One trivial advice that you should always keep in mind: if a cell cannot be black, it must be white! And vice versa. From this comes one simple solving technique: when you conclude that a cell must be painted, circle the numbers in all four adjacent cells to indicate that they must be left unpainted. Now, back to the patterns…

Corners
Check the corners first. Here is an example:

When you have 2 same numbers adjacent to each other in a corner, you can circle at least one other number. Why? It is easy to see that the “1” cannot be black, because one of the “4”s must be black. If the leftmost “4” were black, the “1” would be adjacent to it. If the rightmost “4” were black, the “1” couldn’t be black because those two black cells would isolate the other “4” in the corner, which is against the 2nd rule of Hitori. Got it?


Here are some other similar patterns that occur in corners:
        


And here are some other examples for you to practice:
            

Squeezed in the middle
This is probably the most common pattern in Hitori puzzles.

A number squeezed between two same numbers must be white! If it were black, it would eventually cause two black cells to be adjacent, which is not allowed.
Oh, and don’t be put off by three same numbers adjacent in a row or column. The same rule applies.

Elimination
By the way, once you’ve circled a number, make sure to eliminate all other occurrences of the same number in the same row and column:

No room for three
Another very common pattern is when two same numbers are adjacent to each other and there is another one in the same row/column. Obviously, one of the adjacent “3”s will have to be white (they can’t both be black). Whichever one it is, the 3rd “3” in the same column will have to be black, as you can’t have two “3”s in one column.

Those were the basics. That’s how you start solving any given puzzle. In a few days, I will post some more complex patterns that are used in the more difficult of puzzles.

It can be used when a cell has only one “neighbor” that is an empty cell. In such a case, that cell (with only one empty neighbor), must contain a number that is consecutive to (at least) one of the already filled in neighbors.

Look at the upper left corner. Number 33 could in theory go into any of the 4 cells that are neighbors to both 32 and 34. However, it must actually go into R1C1 (row 1, column 1). Why? Because R1C1 has only one empty neighbor and according to the rule I stated above, in such a case that cell must be consecutive to (at least) one of the existing neighbors.
Why? Because each cell (other than 1 and the largest number) must have both a “+1” and a “-1” neighbor. So, the only options for R1C1 are 31 or 33 or 35. If you put any number other than one of those in R1C1, it would have only one of the “+1”, “-1” neighbors, because there is only room for one number next to it. That is why I also call this technique “dead-end”. But 31 would be too far from 29; and 35 would be too far from 39. Therefore R1C1=33!

Now, look at R1C7 (it’s circled). See if you can figure out why the number in that cell MUST BE 5.

It’s an interesting concept, quite similar to Killer Sudoku, but with all four basic arithmetic operations involved. Another change is that repeats within a cage are allowed if possible.

To help out those of you who are starting to like this game, I have prepared a tool that shows you which combinations of numbers can go into a certain cage. All you need to do is plugin the numbers, click “Calculate” and voila!, you get the list of possible options.

Obviously, this same calculator can also be used for Killer Sudoku and Kakuro puzzles.

Here is the calculator:

UPDATE: I’ve started posting these puzzles on a daily basis. You can find them in the Square Wisdom (CanCan, Kendoku) category.

This technique is somewhat similar to innies/outies which is used for solving Killer Sudoku puzzles, but there is no math involved and, again, it applies to Jigsaw Sudokus (and variations thereof).

First, I’ll explain some terminology that will be used:

LOL can be applied to any number of either rows or columns. How many rows you will use, it’s up to you (depending on the puzzle you are solving). The rows you are using must be adjacent to each other and they can start from the middle. In other words, they can, but don’t have to, be aligned to the edge of the puzzle. The adjacent rows (columns) that you choose to apply LOL, we will call AREA.

Some jigsaw nonets will be completely outside your chosen area, some will be completely inside it and some will have cells both inside and outside your area. Focus on nonets that have cells both inside and outside the area. We’ll call them BROKEN NONETS.

INNIES are cells that are within the area and belong to broken nonets which have fewer cells inside the area than outside of it.

OUTIES are cells that are outside the area and belong to broken nonets which have fewers cells outside the area than inside it.

Ok, this sounds quite confusing. So here is a sample puzzle to make it clear:

By using the common Sudoku solving techniques, you should reach this position:

Now what? Look at the first 3 columns. This will be your area. It is outlined in red in this picture:

If you are paying attention, you will already know why are some cells outlined in green. They are innies and outies. If you are not sure why, read the definition of innies and outies again!

IMPORTANT: The number of innies must always be equal to the number of outies. If your number of innies doesn’t match to your outies, you’ve done something wrong. Go back and look at it again!

Finally, it is time to state The Law of Leftovers:

The set of numbers in innies must be the same as the set of numbers in outies.

Why? Because in your area, which consists of “N” rows, there must be precisely “N” occurences of each digit (1-9 in normal Sudoku case). This is always the case in any Sudoku puzzle. Also, any “N” nonets contain precisely “N” times each digit (1-9). Now, jigsaw nonets that are broken by this area, “borrow” (or “lend”, depends how you look at it) some of their numbers from the rest of the puzzle. Those are innies and outies. In order to keep the puzzle consistent, those numbers that are borrowed must be the same as the numbers that are lent.

Anyway, lets apply LOL to our sample puzzle. Our 3 innies have these candidate numbers (some of them are naked singles, but it doesn’t matter): 7, 2 and 3|9. Our 3 outies can be: 2|8|9, 2|8 and 7. So, number 3 is a candidate in innies, but it’s not a candidate in outies. According to the Law of Leftovers, this cannot be! Therefore, we can eliminate 3 from the list of candidates in innies. Accordingly, we can eliminate 8 from outies (because it does not appear in the list of candidates for innies). There you go! We have solved two cells: R3C4=2 and R8C3=9.

From here, this puzzle can be solved by using the usual techniques. Here is the final solution.

Obviously, LOL can be applied to Jigsaw overlapping puzzles (Samurai, TwoDoku and any other) as well, using the same approach.

The only remaining solving method that I don’t consider trial and error is Swordfish. So, to complete the list of Sudoku solving tips, here is an example of a Sudoku puzzle that can be solved using Swordfish.

Here is the puzzle:

This is how far you can get without using Swordfish:

After you pencil-in all the numbers, you focus on number 4 and you look at columns 1, 5 and 7. In those 3 columns, number 4 appears altogether in precisely 3 rows: 1, 7 and 8. So, 3 columns with candidates in only 3 rows, it follows that (since each number can appear only once in each column and each row) number 4 can be eliminated from rows 1, 7 and 8 except from those cells that belong to columns 1, 5, 7. In other words, number 4 appears in precisely 3 cells from those 7 that I highlighted with a red rectangle. Of course, those 3 cells must be in different columns/rows, but the point is that in rows 1, 7 and 8 number 4 cannot appear in any other cell.

This indirectly solves R1C8 – because it can’t be 4 the only other option is 7.

I hope this makes sense.

Download the final solution.
Download the text file to import to Perfect Sudoku.

These techniques are used in ALL sudoku puzzles – Classic, Killer, Samurai and others. Make sure you fully understand them before you attempt to solve the more difficult puzzles.

I will be using one classic Sudoku puzzle in which I will demonstrate how each of the techniques contributes to the final solution.

Perhaps you’ve already seen this particular puzzle:

1. NAKED SINGLES (aka “elimination”)
I’m sure you know this method, but lets clearly explain it anyway.

Have a look at row 5, column 3 (from now on, this will be marked as simply R5C3).
Numbers 1, 2, 5 and 9 are in the same nonet so they can’t go into that cell.
Numbers 3, 4 and 6 are in the same row so they can’t go into that cell either.
Finally, number 8 is in the same column, so the only remaining number is 7. Therefore, R5C3=7 and this is a naked single. (it’s “naked” because it’s the only number that can go into one cell)

2. HIDDEN SINGLES (aka “singles”)
Having applied “naked singles” a few times, we come to this position:

Now, focus on number 2 in the top part of the puzzle. Red lines indicate where number 2 can’t go:
- R2C7 = 2, so number 2 can’t go anywhere in R2;
- R3C3 = 2, so number 2 can’t go anywhere in R3;
- R5C5 = 2, so number 2 can’t go anywhere in C5.

When you look at nonet 2, there is only one cell that is not covered with red lines – it’s R1C4, so this is the only place in nonet 2 where number 2 can go into. Therefore, R1C4 = 2 – it’s a “hidden single”, because potentially there are other candidates for this cell (4, 7 and 9) and number 2 is hidden amongst them.

If we keep applying “naked” and “hidden” singles, we come to this stage:

Now what?

Alright, now we must start using pencilmarks. For each unsolved cell, pencil in all the numbers that are still possible candidates for that cell (i.e. they are not in the same column, row nor nonet as that cell).

Your grid should look like this:

For each of the remaining 3 techniques, I will be coming back to this image.

3. BOX BREAKS (aka “row/column and nonet interactions”)
We again focus on nonet (“box”) 2 and row 2. Check the possible cells for number 8. In row 2, number 8 can appear only in R2C4 and R2C5. Both these cells are in nonet 2. Whichever one of them ends up being 8, number 8 will be eliminated from the rest of nonet 2. Therefore, we can do that straightaway:

In other words, you make an intersect of row 2 and nonet 2 on one particular value and remove that value from the remainder of that nonet.
It can also work the other way – just exchange words row and nonet and you’ll get it.

So, like all remaining techniques, this technique doesn’t actually solve a cell – it only removes some candidates which then helps you in the solving process.

4. NAKED PAIRS
Have a look at nonet 7 and row 9. In nonet 7, we know that numbers 7 and 8 must go into R9C1 and R9C2. These numbers are “naked” – there are no other candidates for those two cells except two of them. So, two numbers in two cells and no other candidates for those two cells – that’s a naked pair. When you find such two cells and they belong to the same row or column or nonet – you can eliminate those two numbers from other cells in that area.

In this case – there is nothing to remove from nonet 7 because all other cells are already solved.
But, since those two cells also both belong to row 9, there are candidates that can be removed from that row. Therefore, we remove numbers 7 and 8 from the candidates list in row 9.

4. NAKED SUBSETS (aka “subsets”)
This is essentially the same technique as the previous one, except that you take more than two cells at a time.
So, what you are looking for are “n” numbers that are candidates in precisely “n” cells that fall onto the same nonet or column or row.

We have this case in nonet 3/row 1.

NB: What’s important to notice is that each of the marked cells has only two pencilmark values, but there are altogether 3 different numbers there. It could happen that, for example, one cell has 3 values and two cells have 2 values, but it is important that there are N different numbers in N different cells altogether.

4. HIDDEN SUBSETS (aka “subsets”)
So far we were only looking at “naked” subsets – subsets with no other pencilmarks in those cells.

Look at the same row again. Numbers 1 and 8 are penciled-in in only two cells – R3C8 and R3C9. Two numbers, two cells – it’s a hidden pair! When you are sure of that, you convert that hidden pair into a naked pair.

So, first you do this:

Now, the pair 1-8 has become naked in row 3 and you can remove those two numbers from the remaining cells in row 3.

…

After you have applied all these techniques on this puzzle as many times as possible, you should reach this position:

Well, to proceed further, you must learn another technique – “X-Wing”. I’ve already explained that in detail before.

X-Wing will solve this puzzle. The final and most advanced technique is Swordfish which is similar to X-Wing except that it applies to more than two rows/columns.

I consider all other techniques (“xy-wing”, “colouring”, “nishio”, etc.) to be a limited form of trial and error and I don’t use them for my puzzles.

1. For Samurai puzzles, read this article
2. For other puzzles, make sure the puzzle is not an “X” (diagonal)
3. For puzzles from The Washington Post and The Express, read this post (and also read the article from point 1).

If you still think both (all) of your solutions are correct, make a post in the forum in the appropriate topic. The forum is located here.

It is yet to be shown that ANY puzzle I created had more than one solution. A few have tried – but all of them have failed.

Rules:

1. Each Samurai Sudoku puzzle consists of 5 overlapping “classic” 9×9 Sudoku sub-puzzles.
2. Each 9×9 sub-puzzle must be solved according to the rules of Sudoku: each row, column and 3×3 box must contain all digits from 1 to 9 – therefore, digits cannot be repeated.
3. Each Samurai Sudoku puzzle has one solution only.

Warnings:

1. DO NOT ATTEMPT to completely solve each sub-puzzle as an individual puzzle!
2. Each 9×9 sub-puzzle when solved individually could have more than one solution.
3. Never resort to guessing – our puzzles can be solved using deduction logic.

Solving suggestions:

1. Work on the puzzle as a whole.
2. Start with one sub-puzzle and solve as many cells as you can, until you can’t go further (do not guess!)
3. Move on to the next sub-puzzle. Use clues from the previous sub-puzzle.
4. Repeat step 2 for that sub-puzzle.
5. Keep repeating steps 2-3-4 until you solve the whole puzzle!
6. Do not start with the center sub-puzzle. They usually have fewer clues than other sub-puzzles.

Usual classic Sudoku techniques are required to solve these puzzles: naked and hidden singles, naked and hidden subsets (pairs, triplets etc) and “row/column and box interactions”. For the hardest of puzzles X-Wing and Swordfish techniques might be needed.