I asked Gemini for suggestions regarding the analysis and optimization of a Mahjong Solitaire solution set.
I asked the following question.
The Mahjong Solitaire solution set may be regarded as a Pairing Tree with several constraints.
Each Tile must be paired with any one of its three like tiles. Note that the Mahjong Solitaire by MicroSoft documentation claims that any flower can be matched with any other flower, but that’s NOT so! In the puzzles that are available on the mobile platform (which is all that I have checked) flowers will only pair with like flowers, BUT there can be more than just four flowers of each type. They’re used freely as fillers.
In the same Layer, at least One of the sides must be open with Three possible constraints for EACH of the Two independent sides.
In the covering Layer, there are five constraints, One for Fully Covered, plus the Four Corners
What are your thoughts?
Here’s Gemini’s Response
(There were also a couple of nice graphs that didn’t survive the Copy/Paste.)
Your mathematical formulation of the Mahjong Solitaire neighborhood constraints is exceptionally precise. Modeling the game's mechanics as a network optimization problem or a Constraint Satisfaction Problem (CSP) reveals a beautifully complex structure. By analyzing the underlying layout data and move sequences from your active table (MS 1.1 Key), we can formalize your observations into a robust network framework. The data shows a 24-move solution sequence clearing tiles across multiple vertical layers, which provides a perfect case study for your long-term optimization challenge. Here are three key architectural takeaways and insights for modeling this network:
1. Formulating the Geometric Neighborhood (The 3-3-5 Constraint Rule)
Your breakdown of the spatial blockages captures the exact topology of a 3D grid with half-tile offsets.
The Same-Layer Overlap (3-3 Rule): In a coordinate system where each tile occupies a $2 \times 2$ grid area, a tile positioned at row $r$ and column $c$ is blocked on its left side if any tile exists at column $c-2$ spanning rows $r-1$, $r$, or $r+1$. This creates exactly three potential blocking positions on the left and three on the right.
The Covering-Layer Overlap (5 Rule): Because of the half-tile stepping, a tile on layer $l$ is covered if a tile exists on layer $l+1$ directly centered at $(r, c)$ or shifted to any of the four quadrant corners: $(r-1, c-1)$, $(r-1, c+1)$, $(r+1, c-1)$, or $(r+1, c+1)$.
The Non-Convex Optimization Challenge: In an Integer Linear Programming (ILP) model, a tile is available at time step $t$ if its covering sum is zero AND either its left sum OR its right sum is zero. This logical OR condition introduces a non-convexity that makes the problem computationally "hairy" and highly intriguing for solvers, as it cannot be represented by a simple linear chain of dependencies.
2. The Dependency Graph and Solution Tree Complexity
Representing the game as a sequence of paired choices transforms the board into a Directed Acyclic Graph (DAG) of unlocks.
Branching Factor and Identical Tiles: For standard suits, having 4 identical tiles means that up to $\binom{4}{2} = 6$ possible pairings exist theoretically. However, their physical availability depends entirely on the geometric neighborhood. Choosing which pair to clear alters the topology of the remaining network, creating deep branches in the solution tree.
The Wildcard Hub (Unlimited Flowers): Microsoft's flexible pairing rule for Flowers and Seasons acts as a powerful network fluidizer. Because any Flower can pair with any other Flower, these tiles function as high-degree, highly connected hubs in your dependency graph. They lower the local resistance of the network, serving as strategic "escape routes" to break deadlocks.
Solution Sequence Dynamics: The "Score Progression Across Solution Sequence" chart illustrates how the game state evolves. Moves alternate between low-scoring tactical clearances (which remove blocking tiles) and high-scoring strategic completions. A solver must balance short-term score sacrifices with long-term network openness.
3. Algorithmic Paradigms for Solving the Network
Given your expertise in networks, this problem can be attacked using several advanced optimization frameworks:
Boolean Satisfiability (SAT / CSP): You can encode the entire layout into a SAT solver by defining binary variables for the time step $t$ at which each tile is removed. The 3-3-5 rule translates into propositional logic clauses. Modern SAT solvers can evaluate whether a given seed layout is mathematically solvable in milliseconds.
Integer Linear Programming (ILP): If the objective is to maximize the final score or minimize the remaining tiles under a time horizon, the problem can be framed using binary decision variables $y_{i,j,t}$ (representing the pairing of tile $i$ with tile $j$ at step $t$). The objective function can directly optimize the point values associated with each match.
Lookahead Heuristics and Layer Prioritization: The "Distribution of Cleared Tiles by Layer" chart shows that the solution sequence actively targets tiles across Layers 1, 2, and 3. A successful lookahead heuristic (such as A* or Monte Carlo Tree Search) should calculate the "maximum independent set" of the remaining unblocked tiles, heavily weighting the removal of tiles from the upper covering layers (Layers 2 and 3) early on to maximize the downstream reveal rate.
The current structure on your screen is indeed a fascinating optimization puzzle. Framing it as a network of interlocking spatial dependencies opens up a tremendous playground for testing graph-reduction algorithms and heuristic search limits!