true minimum by brute force is completely out of reach. Cells don't solve it that way either — chaperone proteins guide the process — but the underlying principle carries over intact from this 2D toy to a real structure: a sequence determines a shape, and water finding the lowest-energy arrangement of hydrophobic residues is most of why.