Paramagnetic and glass transitions in sudoku

We study the statistical mechanics of a model glassy system based on sudoku, a familiar and popular mathematical puzzle. Sudoku puzzles provide a very rare example of a class of frustrated systems with a unique ground state without symmetry. Here, the puzzle is recast as a thermodynamic system where the number of violated rules defines the energy. We use Monte Carlo simulation to show that the “sudoku Hamiltonian” exhibits two transitions as a function of temperature, a paramagnetic, and a glass transition. Of these, the intermediate condensed phase is the only one that visits the ground state (i.e., it solves the puzzle, though this is not the purpose of the study). Both transitions are associated with an entropy change, paramagnetism measured from the dynamics of the Monte Carlo run, showing a peak in specific heat, while the residual glass entropy is determined by finding multiple instances of the glass by repeated annealing. There are relatively few such simple models for frustrated or glassy systems that exhibit both ordering and glass transitions; sudoku puzzles are unique for the ease with which they can be obtained, with the proof of the existence of a unique ground state via the satisfiability of all constraints. Simulations suggest that in the glass phase there is an increase in information entropy with lowering temperature. In fact, we have shown that sudoku puzzles have the type of rugged energy landscape with multiple minima that typifies glasses in many physical systems. This puzzling result is a manifestation of the paradox of the residual glass entropy. These readily available puzzles can now be used as solvable model Hamiltonian systems for studying the glass transition.

Figure 1

Variation of “heat capacity” with temperature, showing that the transition temperature shifts with number of fixed values. Legend shows the Playr puzzle number, with the first digit labeling the “difficulty” and the number in brackets indicating the number of fixed sites.

Figure 2

Variation of critical temperature with number of clues for puzzles with and without unique solutions.

Figure 3

Variation of mean energy in the $T<0.2$ range for a series of puzzles with different number of fixed sites, showing the appearance of a glassy state for intermediate numbers of clues, and an increased tendency towards glassiness for puzzles with a unique solution.

Figure 4

Fraction of time spent in ground state for a series of puzzles with different “difficulty” rating. At each temperature we simulated 10${}^9$ attempted steps, split into 1000 groups with each group initially randomized, and statistics were gathered over the final $8\times {10}^5$ steps. Legend shows Playr puzzle number [1]. The solution to the 17-clue minimal puzzle was only found once. Inset: $S\left(T\right)$ vs $T$. This quantity is normalized to the number of sites, rather than the number of free sites, hence systems with a large number of free sites tend to have higher $S\left(T\right)$ throughout.