Sudoku via simulated annealing

The Sudoku puzzle in this Sunday edition of Le Monde was horrendously difficult, so after spending one hour with only 4 entries filled, I decided to feed it to the simulated annealing R program I wrote while visiting SAMSI last year. The R program reached the exact (and only) solution in about 6000 iterations, as shown (?) on the graph above. The Sudoku grid is defined in the R program by a 9×9 matrix s and the simulated annealing target function counts the number of duplicates

for (r in 1:9){

After pruning out the deterministic entries (3 in my case!), the R program uses the temperature sequence


to weight the target function. and it runs over the 10,000 iterations random moves on some of the unallocated sites. On the graph above, the green dots correspond to accepted moves. The yellow dots correspond to accepted proposals to move a single site. These choices lead to a correct solution most of the time, the other cases most often producing a penalty of two. (Please note there is nothing optimised about my code. It takes ten to twenty minutes to produce the graph above. a far cry from the fastest Sudoku solvers!)

19 Responses to “Sudoku via simulated annealing”

  1. […] the first issue of Le Monde magazine, including the solution to puzzle #52 I solved just in time by simulated annealing! The trick is in using the following theorem: Iter(1,x,y) is divisible by 10x-1 if and only if y is […]

  2. […] completely at rest!). However, this sounds like the last resort solution and I thus tried first a simulated annealing approach already tested for the sudoku problem a few years ago… (This puzzle is actually of […]

  3. […] annealing” methods have succesfully been used to solve the Sudoku problem though: see Christian Robert blog for example. I am also planning to continue my reading of the fascinating book […]

  4. Interesting approach to solve a sudoku puzzle using simulated annealing. I have not seen that before. Very good.

    However, every sudoku can be described as an instance of the exact cover problem. This allows both for a very elegant description of the problem and an efficient solution using a backtracking algorithm called the Dancing Links algorithm, also known as DLX suggested by Donald Knuth. This Algorithm X is a recursive, nondeterministic, depth-first, backtracking algorithm that finds all solutions to the exact cover problem. Some of the better-known exact cover problems include tiling, the N queens problem, and Sudoku.

    I use this method in my Sudoku Instructions program, which can solve every sudoku instantly.

    However, the Sudoku Instructions program can also be your helpful assistant during solving of the puzzle by providing hints, tips and comments at every step.

    Best Regards.

    Erik Christensen

    • Erik: Thank you for the pointer. Again, the sudoku solver has no claim to efficiency, it is a simple illustration of simulated annealing for my R course…

  5. […] more random than random! Darren Wraith pointed out this column about sudokus to me. It analyses the paper by Newton and De Salvo published in the Proceedings of the Royal […]

  6. Neat! (And slow :-) )

    There is a ‘}’ missing at the end of the program code.

  7. ..pretty good idea.

  8. […] kirjutas üks R-bloggeri aktiivsemaid postitajaid Xian [3] , kuidas ta lahendas  simuleeritud karastamise algoritmi […]

  9. […] here is the Sudoku that started my blog on simulated annealing, nicely represented on Revolutions. (Although I cannot see why the central […]

  10. Can you please post your Simulated Annealing code.

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