![]() ![]() ![]() And they're certainly not isomorphic to the grid in your question, which has no rectangle whose four corners match diagonally in this way. Which, while clearly closely related, I don't believe to be actually isomorphic. Other types of puzzle Its not just Sudoku-X you can solve online at puzzlemix. In these cases you must place 1 to 9 and A to C, or 1 to 9 and A to G respectively, into each row, column, bold-lined box and marked diagonal. I've set a computer program to generate solutions which match the constraints (and the additional constraint that the first row is $123456789$) so far it has found more than $20000$. Occasionally puzzlemix will feature a larger Sudoku-X 12x12 or even Sudoku-X 16x16 puzzle. I'm only counting $3!^2$ permutations of the rows rather than $9!$ because most of those $9!$ will break one or more of the $3\times 3$ squares with a Sudoku constraint. In the general case, a grid will be in an isomorphism equivalence class of $9!\cdot 3!^4 \cdot 2$ corresponding to permutations of the symbols, permutations of the rows and columns, and the symmetry of the square which is not already covered by permutation of the rows and columns. The diagonally adjacent givens are highlighted - there may be more! ![]() So I am asking if there is a diagonal-free grid with a quintessentially different structure to either of the two grids given here.įor reference here is a Sudoku grid with diagonals, from Conceptis Puzzles: No Internet required - play offline full screen. The torus example is derived from this one using only the operations defined above. Play Sudoku from 4x4 up to 16x16 grids Click for free trial download. We can see that the $2$ on the base line is diagonally related to the $2$ on the top line, and so this grid is not a torus. This can create grids with no internal diagonals but that are not torii, for example (this is my original grid by the way!): Isomorphism in this instance implies we can change the orientation, change the permutations of the numbers, perform row/column swaps as long as the conditions still hold. If we wrap the grid into a cylinder, and then bend the tube into a torus, the diagonal property still holds, so, for example, the $9$ on the base row is considered to be diagonally adjacent to the $4$ and $7$ on the top row. This does NOT imply the the whole diagonal is distinct (as in X-factor). To explain the diagonal-free property, if we have: My question is up to isomorphism, is the grid unique? Sudoku Dream is a Windows version of the famous game, sudoku.You can generate puzzles in sizes from 4x4 to 16x16 and difficulties from very easy to very hard. I have a Sudoku grid with the property that diagonally adjacent elements are distinct (it is also a torus under the same property). ![]()
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