it does well on puzzles with very few clues (from 0 to 16). PSS is generally able to solve ambiguous puzzles I have also tested PSS against many "ambiguous" puzzles, " top1465" ("1465 hardest sudokus sorted by rating ")ģ6,628 17-clue puzzles posted by Gordon Royle at " topn234" ("234 hardest sudokus sorted by They are here mainly for historical reasons") " top95" ("95 hardest sudokus sorted by rating: Included from Gordon Royle's list of 18-clues-sudokus ") They are hard for the new as well as for the old program. " topn87" ("87 hardest sudokus sorted by rating , Many lists of difficult puzzles have been posted at Puzzles, which is consistent with the above statistic: that PSS solves about 97% - 98% of This effort by Andrew Stuart of, PSS solved 573 - about 52%.Īccording to Andrew these puzzles represent 5% of their randomly generated Of 1106 "unsolvable" puzzles generously donated to Puzzles, all but about 1.3% of the hard puzzles, and 60-70% of the Randomly generated puzzles it solves all of the very easy, easy, and medium These tests show that PSS solves about 97% - 98% of percent solved = 72.13 Worked 2000 total puzzles. percent solved = 97.90 Worked 17 very easy puzzles. percent solved = 61.54 Worked 1000 total puzzles. percent solved = 97.60 Worked 9 very easy puzzles. Here are the results: Worked 500 total puzzles. Program) The first test run consisted of 500 puzzles, the second of 1000,Īnd the third of 2000. ![]() Rated for difficulty (with much thanks to Michael Kennet for his excellent public domain sudoku I tested PSS against 3500 randomly generated puzzles "unsolvable," "fiendish," or "bifurcating" puzzles. However, PSS is not perfect - it does fail on a significant number The algorithm it is rather surprising that PSSĭoes so well. Solving any new cells, it gives up. In general I find that it solves almost all puzzles intended for humanĮntertainment, from very easy through very hard. If PSS goes through more than about 100 loop iterations without For example, it will fail on inconsistent puzzles that have no However, PSS is not always able to accomplish this When all the probabilities have been locked (to 0.0 orġ.0) the puzzle is solved. "backtracking." Once PSS decides that a digit goes (or can not go) in aĬell, it never changes it's mind - there is no trial and error involved in Locked at that value so PSS can no longer change it. When any probability becomes either 0.0 or 1.0 it is Probability (of all 729) is incremented by. The clipping process is very simple: The highest Nudging them to converge to a (nearby) state in which the conservation laws are The evolution process perturbs the probabilities, gently Probabilities will not satisfy these conservation laws (assuming the problem is After initialization (or after clipping), the Must equal 1.0, and, for each character k the sum of the probabilities of k Probability to be conserved, the sum of the digit probabilities for each cell ![]() Incrementally in order to conserve probability within the game state. The evolution process adjusts the probabilities I call the two processes in the loop "evolution" and "clipping." Two step loop which repeats until the puzzle is solved (or fails to be solved). ![]() Sums to 9, and the probabilities of the digits in each cell sums to 1.Īfter the probabilities are initialized, PSS enters a Probabilities are set so that the total probability of each digit in all cells Initialized by setting probabilities to 1.0 or 0.0 in this manner as dictated by (the row, column, or region) to which C belongs is 0.0. The probability of k in all other cells belonging to any of the three groups Thus, if a character k definitely occupies a cell C then the probability of k inĬ is 1.0 and the probability of the other 8 characters in C is 0.0 furthermore, One for each possible digit (1 thru 9) in each possible cell (of the 9x9 grid). The puzzle state is represented by 729 probabilities, Here is a functional description of the algorithm: You could say that it works by making well-educated It does not workĪt all like a human being indeed, it does not use any form of logicalĭeduction or backtracking. Probabilistic Sudoku Solver (PSS) Algorithm A Mathlink (Mathematica) executable, PSSML.EXE.A Visual C++ project with an implementation of the.A description of my probabilistic algorithm for.Gottlieb's unorthodox probabilistic sudoku Probabilistic Sudoku Solver The Probabilistic Sudoku Solver (PSS)
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