A natural starting point is to implement the intended construction from the original Holmes and Watson's Bet, and then run a more carefully engineered simulated annealing search on top of it.
Once reaches the scale of billions, evaluating a single candidate board by direct simulation becomes expensive. At the same time, changing only one mirror can drastically reduce the score, so the search easily gets trapped in a local optimum. Both the evaluation cost and the locality of the search must therefore be addressed.
With the K-structure from the original intended solution, an excursion from the central region to the outside and back has a regular form. This allows a candidate to be estimated without simulating every move directly: combine the number of times the K-structure is traversed, the average number of moves spent in that structure, and the number of moves inside the central region.
Long-period infinite-loop boards are also useful for escaping local optima. By changing a small number of cells, such a cycle can sometimes be broken into a long finite trajectory or transformed into an even longer cycle. However, a long state-space period is not the same as visiting new cells for a long time. If the robot repeatedly visits the same cells, placing the start and finish near the ends of the state cycle may still produce a much smaller actual .
Keeping long-cycle boards in a separate pool and reusing them as material for simulated annealing helps the search leave one local basin and move into a different region of the state space.
In the original problem, plain simulated annealing could reach an accepted score even without the K-structure. For the Large version, directly searching an unstructured interior all the way to is impractical. Structural acceleration and stronger search techniques are both important.
Solution written by GPT5.6
