14

Some initial deductions around the edges already give some regions: We can break in some more with internal clues: Some more clever deductions can lead to the next step: That gives a break-in to the rest of the puzzle: So the solution to the puzzle is:


10

Step 1: Step 2: Step 3: Step 4: Step 5:


10

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7/solution:


10

Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Step 7/solution:


7

Solution Step 1 Step 2 Step 3 Step 4 Step 5


7

Suppose we have a solution, which is a tree graph with vertices of degree 1 and 3 only. This graph has the following property: This can be proved by induction: This property has some consequences for the $M\times N$ grids that can be filled with Ts and bulbs. For square grids $N\times N$ this means that Edit: Here are pictures of general solutions for ...


6

The completed grid: Step-by-step solution: Now comes the first deduction which is a bit trickier: Next, we look at Now we are almost done!


5

To complement Jaap's excellent answer, a family of non-toroidal grids for odd length $4k+3$: EDIT: And a family covering all the toroidal grids:


5

Let's take one more step, with logic. Or, look at this picture: Therefore, But from here, there's no way to progress by logic, because there are two solutions. (Thanks to @venus in the comments for alerting me to this!)


4

What an interesting and rewarding puzzle. There is indeed a mistake in the puzzle, but one that is easily rectified. It was a bit tedious doing this on paper, I wish there was an app to integrate something like this together. Anyway, the first step is obviously to start solving the sudoku. I will not post every single step, just the key ones that will allow ...


2

I believe this is the solution (apologies in advance, not sure how best to format this): Edited to include explanation of how I arrived at the solution:


2

It is possible for a 9×9 Sudoku if you drop the requirements that the diagonals have the same sum; for example: 5 7 3 2 9 4 1 8 6 1 6 8 7 5 3 9 4 2 9 2 4 6 1 8 5 3 7 6 8 1 3 7 5 2 9 4 2 4 9 8 6 1 7 5 3 7 3 5 4 2 9 6 1 8 4 9 2 1 8 6 3 7 5 3 5 7 9 4 2 8 6 1 8 1 6 5 3 7 4 2 9 Of the 18 diagonals, only 6 have sums not equal ...


2

The next step is to That then allows you to fill in some more numbers. After that, you must choose one of the five possible solutions.


1

I thought that I'd found counter-examples to the proofs given in the other answers: However, I then realised that Instead we can In particular,


1

As the comments say, the smallest number of clues found for a 16x16 sudoku (with 4x4 squares) is 55. It has not been proven that this is the smallest number, but no one's found a better one yet. I suspect that less research into a minimum number has occurred for the larger variants because they are less common and would require more computing power to brute-...


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