## Notice

I am no longer posting new puzzles to this blog. For all of my Sudoku puzzles, old and new, please visit Sudoku in another section of this website. I will still create and offer new puzzles, in batches of a couple of hundred, once a week or so.

## Sudoku #60

Category: Sudoku
Sun, 21 May 2006, 19:51

In my last Sudoku weblog posting, I described another solving technique I recently stumbled upon. Of course, I added the technique to my program. I was initially quite pleased to see my program generate lots of puzzles that needed to use this technique. However, additional analysis turned up something. Many of the generated puzzles had the undesirable property of having multiple solutions.

To begin with, here's a quick summary of the new method: Look at the four cells at the intersection of two rows and two columns. If three of the cells have possibilities {ab} and the fourth has possibilities {abc}, the fourth cell must be c. Otherwise, the puzzle would have multiple solutions.

The following is a fragment of what can be produced when the technique is applied to a randomly generated board:

In this case, there are two possible solutions:

Why does this happen? This solving technique works so long as we know there is only one solution. Unfortunately, while we are still generating a puzzle, we can't yet be sure that we have a unique, solvable puzzle.

Does this mean one can't reliably generate a puzzle that uses this technique? No. It just means that some additional checking is needed to ensure that you end up with a puzzle with a unique solution. At least, you can use brute-force back-tracking analysis to try to generate all possible solutions. I suspect that may be overkill, though.

Or at least, I hope that brute-force back-tracking is overkill. I tried to weed out invalid puzzles by repeating the new technique backwards to see if the same solution comes out. But while this can find some cases of invalid puzzles, it can still let through other invalid puzzles.

The bottom line is that more work is needed before I can post puzzles that need this new technique.

Hans

## Sudoku #59

Category: Sudoku
Sun, 07 May 2006, 17:09

After doing Sudoku puzzles for about a year, you might think there's nothing left to learn about the puzzle. On the contrary, I never believed for a moment that I knew all there is to know about Sudoku. Actually, I always suspected that there was yet another solving technique to be discovered. And finally, this past week, I came across that technique.

Here's part of my solution and notes to puzzles S72.60 in the Standard Puzzles:

Note that the cells marked with yellow have the possibilities {14}. But the cell to note is the cell marked in blue with possibilities {149}. Let's assume that the blue cell does not have the value 9. All the yellow and blue cells would then have the possibilities {14}. This would mean that the puzzle would have two solutions! Since valid Sudoku puzzles must have unique solutions, our original assumption must be wrong, and the blue cell must have the value 9.

The following diagram illustrates the technique:

Consider the four cells at the intersection of two rows and two columns. If three of the cells have possibilities {ab} and the fourth has {abc}, then possibilities {ab} can be eliminated from the fourth cell. Note that c might be one or more possibilities.

Also note that, unlike the other advanced techniques described in How to Solve Sudoku, this one can result in the direct determination of the value of a cell.

Anyways, what does this new technique mean to my puzzles? Clearly, none of my 150,000 puzzles require this technique. I said I wasn't going to post any more new puzzles. But I may have to reconsider that decision. I'll add this new technique to my program, and then I'll see if it's possible to generate puzzles that do require this technique. If so, then I may have to add some more puzzles.

Hans

## Sudoku #58

Category: Sudoku
Sun, 09 Apr 2006, 20:08

That's it, I've had enough of creating Sudoku puzzles.

I just posted another batch of more than 10,000 puzzles, putting the total number of puzzles available from this web site at over 150,000. Even if you solved five puzzles a day for the rest of your life, you'd still probably not solve all of these puzzles.

If you like these puzzles, don't worry - I'm not taking them away. I'm just not going to add any more.

Anyways, it's been fun and interesting playing with the algorithms for solving and creating puzzles, but enough is enough. There's only so far you can go with this silly puzzle. I thought about branching out into other puzzles, such as Kakuro (cross sums). After all, Kakuro is very similar to Sudoku and many of the same algorithms can be adapted. But then again, it's too similar. If I do play with other puzzles, I'll go with something completely different.

Hans

## Sudoku #57 - January Web Stats

Category: Sudoku
Sat, 11 Feb 2006, 17:50

I just looked at my web stats for last month. My web site served up a total of 27,206 PDF files. At a hundred puzzles per PDF, that's a grand total of 2,720,600 puzzles printed out during the month!

By a large margin, the most popular group of Sudoku puzzles on this site are the Nonomino Puzzles. With over 3500 visits during January, it's actually the 5th most visited page on boldts.net!

The second most popular group of puzzles are the Advanced Puzzles, followed close behind by the Beginner Puzzles. The Standard Puzzles and Hard Puzzles are lower down in popularity.

When I first started creating Sudoku puzzles, I wanted to offer puzzles that were more challenging than the usual fare printed in the local newspapers. Looking at the January stats, I think there's a real demand for the more difficult puzzles. Sometimes it's nice to relax and zip through a bunch of easy puzzles, but that can get tiring. On the other hand, a difficult puzzle can keep you occupied for much longer. And when you complete it, you feel like you've really accomplished something.

If you want puzzles that are excruciatingly difficult, try out the Diagonal Puzzles which I posted just a few days ago. In these puzzles, you have two extra groups to deal with. In addition to the nine rows, nine columns and nine 3x3 squares, you also have to place the digits from 1 to 9 in the two diagonals. No big deal? Think again. These puzzles all have just 16 seed values, and the ones I've tried so far have been killers. I didn't tell my program to produce puzzles of a particular difficulty, but the vast majority of them require the use of the advanced solving techniques.

Anyways, over the next few days, watch for yet another batch of 20,000 puzzles. I haven't tried many of them myself yet, so it's too early to tell how difficult they are. I'd like to see how far I can go in producing really challenging Sudoku puzzles. But to go further, I think I now need either a better puzzle generation algorithm or a faster computer.

Hans

## Sudoku #56

Category: Sudoku
Fri, 11 Nov 2005, 10:14

In #55, I commented about the 2001st puzzle posted to my Sudoku Puzzles. I said that it was perhaps time to slow down a bit. Well, two and a half months later, I just posted the 32,900th puzzle!

A few things changed since August. To make things easier, I decided to post puzzles in PDF form only, and put 100 puzzles in each PDF. After all, the best way to solve these silly puzzles is with pencil and paper. Since I can produce about 200 advanced puzzles a day, or several thousand standard puzzles a day, it's not much trouble to create and upload so many puzzles.

Where will it end? Will I reach 50,000 puzzles? or 100,000 puzzles? Who knows? I first thought 2000 puzzles was a lot. With 32,900 puzzles, no one in their right mind can expect to solve them all!

Hans

## Sudoku #55

Category: Sudoku
Sun, 28 Aug 2005, 19:04

It hasn't been long since I started making my puzzles available in a new format. I had expected to post 10 or 20 once every week or so. But today, I just uploaded my 2001st puzzle. If you solve one of these puzzles a day, you now have enough puzzles for five and a half years!

I think it's now time to slow down a bit, and really get back to my original plan. Or perhaps I'll leave it at 2001 puzzles for a while. After all, this is just another silly puzzle.

Hans