# A set based T-SQL Solution for Sudoku Puzzle

• Comments posted to this topic are about the item A set based T-SQL Solution for Sudoku Puzzle

• Just want to be sure I'm using this right. So, do I enter the entire puzzle number set beginning from the top left corner, reading a whole row, and then moving to the next row and continuing until I've reached the bottom right corner? Though this might seem obvious to the writer, not having this instruction may have people confused as to how to enter their puzzle problem set.

• Hi Kevin,

I really enjoyed reading your script, it was good to see TSQL used in a completely different way than usual. I would never have thought of using the bitwise operator to solve this problem.

I like to see something different because seeing how other people approach a problem and write code is the best way of learning how to improve your own skills.

Could you explain a bit about why you chose to work with the numbers in binary? was it your first choice or did you experiment with other ways first?

The problem with sudoku is that for some of the puzzles you can only get so far by counting the numbers, eventually you may be forced to take a best guess at the content of a square. This is something that is missing from your script, do you intend to extend it in the future?

I had a go by using recursion, but there is probably a better way, I modified the script to return 3 states: complete, Fail, Try Again. When the Try again state was reached the code would generate a new value for @t with one of the 0's replaced with a guess to see if that would lead to the completion of the puzzle.

I identifed a square that had the minimum number of possible values and then created a new @t values for each of the possible values. I limited this to two levels of guesses.

Here are the results I got for this sudoku:

080700000902000000300090020060800200750109043009004070040050009000000706000007030

_C1C2C3C4C5C6C7C8C9

R1_8_7329__

R29_2___3__

R33___9__2_

R4_6_87_29_

R5758129643

R6__9_64_7_

R7_47_5___9

R8______756

R9_____7432

_C1C2C3C4C5C6C7C8C9

R1_8_73291_

R29_2___36_

R33___9__2_

R4_6_87_29_

R5758129643

R6__9_64_7_

R7_47_5_189

R8______756

R9_____7432

_C1C2C3C4C5C6C7C8C9

R1586732914

R2972481365

R3314596827

R4463875291

R5758129643

R6129364578

R7647253189

R8231948756

R9895617432

I would be very interested to see how you would solve this problem.

Thanks.

Ian.

• Hi Ian,

Thanks for your posting and your detailed suggestion. It’s my pleasure to have discussion on the “ugly” script that I would never have expected anyone be patient reading through 😉 This script was created to be used in my SQL Server class to illustrate:

•The new T-SQL syntax in SQL Server 2005.

•The set-based approach for solving problems that have unique constraint on each set (specifically in Soduko puzzles, the rows, columns and blocks).

•The performance comparison.

For Soduko puzzles, there are two main types of solutions. One is iteration approach, guessing and verifying, as you suggested; the other one is set-based approach, eliminating and exclusion. Because RDBMS are very good at set-based approach, and the uniqueness constraint (in row, column and block) is a perfect scenario for using bitwise operators (using logical AND and OR to find and subtract all duplicates in one shot), I used this as a coaching example. For the purpose of illustration, I didn’t use any of the iteration methods in the script. Actually, I have the scripts using iteration approach in T-SQL, Java and JavaScript, they are posted below.

In terms of bitwise operation, it’s not mandatory for the set based approach. I have a script using character string values instead of binary values(script attached below), but I needed an extra loop to implement the elimination rules. I am a big fan of binary data structure, and it's always been my first choice when applicable, say flag related problems(but unfortunately, SQL Server does not support bitmap index unless in the OLAP cube).

A couple of extra notes on the set-based script:

•There should be only one solution for the puzzle; otherwise the eliminations will be confused and stopped at certain point. But you can find out all solutions by using iteration approach easily.

•The elimination rule has to be complete (which is not in my script, see the exception I gave in the posting. But I believe there ARE some elimination rules that can be added to make it perfect).

On the performance side, SQL is faster in set-based approach than iteration approach in a lot of cases (BTW, Itzik Ben-gan's SQL books are very good resources on the in-depth SQL skill, especially in this type of comparison). But using iteration approach in procedure languages such as Java, C++ or even JavaScript is even much faster. That is why I usually strongly recommend implementing the business logic at the application (Middle) tier other than doing it inside the databases, which is the most expensive resource in the application. Scalability is also a big concern (you could have tens of application servers running the iteration code).

Yes, I should have made the scripts perfect even though I just used it for teaching. If you have any good ideas or suggestions, I am more than happy to learn.

Thanks,

Kevin

Following is the iteration approach in T-SQL.

`USE [tempdb]`

`GO`

`IF OBJECT_ID(N'dbo.uspSolveSudoku_2', N'P') IS NOT NULL DROP PROCEDURE dbo.uspSolveSudoku_2;`

`GO`

`CREATE PROCEDURE [uspSolveSudoku_2] ( @t VARCHAR(81), @Trace INT = 0 )`

`AS`

`SET NOCOUNT ON`

`DECLARE@datetime DATETIME,`

`@ii BIGINT,`

`@s-2 VARCHAR(max),-- temp variable for building sql query strings`

`@SQL1 VARCHAR(max),-- query for basic elimination rules`

`@SQL2 VARCHAR(max),-- first query for second elimination rules`

`@SQL3 VARCHAR(max)-- second query for second elimination rules`

`SET@datetime = GETDATE();`

`----------------------------------------------------------------------------------`

`--Step 1: Setup working table #SDK.`

`CREATE TABLE #SDK ( i INT, -- Working table, one record per cell, with row/column/block lable and current possible values`

`--DECLARE #SDK TABLE ( i INT, -- Working table, one record per cell, with row/column/block lable and current possible values`

`r INT, -- Row #`

`c INT, -- Column #`

`b INT, -- Block #`

`v INT);`

`WITH n (i, r, c, b, v) AS (`

`SELECT1, 1, 1 , 1,`

`CASE WHEN substring(@t,1,1)>0 THEN -CONVERT(int, substring(@t,1,1)) ELSE 0 END`

`UNION ALL`

`SELECT i+1, i/9 +1, (i % 9) +1, (i / 27)*3 + ((i%9)/3+1) ,`

`CASE WHEN substring(@t, i+1,1)>0 THEN -CONVERT(INT, substring(@t, i+1,1)) ELSE 0 END`

`FROM n WHERE i 0 AND @i<82 BEGIN`

`SET @ii = @ii+1`

`IF @Step0 SELECT @v-2 = V, @r = R, @C = C, @b-2 = B FROM #SDK WHERE i= @i`

`-- Given cell, go forward or backward to skip`

`IF @v-2 <0 BEGIN`

`SELECT @i = @i + @Step`

`END`

`ELSE `

`-- Already hit max, reset to zero and getting backward`

`IF @v-2=9 BEGIN`

`UPDATE #SDK SET v=0 WHERE i= @i`

`SELECT @Step = -1, @i = @i - 1`

`END`

`ELSE `

`BEGIN`

`SET @v-2 = @v-2 +1`

`UPDATE #SDK SET v = @v-2 WHERE i= @i`

`IF EXISTS (select * FROM #SDK WHERE i@i AND ABS(v)=@V AND (r=@r OR c= @C OR b= @b-2)) BEGIN`

`SET @Step = 0`

`END`

`ELSE BEGIN`

`SELECT @Step = 1, @i=@i+1`

`END`

`END`

`--SELECT @step`

`IF @ii = -1 GOTO OutputSudoku`

`END`

`SELECT @ii`

`----------------------------------------------------------------------------------`

`--Step 4: Output the Solution`

`OutputSudoku: -- you can add "GOTO OutputSudoku" anywhere above to stop the calculation and see the interim results!`

`WITH a AS (SELECTr AS 'Row',`

`c,`

`v`

`FROM#SDK`

`) SELECT * FROM a pivot ( max(v) for c in ([1],[2],[3],[4],[5],[6],[7],[8],[9]) ) AS p`

`SELECT DATEDIFF(ms, @datetime, GETDATE())`

`GO`

`EXEC uspSolveSudoku_2 '080001000030750000100000027000004301400000006701300000520000004000049070000800090',1 -- an evil level puzzle`

`--EXEC uspSolveSudoku_2 '080700000902000000300090020060800200750109043009004070040050009000000706000007030',1`

`--EXEC uspSolveSudoku_2 '200060000000900871740008006006080030003000100090030400300700018972005000000090002' , 1`

`--EXEC uspSolveSudoku_2 '200060000000900871740008006006080030003000100090030400300700018972005000000090002' ,1`

`--EXEC uspSolveSudoku_2 '029000008030000010000520097070056100000000000006310070760041000050000020800000630' , 1`

`--EXEC uspSolveSudoku_2 '080700000902000000300090020060800200750109043009004070040050009000000706000007030',1`

`--EXEC uspSolveSudoku_2 '060104050008305600200000001800407006006000300700901004500000002007206900040508070',1`

`-- Code Cleaning:`

Following is the iteration approach in JavaScript.

`//Create two arrays:`

`// a: Holds the puzzle value.`

`// b: Holds the predefined flag(non-zero)`

`a = new Array(0,0);`

`b = "060590004"`

`+"005004302"`

`+"290700100"`

`+"000017238"`

`+"400000006"`

`+"378920000"`

`+"002001049"`

`+"507600800"`

`+"800035010";`

`for (i = 0; i < b.length; i++){`

`a = parseInt(b);`

`}`

`//Loop through each none predefined cell`

`L: for (k = FindNext(-1); k= 0){`

`if (b==0){`

`if (a<9){ return i;}`

`else{`

`a = 0;`

`}`

`}`

`}`

`return -1;`

`}`

`//Validate the cell x`

`function Validate(x){`

`xr = Math.floor(x/9)*9; //Row offset`

`xc = x % 9; //Column offset`

`xb = (Math.floor(x / 27)) * 27 + ( (Math.floor((x % 9)/3)) *3 ); //Block offset`

`for (i = 0; i < 3; i++){`

`for (j = 0; j < 3; j++){`

`if ((((Xr+i*3+j) != x) && (a[Xr+i*3+j] ==a[x]))`

`|| (((Xc+(i*3+j)*9)!= x) && (a[Xc+(i*3+j)*9] ==a[x]))`

`|| (((Xb+i*9 + j) != x) && (a[Xb+i*9 + j] ==a[x])))`

`return false;`

`}`

`}`

`return true;`

`}`

Following is the iteration approach in Java.

`public class Sudoku {`

` public int[][] s = {`

` {0,0,9,0,2,8,0,0,0},`

` {0,0,0,7,0,0,0,0,2},`

` {0,0,0,0,0,0,1,3,4},`

` {0,0,0,0,6,0,0,2,8},`

` {8,0,0,0,4,0,0,0,9},`

` {3,5,0,0,7,0,0,0,0},`

` {6,8,4,0,0,0,0,0,0},`

` {9,0,0,0,0,1,0,0,0},`

` {0,0,0,5,9,0,7,0,0}`

` };`

` private int[][] f = new int[9][9];`

` public int row = 0, col = -1;`

` private String msg = "Still Trying...";`

` public void solveIt(){`

` for (int i = 0; i < 9; i++){`

` for (int j = 0; j < 9; j++){`

` f[j]=s[j];`

` }`

` }`

` while (moveNext()){`

` while (!this.findNextNumber()) {`

` if (s[row][col]==10){`

` if (!moveBack()){`

` msg = "No Solution Found";`

` return;`

` }`

` }`

` }`

` }`

` msg = "Congratulations! Found Solutions!";`

` }`

` private boolean findNextNumber(){`

` while (++s[row][col]<=9) {`

` if (checkCell()) {return true;}`

` }`

` return false;`

` }`

` private boolean moveNext(){`

` do {`

` if (++col ==9) {row++; col = 0;}`

` }`

` while ( row<9 && f[row][col]!=0);`

` return (row=0 && f[row][col] != 0);`

` return (row < 0 ? false : true);`

` }`

` private boolean checkCell(){`

` for (int i = 0 ; i < 9; i++){`

` if (i != col && s[row]==s[row][col]) return false;`

` if (i != row && s[col]==s[row][col]) return false;`

` }`

` for (int i = row/3*3 ; i < row/3*3+3; i++){`

` for (int j = col/3*3 ; j < col/3*3+3; j++){`

` if (!(i == row && j == col) && s[j]==s[row][col]) return false;`

` }`

` }`

` output();`

` return true;`

` }`

` public void output(){`

` System.out.println(msg);`

` for (int i = 0 ; i < 9; i++){`

` for (int j = 0 ; j 0 then substring(@t,1,1) else '123456789' end`

` UNION ALL`

` SELECT i+1, i/9 +1, (i % 9) +1, (i / 27)*3 + ((i%9)/3+1) , case when substring(@t, i+1,1)>0 then substring(@t, i+1,1) else '123456789' end`

` from n where i < 81`

`)`

`insert into #s (i,r,c,b,v) select * from n`

`OPTION (MAXRECURSION 80);`

`---------------------------------------`

`--Step 2: Create sequential number`

`---------------------------------------`

`--Create table #num ( vInt int, vBinary int);`

`WITH n (i ) AS (`

` SELECT 1`

` UNION ALL`

` SELECT i+1 from n WHERE i 0 begin`

` update #s set v = replace(#s.v, b.v, '''')`

` from (select Range, v from #s where d = 1) as b`

` where #s.Range= b.Range and #s.d >1 and charindex(b.v, #s.v)>0;`

`end;'`

`-- query to determin the cell that solely contains a possible value in the same range`

`Set @SQL2 =`

`'Update #s set v = p`

`from ( select Range, p`

` from #s cross apply (select P from #num where charindex(#num.p, #s.v)>0) as n`

` where d >1`

` group by Range, p having count (*) =1`

` ) as t`

`where #s.Range = t.Range and #s.d >1 and charindex (p, #s.v) >0; '`

`-- if 2 cells in the same range have the same pair of possible values, eliminate the pair of values from other cells in the same range`

`Set @SQL3 =`

`'Update #s set v = replace(replace(v, substring(v2,1,1), ''''),substring(v2,2,1), '''')`

`from ( select Range, v as v2`

` from #s`

` where d = 2`

` group by Range,v having count(*) =2`

` ) as t`

`where #s.Range = t.Range and #s.d>1 and (charindex(substring(v2,1,1), #s.v)>0 or charindex(substring(v2,2,1), #s.v)>0) ; '`

`Declare @Count1 smallint, -- Count of determined cells, before updates`

` @Count2 smallint, -- Count of determined cells, after updates`

` @CountSecondRuleExecTimes int ;`

`Select @Count1 = 0, @CountSecondRuleExecTimes = 0;`

`Select @Count2 = Count(*) from #s where d=1`

`While @Count2 < 81 and @CountSecondRuleExecTimes < 100 Begin`

` While @Count1 @Count2 and @Count2 < 81 Begin`

` Select @Count1 = count(*) from #s where d = 1`

` Set @SQL = replace(@SQL1, 'Range','r'); Exec (@SQL);`

` If @Trace= 1 print Char(13)+Char(10)+@SQL`

` Set @SQL = replace(@SQL1, 'Range','c'); Exec (@SQL);`

` If @Trace= 1 print Char(13)+Char(10)+@SQL`

` Set @SQL = replace(@SQL1, 'Range','b'); Exec (@SQL);`

` If @Trace= 1 print Char(13)+Char(10)+@SQL`

` Select @Count2 = count(*) from #s where d = 1`

` End`

` If @Count2 < 81 Begin`

` Set @SQL = replace ( case when (@CountSecondRuleExecTimes % 6) <3 then @SQL2 else @SQL3 end,`

` 'Range',`

` case (@CountSecondRuleExecTimes % 3) when 0 then 'r' when 1 then 'c' when 2 then 'b' end)`

` If @Trace= 1 print Char(13)+Char(10)+@SQL;`

` Exec (@SQL);`

` Select @Count2 = count(*) from #s where d = 1`

` Select @CountSecondRuleExecTimes = @CountSecondRuleExecTimes + 1`

` if @Count1 @Count2 AND @Trace = 1 Print replace(@SQL, char(13)+char(10),'')`

` End`

`End ;`

`---------------------------------------`

`OutputSudoku:`

`---------------------------------------`

`with a as ( select r , c, v from #s ) select * from a pivot ( max(v) for c in ([1],[2],[3],[4],[5],[6],[7],[8],[9]) ) as p;`

`If @Trace = 1 select datediff(ms, @datetime, getdate()), @CountSecondRuleExecTimes`

`GO`

`exec uspSolveSudoku_C '200060000000900871740008006006080030003000100090030400300700018972005000000090002' , 1`

• Hi Dave,

Sorry for my overly simplied comments. Yes, you are right. The input string is composed by the contents in the cells, from left to right, from top to the bottom, it should be exactly 81 characters in length. For non-predefined (empty) cells, use 0's.

You can jump to the output script at the bottom anywhere after the input is read, to see the puzzle values.

Thanks,

Kevin

• This looks like fun.

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