Sudoku: The mathematical legend of this millenium

[Guille]

Sudoku is definatelly the game involving mathematics in a direct way which is most known and followed in the world now. You can look information about the game and how to play it in wikipedia pages.

if you find challenging sudoku puzzles send them to me. I've become a really big fan of them. Any other shares this hobby with me?

Also, do you know any mathematical "trick" to apply in sudoku?

[Guille]

As nobody posted I've chosen to start the math myself.

For me, the best thing of sudoku is that it shows that logic is not reducible to mathematics. It does that by proving to be a game which requiers absolutelly logical thought of many kinds: logical organisation, logical memory, logical order, logical deduction, logical spatial analisation...etz. But there is no mathematics needed, or at least no mathematical knowledge is requiered, apart from knowing the 9 different positive integers natural numbers, but we could replace these by any nine symbols.

The best way of solving sudoko games is the following: 1. Take the smallest number that you haven't done, if you are starting the game now, it is the 1, 2. Look at each 3x3 box and observe which has the number and which doesn't, if it does have the number, leave it, if it doesn't, then do what I'll explain now. By the way, the best logical way of going through boxes is from the top left, to the top middle, to the top right, to the middle left, to the middle middle, to the middle right, to the bottom left, to the bottom middle, and finally to the bottom right. Easy. 3. If the box doesn't have the number, look at all the numbers that are already in each column, from left to right, and if the number you have is there, discard, if it does, take a mental note. If there is more than one column without the number, then go and look at the bars, if a horizontal doesn't have the number, make a mental note. Now look at the squares which enter in both a column and a bar without the number, if there is more than one, put a small sing of that number at an edge of the square, and leave the box to the next. When you go through the game, and you come along a number which satisfy one of the marked squares, then put it in, and cross or erse the small number note, at the end, you will only have one square left, it will be for it. The reason you ahve to look first at the columns and the at the bars is because we, our mind and body, has vertical simetry, so it is much more familiar and easier and fast than horizontal symetry. 4. When you finish all the boxes for a numbe,r go to the next, until you finish all the numbers. This is the best logical manner to solve sudoku games.

[Guille]

Although I agree with the statements I've done in my previous post, I do believe mathematical knowledhe and tools can be applied to the solution of sudoku games. If anybody else has noticed other mathematics related things in sudoku games, please post them, doesn't matter if they are equations, rules, sequences, properties, etz.

Ok, as in each box, column and bar there must be each of the different natural positive integers, then the total value of each must be 1+2+3+4+5+6+7+8+9=45. We know that 9x5=45 and that 4+5 are the two nearest integers that can equal 9. I also mentally derived that this beatifull mathematical relationship exists:

[9!/(4!x5!)]-45=81=9x9=(4+5)x(4+5)

Where ! is the factorial, the 4 and 5 are the digits of 45 and the 81 is:

(45x2)-9=81

Where x2 is because 45 has 2 digits, -9 because 5x9=45.

But this is only one of the many mathematical singularities there are. Now, we know that 1x2x3x4x5x6x7x8x9=9!=362,880. And that 9!-(4!x5!)=360000 thus 4!x5!=2880, and as 4!+5!=144, and 2880-6!=2880-720=2160, we know that 160-144=16=4^2, so 2000 because 2 for 45 has two digits, and 000 because it is the number of digits, 3, that has 6!.

And, whoever really has followed by explenation, which is bad, I know, but I'm not able to improve, will know that it is the secret to solve absolutelly any sudoku game. Good luck if you haven't understood!

[Cubola Zaruka]

The sum of the values of the numbers is irrelevant in classic sudoku because they could be permuted or replaced with other symbols. I initially search for places where there are two of the same number in any third and see if there is enough infomation to determine the third one. I also look for boxes in which a larger proportion of the spaces are filled and most of the remaining ones are in the same row/colmumn, then the other spaces in the box and the row/column are likely to be determinable. The method which you have given works, but I find it to be slow and use a more intuitive approach. http://uk.games.yahoo.com/sudoku.html