**A “Very Hard” Sudoku that turned out to be pretty easy …**

The *San Francisco Chronicle’s* Saturday Sudokus are rated “6 stars”, and are considered very hard. Many weeks, they won’t even “validate” when I load them into my Sudoku program, which supposedly indicates there is no unique solution. Some weeks, I don’t even bother trying these puzzles. My experience with this one was different.

In solving Sudoku, the generally accepted approach is to begin by filling out the obvious entries that you can “eyeball”. Once you’ve exhausted the empty squares that can plainly hold only one number, you’re left with squares that could be any two or more different digits, but, usually could not be some other subset of digits without violating a rule.

To the left, it is easy to see which numbers are “given” in the puzzle, and which numbers I have penned in. For these, I use a non-erasable black pen because it is easy to see, and because there is very little chance of making a mistake at this stage. I’ve been playing the game for about half a year, but I was surprised at how many numbers I was able to fill in this way today.

Once you fill in a number, it forms a new clue for our puzzle, and you can build on that. The ‘2’ in the top right cell (3×3 grid) is obvious: a ‘2’ can only go in that cell’s middle row, and there is only one open space in that row.

**The ‘2’ in the lower right** is much less obvious and you can skip this if it gets difficult. Where can the ‘2’ in the middle right cell go? I don’t know, but it can’t go in the middle row because I already had a ‘2’ in that row. It can’t go in the left column because of the ‘2’ in the top right that we just added. So it could be in either of the empty squares in the middle column.

**Looking at the bottom right 3×3 cell again**, now we know the ‘2’ can’t go in the middle column or the left column, and it can’t go in the bottom row. We already had the ‘9’ position. The ‘2’ has to go in the upper right square. It takes a little practice before you can “see” that without penciling in the possibles with teeny little numbers.

The newspaper scan above represents my best ability to jumpstart the solution. Sharper or more experienced players will undoubtedly do better. The next step is to pencil in “possibles”: the possibles for the lower right square of the lower right cell are, by simple elimination, {7,8} at this point.

### (1) Pencil Tips filled in:

However, my Pappocom PC puzzle solver doesn’t think this is a fair puzzle. What an ego boost! But not every program can be perfect. At least it accepted this puzzle. For this article, I used the program to fill in all my “pencil tips” at once. If your eyes are very good, you can do this on the newpaper or puzzle book, but my eyes aren’t that good, and I make too many mistakes.

I already solved this puzzle, but here I have reverted it and re-filled in my pencil tips with the “possibles”. You will get a few “freebies” like the ‘8’ in the middle 3×3 cell, since that middle square can’t contain any of the other numbers. [Why not ‘7’? See “(3) Tricks and Rules” below].

This means you can eliminate any other 8’s in that middle 3×3 cell, and in that middle row, and in that middle column. In the row where we can fill in the ‘8’, we now also know which square is the ‘1’ and which is the ‘7’, further eliminating 1’s and ‘7s in their associated 3×3 cells, rows, or columns:

### (2) Partial Solution:

Filling in the ‘1’ and ‘8’ we found for the middle row (fifth row down), we’ve eliminated the associated 1’s and 8’s, since a number can only occur once in any row, column or 3×3 cell.

This reveals a definite solo ‘4’ and a definite ‘8’ in the middle and bottom right-hand cells, respectively.

Filling these in, and further eliminating possibles that are no longer possible, allows us to complete the middle right 3×3 cell. Given the single ‘4’, can you see which is the ‘2’ and which is the ‘8’?

We’re well on our way to solving this puzzle. There is not much more we can do with the right-hand 3×3 cells until we complete some of the other grids. In the upper left cell, there is an extra ‘4’ pencil tip that should not be there, in the middle square, since there is already a ‘4’ in that column. But fixing that error does not give us a new grip on the puzzle.

If you are intending to try to complete this on your own, I might next look for the ‘2’ in the fourth row. While you’re in that middle cell, look for the ‘7’.

When you complete a square, don’t ever forget to rid your pencil tips of eliminated possibles right away, or the patterns will be much harder to spot.

### (3) Sudoku Tricks and Rules

There are a number of logical rules one can memorize and practice to help speed solutions. I am learning them now, and at some point will publish my own notes. But they are just rehashes of compilations I have already found elsewhere. Visit www.sudoku.com for an excellent set of tips, and through this site, or Google, you can also find other excellent sites with tutorials.

I do not think I found it necessary to use many of these tips to solve this puzzle. There are reports of young kids who solve these puzzles without tricks or tips just by looking at the possibles and eliminating impossibles, one square at a time. There is also an excellent article in the current Scientific American, The Science Behind Sudoku, which gives the clearest expression of the general principles that I’ve seen yet.

The only example I will discuss here is PAIRS:

*Naked Pairs:*

If two cells in a group contain an identical pair of candidates and only those two candidates, then no other cells in that group could be those values. [Solving Sudoku, Angus Johnson]

(The term “group” is always used to refer to any row, column or 3×3 cell of 9 squares.)

In our Sudoku partial solution above, we have a pair of {4,5} possibles in the middle 3×3 cell. We don’t know which square is the ‘4’ and which the ‘5’, but we do know that no other square in the cell can be a ‘4’ or a ‘5’. The pairs rule means we can eliminate the ‘4’ in the {2,4,7} square, so this square is really only a {4,7} — it can really only contain a ‘4’ or ‘7’.

But, this is the only square with a ‘7’ in the middle cell, so, the pairs rule doesn’t help tell us (this time) which is the ‘7’. We already knew that, since no other square in that cell (or in that row) was a possible ‘7’.

In reviewing this article, I found a **PAIR** which was used to solving the puzzle. Look at figure (1) again where we have penciled in a lonely ‘8’ in the center 3×3 cell:

We know that this square can’t be a 2,3,9 or 6 because its row already contains these values. It can’t be a 1, 4 or 5 because its column already contains those values. That leaves a ‘7’ or ‘8’, so why can’t it be the ‘7’?

This column contains a pair of {6,7} possibles. By filling in our pencil tips for the column, we have learned that one of those squares must be the ‘6’, and the other, the ‘7’, though we don’t know which is which yet. Since one must be the ‘6’ and one must be the ‘7’, we do know no other squares in the column can be a ‘6’ or a ‘7’, as the Pairs rule formalizes in logic. So, by elimination, we know this square can’t be a ‘7’. It has to be the ‘8’.

### (4) Solved Sudoku:

I finished this one in the PC puzzle application 15 minutes 29 seconds, which is pretty nearly a personal record for me for any difficulty level. That doesn’t include “pencil time” for “obvious” entries on the newspaper itself. There is no exact order required for a solution. I don’t always fill in all the pencil tips at once, but only for the cells, rows or columns on which I am actively working. This helps reduce clutter. When I can go no further any other way, I continue filling in pencil tips until the whole puzzle is filled.

I don’t know why this puzzle was rated “very hard”. There must be a mathematical algorithm or reason for the difficulty ranking. Whatever the reason, it is a great confidence-builder. Any puzzle which does have a unique solution can be solved by using the simplest logic and rules.

It is possible to use the PC Sudoku program to “cheat” by guessing; this particular program will flag an erroneous entry in red. But I try not to do that, and didn’t have to guess to complete this solution.

I loaded the original newspaper puzzle into the Pappocom Sudoku program. The program displays the original puzzle clues in blue letters. My solved entries are in dark gray.

*Alex Forbes ©May 27, 2006*

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