Teach Yourself Sudoku

Learn the secrets that let you easily solve Sudoku puzzles faster!

Lesson #11. What if?


We introduced the “What if?” concept in Lesson 9 to illustrate the SHARED PAIR logic.  Since “What if?” is a form of guessing, using it to attempt to solve a Sudoku puzzle is generally frowned upon because Sudoku is a game of logic and you should be able to definitely determine the answer for each cell as you work your way through the puzzle.  Taking a guess at a number and then building on that possibly wrong guess could ultimately lead you into a dead end with such a complicated mess that all you can do is erase everything and start over.  Obviously, “What if?” is not the best way to solve Sudoku puzzles.


So why bother to discuss it all?


The answer may surprise you.  Here it is:  not all Sudoku puzzles are created equally.  There are some Sudoku puzzles that require that at some point, you’ll have to switch to the “What if?” technique in order to go any further.


Trouble in Sudoku-land?


Believe it or not, there’s actually a controversy in the Sudoku community about these “What if?” puzzles.  Sudoku purists consider them to be inferior, poorly designed, “not worthy of playing”, reject puzzles because they can’t be solved by the classic Sudoku logic methods.  Others claim that they are absolutely valid Sudoku puzzles and that “What if?” is just another form of logic that you can use to solve these puzzles.


Inherent in this controversy is that fact that there are only so many ways to make Sudoku puzzles harder.  Basically, you can design a Sudoku puzzle so that if has fewer numbers that are possible to solve (ultimately bringing it down to just a single number that can be solved at any one point in time) and / or making the logic required to solve the next cell more sophisticated.  But once a player learns some of the advanced logic techniques discussed here, they can solve just about any standard Sudoku puzzle regardless of its level of difficulty.  It may require a careful eye to spot the one single pattern that can resolve the next number and that might mean more time is needed but in general, there is a limit to how challenging Sudoku puzzles can be. Hence, the Sudoku variants that use larger grids (16 x 16) and other structural changes to provide additional levels of difficulty.


For the standard 9 by 9 grid, one of the ways to make Sudoku puzzles harder to solve is to require the “What if?” solution method.


Can I have both, please?


While I understand the protests of the purists, I also think that the “What if?” fans have a point.  Here’s where I see some common ground that could be established between the two camps.


The purist are right. Playing a “What if?” Sudoku puzzle is not the same as playing a standard Sudoku puzzle that can be solved using classical logic techniques.  To them, “What if?” isn’t even a valid form of logic.

The “What if-ers” are right.  Playing a “What if?” Sudoku puzzle is just another way to play Sudoku.  It expands on the classical methods of solving the Sudoku puzzles.  To them, “What if?” is just as valid a form of logic as any other solving technique.

While I’m not sure that either side could ever convince the other about whether or not “What if?” is a valid form of logic for solving Sudoku puzzles, I do know that it does work. So to me, it isn’t a question of whether “What if?” is a valid or invalid method of logic, but rather that it’s a different way to solve Sudoku puzzles. If both sides could acknowledge that, it would be helpful.

Given that it is different, if you tag each “What if?” puzzle with “Requires What if? Logic” then players can decide for themselves if they want to work on that puzzle or not. Those who like (or would like to try) “What if?” puzzles can give them a go, those who don’t can move on to the next puzzle (Lord knows there’s no shortage of Sudoku puzzles in the world.)  To be fair, I’d also recommend that most “What if?” puzzle should be tagged with the number of unresolved cells left in the puzzle when the “What if?” technique is required to move forward.  That way, players will have a clear cutoff point and know when to switch techniques.  If you don’t do this, then I doubt that many new people would bother to try “What if?” puzzles as they would be too frustrating for the average Sudoku player.  Having a few puzzles that don’t provide the cell count and / or the “What if?” tag would then give advanced players a real challenge and these puzzles can be flagged as “Extremely Difficult” or some equivalent rating.


Hopefully, these ideas can contribute to restoring some peace within the Sudoku community.


OK, enough diplomacy.


As for actually playing a “What if?” Sudoku puzzle, here are my tips and suggestions:


- Use a blank Sudoku grid to make a copy of the puzzle at the point when you are going to switch over to “What if?” logic. To me, this is the biggest change I had to make in order to use this technique.  Some people prefer to keep their  “What if?” choices and the ensuing implication in their heads, eliminating the need for writing things down separately.  I tried that and it didn’t work for me but more power to you if you can.  Others say that you can circle your guesses while leaving all of the other POSSIBLES in place.  I tried that and found it to be messy and confusing especially when I had to go back and try another number and erase all of previous results. As always, if you find that any of these techniques work for you, by all means use them.

- Copy just the Cells that have a final answer over to the blank grid.

- Now go back to the original puzzle grid and look for TEAMS of 2 as potential starting points.

- If there are multiple TEAMS of 2 to choose from, pick the one that has a number in it that already appears as an answer the highest number of times.  In other words, if you have TEAMS of 2 for the pairs (3,4) and (1,7) see how many times 1, 3, 4, and 7 appear as ANSWERS on the grid. Then pick the number that appears as an answer most frequently as the number that you will start with.  Remember, you are still guessing but you’re trying to put the odds in your favor by choosing wisely and a number that already appears frequently gives you better odds than a random guess.  If you don’t have any TEAMS of 2, look for TEAMS of 3, 4, etc. and apply the same process of identifying the number that already appears the most

- Assuming that you were able to find a TEAM of 2 to start with and one of the number occurs more frequently than any other, write that number down in one of the cells containing the TEAM of 2 and put a circle around it.  If there’s a tie (i.e. two or more numbers occur the same number of times), just pick and try it.

- Glancing back at the original grid, write down the contents of any cells impacted by your initial choice. If you can’t see any obvious next moves, transcribe the contents of all of the other unsolved cells onto your copy grid.

- From here, you’ll work the puzzle just like any ordinary Sudoku puzzle using all of the standard techniques.

- Typically, one of two things will quickly happen: you’ll be able to solve the puzzle and you can then transcribe the answers over to your original grid or you’ll encounter an error (2 or more of the same number in a Row/Column/ Block, etc.)  If the latter occurs, you’ve run aground and there’s no sense in going forward based on your initial number choice.

- If you do run aground, the good news is that you just learned (paraphrasing Thomas Alva Edison) one of the possible configurations that doesn’t work.  If you were lucky enough to have a TEAM of 2 to start with, the news is even better.  The answer for the cell that contained your first guess (remember you circled that number when you started so it will be easy for you to find it) is the other number in that TEAM of 2. So erase everything on your copy of the grid except those numbers that are resolved on the original, write down the other number in the starting cell and start the process again.

- In discussing “What if?” logic with other players, I’ve found that this “If at first you don’t succeed, guess, guess again” technique for playing Sudoku appeals to some and not to others with the vast majority giving it a thumbs down.  I think that’s because it is such a very different way to play the game and it feels so foreign to what players have become comfortable with that it just doesn’t seem like Sudoku to them anymore. Truth be told, everyone’s entitled to their opinion.  Bottom line:  If the idea of using “What if?” logic appeals to you, then give it a try.  You just might enjoy it.  On the other hand, if you’d rather not, then simply move on to the next Sudoku puzzle.

- One final comment: You’re first guess for the starting point might not always be right place to start even if the number is correct.  You may find that you’ve hit a dead end and can’t solve any of the remaining cells although everything you’ve entered so far appears to fit perfectly.  No worries.  This is to be expected when using  “What if?” logic.  Simply pick another cell on the updated grid as your new starting point for the remaining cells and apply “What if?” process again.  With a little practice, you can improve your ability to pick places to start although there’s always a good chance that you’ll need several “What if?” attempts to solve this kind of Sudoku puzzle.



Copyright 2006 Gary Ward All Rights Reserved