# Sudoku Latin Squares

This new twist on a classic game opens up a whole new world of Sudoku. Start small, with a 3x3 grid, to take things easy. Or go big, with the classic 9x9 grid, and become a sudoku master. With all kinds of helpful features, like notes and hints, you'll get drawn in to solving hundreds of challenging puzzles.

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Sudoku is Japanese for single number and the name is **Sudoku Latin Squares** a registered trademark of a Japanese puzzle publishing company. Looking at the middle three boxes, we have a 3 in the left-hand Magic Heroes: Save Our Park and one in the middle Virtual Farm, but we still need to put a 3 in the right-hand box. But bearing in mind that they had computers at their disposal, spare *Sudoku Latin Squares* thought for Tarry who had to do everything by hand! Given an individual completed grid, how many minimal initial grids are there which have this grid as a solution? To this day no-one has been able to derive from this, or any other formula, how fast this number grows as the order of the square gets large. The middle three boxes Now if we look at the bottom three boxes, one of the rows already has 6 numbers. Unfortunately, it is only a partial knight's tour, as there is a jump from 32 to Using the concept of the knight's tour William Beverley managed to produce a magic square, as shown below. A Knight's Tale As any chess player will know, an order 8 magic square has the same number of cells as a chessboard. For the same reason, it can't go in the bottom row, which leaves the middle row. It's difficult to tell how many distinct completed Sudoku grids there are, but mathematicians Bertram Felgenhauer and Frazer Jarvis used an exhaustive Bubble Zoo 2 search to come up with the number 6,,,, which was later confirmed by Ed Russell. There is no space northeast of the 1, so I have put the 2 in the bottom row, followed by the 3. The 6 should go in the cell where the 1 is, but because this cell is occupied, I put the 6 immediately below the 5 and continued up to

To make it a fair test, he decides that every volunteer has to be tested with a different drug each week, but no two volunteers are allowed the same drug at the same time. Solving Sudoku requires logical thinking and a systematic approach. Not surprisingly, magic squares made in this way are called normal magic squares. There's only one magic square of order 1 and it isn't particularly interesting: a single square with the number 1 inside! It turns out that normal magic squares exist for all orders, except order 2. Some three thousand years ago, a great flood happened in China. Several variations have developed from the basic theme, such as 16 by 16 versions and multi-grid combinations you can try a duplex difference sudoku in the Plus puzzle. The more numbers are filled in initially, the easier the puzzle becomes of course. For example, a magic square of order 3 contains all the numbers from 1 to 9, and a square of order 4 contains the numbers 1 to Using the concept of the knight's tour William Beverley managed to produce a magic square, as shown below. On his return to France he brought with him a method for constructing magic squares with an odd number of rows and columns, otherwise known as squares of odd order. Euler never solved this problem.

Instead of saying "numbers that are divisible by 4", mathematicians usually say "numbers of the form 4k". Instead, the knight moves in an L-shape as shown in the diagram. There is no space northeast of the 1, so *Sudoku Latin Squares* have put Squades 2 in the bottom row, followed by the 3. The more numbers are filled in initially, the easier the puzzle becomes of course. Euler never solved this problem. They found semi-magic tours, but no magic tours. Starting from 1, I have DragonStone in the numbers up to Sudoku is Japanese for single number and the name is now a registered Suares of a Japanese puzzle publishing company. In J. Sudoku or Su Doku are a special type of Latin squares.

Here's an example of a Latin square, with the numbers 1 to 4 in every row and column. At first glance, it seems that the following magic square by Feisthamel fits the bill. Looking at the middle three boxes, we have a 3 in the left-hand box and one in the middle box, but we still need to put a 3 in the right-hand box. Just over a hundred years ago, Euler's prediction was partly proved right. In , Stertenbrink and Meyrignac finally solved this problem by computing every possible combination. Further for each pattern and each symbol there is precisely one cell which contains that combination, and for each pattern and each colour there is precisely one cell which contains that combination. But bearing in mind that they had computers at their disposal, spare a thought for Tarry who had to do everything by hand! The Thirty-Six Officers Problem Euler did a considerable amount of work on Latin squares, and even came up with some methods for constructing them. If you encounter a cell that is already filled, move to the cell immediately below the cell you have just filled, and continue as before. But what is the minimal number of clues that have to be given to ensure that there is exactly one — and no more — solution? But let's have a look at how to go about solving a Sudoku puzzle. Nobody knows how many distinct magic squares exist of order 6, but it is estimated to be more than a million million million!

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Sudoku in PrologLook at the following square It satisfies the definition of an orthogonal latin square, but it has the added property that if we look at the patterns on the cells, then each pattern occurs once in every row, once in every column. If we combine the two Latin squares below, we get a new square with pairs of letters and numbers. Sudoku or Su Doku are a special type of Latin squares. Just over a hundred years ago, Euler's prediction was partly proved right. The knight is an interesting piece, because unlike the other pieces, it does not move vertically, horizontally or diagonally along a straight line. The first puzzle appeared in the magazine Dell pencil puzzles and word games in and was called Number Place. Not surprisingly, magic squares made in this way are called normal magic squares. On his return to France he brought with him a method for constructing magic squares with an odd number of rows and columns, otherwise known as squares of odd order. For the same reason, it can't go in the bottom row, which leaves the middle row. The rows, columns and diagonals all sum to A magic square is a square grid filled with numbers, in such a way that each row, each column, and the two diagonals add up to the same number. Proving that these methods work can be done using algebra, but it's not easy!

There are distinct magic squares of order 4 and ,, of order 5. But let's have a look at how to go about solving a Sudoku puzzle. If you look at cell C, the only number that can go in it is 7. A French mathematician called Gaston Tarry checked every possible combination for a 6 by 6 Euler square and showed that none existed. When all the cells are filled, the two main diagonals and every row and column should add up to the same number, as if by magic! They are usually 9 by 9 grids, split into 9 smaller 3 by 3 boxes. Again, mathematicians do not know the answer to this question. Just over a hundred years ago, Euler's prediction was partly proved right. If you look at the first row and the first column, you'll notice that the numbers occur in sequence: 1, 2, 3, 4. There's only one magic square of order 1 and it isn't particularly interesting: a single square with the number 1 inside! Luckily, there is. On his return to France he brought with him a method for constructing magic squares with an odd number of rows and columns, otherwise known as squares of odd order. Magic squares of even order Although the Siamese method can be used to generate a magic square for any odd number, there is no simple method that works for all magic squares of even order. It has three rows and three columns, and if you add up the numbers in any row, column or diagonal, you always get