12/30/2023 0 Comments Sudoku 9x9 dThe player then moves the tiles on the grid to combine tiles with the same number, ultimately aiming to create the tile with the number 2048. The game starts with two tiles displaying the number 2, which can be combined to form a tile with the number 4. The game is played on a 4x4 grid and the objective is to combine tiles with the same number to create a tile with the number 2048. but, as I said, I'm not sure I have a proof there.2048 is a popular single-player puzzle game that was created by Italian web developer Gabriele Cirulli in 2014. The thing is that this pattern does not happen in any of the latin squares because that would force the third element of the square to be in the same row or column as itself, which is a violation of the definition of the latin square. The key part is in order to form a rectangular fence, I need to swap two columns somewhere (either in a single 3x3, or in the over grid): ? A B Though, while I am confident that these do not have rectangular fences, and the generation method precludes them, I am not sure that such is the case. This next one (another one grid generation) is one that is probably closest to the other solutions that have been provided and allows a clear picture of the patterns that are formed in this type of generation. If you have the process only use one latin square, the grid looks like: 1 2 3 4 5 6 7 8 9 The fact that its a latin square on the over board too, then makes the swaps such that I believe that the rectangular fence cannot form. if you note, the 7, 8, and 9 are always in the same set of three, as are the 1, 2, 3 and the 4, 5, 6. There are some properties of these puzzles. I'm taking the shortcut of not doing the replacement in my code. ($final, $final) = ($final, $final) Īnd running it a few times I get: 7 9 8 2 1 3 5 6 4 I am a programmer though, and so I whipped up a quick bit of code:ģ * $outer-> + $board + 1 Unfortunately, I don't have the math background to be able to prove that that such is the case. Here's where I'm guessing, and I haven't done the full analysis of the resulting board. Each value is then read as trinary when converting to base 10, and then some specific swaps are done. Of these 12, you select 9 with replacement that are placed on a 3x3 board and then a 10th is the 'over board'. You take the 12 order three latin squares (using values of 0, 1, and 2). There's a method described in Sudoku: Bagging a Difficulty Metric & Building Up Puzzles for generating a solved Sudoku board. You may use a program to find the answer, but there are many, many possibilities, so you will still have to use some logical approach to it. So, minimizing the number of such occurences would make it possible to create lesser-hint sudoku's and maybe throw some light on why 17 hints are still required.įor puzzlers, the question in section 1 is complete by itself you need not bother about the rest. I realized that if there exists a symmetric rectangular fence (as I have termed it), then it becomes necessary to give at least one of the 4 cells as a hint. I have been trying to prove that there exists no 16-hint solvable sudoku. Make a 9x9 completely-filled sudoku that has a minimum number of symmetric rectangular fences. You can visualizeĪ symmetric rectangular fence is a rectangular fence where oppositeĬorners hold the same value. A rectangular fence is a set of 4 cells on a solved sudoku grid withĬo-ordinates $(i,j)$, $(m,j)$, $(m,n)$ and $(i,n)$.
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