Outside Sudoku Puzzles With Answers? Best 51 Answer

Samurai Sudoku (also known as Gattai-5 or simply Samurai) is a Sudoku variant in which there are five overlapping grids to solve. These five grids overlap to form an X shape as shown below.

In most versions of Samurai Sudoku, each of the five regular Sudoku grids can be solved independently.

Although it may seem daunting because each of the four corner 3×3 regions must match two different Sudoku grids, the Samurai Sudoku puzzles are actually easier to solve than regular Sudoku. They’re just a lot more time consuming!

This is because you can use clues from two different Sudoku grids to solve which numbers fit into those four corner areas in the central grid.

Of course, the complexity of the puzzle depends on how many numbers are initially given and what Sudoku techniques are required to complete the solution.

Some more challenging versions will make it so that each 9×9 grid cannot be solved independently. This means that when solving the 3×3 corner regions, you must consider both grids.

Where can I find samurai sudoku puzzles?

There are many places online where you can play samurai sudokus, as well as many books to buy.

Some of the online places are:

Samurai Sudoku Books

If you prefer solving your puzzles with physical books that you can write on and take with you, then check out the following books:

Looking for more Sudoku variants?

If you’re looking for something more interesting, exciting, or novel than regular sudokus, then there are a few other variations you might want to try as well:

Sudoku 1

Here’s one way we could solve this puzzle.

Here are the 16 equations:

\(c+m+h=19\) \(f+e=10\) \(k+g+m+c=23\) \(g+p=11\) \(h+f=14\ ) \(g+m=13\) \(a+e+k+h=11\) \(k+c+f+a=22\) \(c+g+k=17\) \(f +g+a=19\) \(m+k+c=16\) \(p+e=5\) \(g+m+f=22\) \(h+m+a=14\) \(e+c+h+k=16\) \(f+h=14\)

We first consider the simplest equations with only two unknowns.

Equation 12 states that \(p+e=5\). This means that both \(p\) and \(e\) must be less than or equal to \(4\).

Equation 2 says that \(f+e=10\), and since we know that \(e\) is less than or equal to \(4\), this means that \(f\ge6\).

Equation 4 says that \(g+p=11\), so we must have \(g\ge7\).

Equation 5/16 tells us that \(h+f=14\), so \(h\le 8\).

Now consider equation 7, \(a+e+k+h=11\). The only four different positive integers that can add up to \(11\) are \(1+2+3+5\). So we must have that each of \(a\), \(e\), \(k\), \(h\) is one of \(1\), \(2\), \(3\ ) , \(5\). We know that \(e\le4\), but we now know that \(e\le3\), so \(2\le p\le4\). Also, since \(h\le 8\) and with a number less than or equal to \(9\) must be added to give \(14\) by Equation 5, we have that \[h=5.\] So by Equation 5, \[f=9.\] But now by Equation 2 \[e=1,\] and by Equation 12 \[p=4,\] which we can plug into Equation 4 to find that \[ g=7.\] We now use Equation 6 to derive that \[m=6.\]

It remains to derive the values ​​of the letters \(a\) and \(k\) (which we know each must take one of the values ​​\(2\) and \(3\)) and \(c\) . So the only number left for \(c\) is \(8\) – we can use Equation 1 to check \[c+m+h=c+6+5=19 \implies c=8 \; \check mark\]

We use Equation 14 to check what value \(a\) takes: \[h+m+a=5+6+a=14 \implies a=3.\]

So we must have \(k=2\), but we can check it again: by equation 11, \[m+k+c=6+k+8=16 \implies k=2 \checkmark\]

Letter Number \(a\) \(3\) \(c\) \(8\) \(e\) \(1\) \(f\) \(9\) \(g\) \(7 \) \(h\) \(5\) \(k\) \(2\) \(m\) \(6\) \(p\) \(4\)

This is what the Sudoku looks like now

The solution of the Sudoku is

Sudoku 2

For the second Sudoku we have the following 10 equations:

\(m+n=a\) \(g+n+f=g+c\implies n+f=c\) \(g+d=n\) \(g+m=m+f+d \ implies g=f+d\) \(g+c+e=a+e\implies g+c=a\) \(b+g+f=a+g\implies b+f=a\) \( n+g+c=b+c\implies n+g=b\) \(a+d+e=2a\implies d+e=a\) \(g+f+a=m+a\implies g +f=m\) \(e+n+m=a+b+d\)

Equations 3 and 4 share two of the same unknowns, so this seems like a good place to start. Eliminating \(g\) gives the equation \[f+d=n-d \implies n=2d+f.\]

From equation 2, \(n=c-f\), we further see that \[c-f=2d+f \implies c=2d+2f.\] From this equation we see that \(c\) is even and \ (c\ge 6\) (since the smallest values ​​that \(d\) and \(f\) can take are \(1\) and \(2\)), so we have either \(c= 6\ ) and \(d\) and \(f\) are equal to \(1\) and two or \(c=8\) and then \(d\) and \(f\) are equal to \( 1 \) and \(3\).

Now consider Equation 5, \(g+c=a\). The largest value that \(g\) can take is \(3\) when \(c=6\). The only other possibility is therefore \(g=2\), since we have already established that one of \(d\) and \(f\) must be \(1\). But since we must have this \(a\le9\), it means that we must have \[c=6,\] and therefore each of \(d\) and \(f\) must take one of the values ​​\ . (1 and 2\). This rules out the possibility of \(g\) taking the value \(2\), so we must have \[g=3,\] which tells us that \[a=9\] by Equation 5.

Equation 1 states that \(m+n=a=9\). Since the values ​​\(1\) to \(3\) are already taken, we can only make \(9\) from two numbers using \(4\) and \(5\). So \(m\) and \(n\) must each have one of the values ​​\(4\) and \(5\). The other letters and numbers we have are \(b\) and \(e\) and \(7\) and \(8\). We’ll look at Equation 10, \(e+n+m=a+b+d\), to determine what values ​​they should take. We know that \(n+m=a=9\), so we can simplify this equation to account for \(e=b+d\), which says \(b < e\), since we so both knew \(b\) and \(e\) were one of \(7\) and \(8\), we now know that \[b=7 \text{ and } e=8,\] , which further tells us that we must have \[d=1,\] and therefore also \[f=2.\] Equation 3 now says that \(3+1=n\), so we finally know that \[n= 4 \text{ and } m=5.\] Letter Number \(a\) \(9\) \(b\) \(7\) \(c\) \(6\) \(d\) \(1\) \(e\) \(8 \) \(f\) \(2\) \(g\) \(3\) \(m\) \(5\) \(n\) \(4\) How to solve the Sudoku The solution of the Sudoku is

According to scientists, brain training games do not make you smarter.

Practicing a game like Sudoku or using a brain-training app might improve you at it, but it won’t boost your IQ or overall brain power, a study claims.

And instead, researchers suggest that people get more exercise, socialize, and make sure they get enough sleep if they want their minds to be sharper.

Experts expected people to do better on tests if they practice a similar exercise for a long time, but they were wrong.

In fact, those who do brain training don’t do any better than people who don’t, suggesting that the numerous mobile apps aren’t working.

Games like Sudoku or online brain-training apps don’t improve people’s overall IQ or even their ability to do well on similar tests, Western University scientists say

Researchers from Western University in London, Ontario examined the effects of brain training games in an experiment involving 72 people.

People in a group played a brain-training game on a computer for 13 hours over a 20-day period aimed at improving the people’s working memory.

Working memory is the part of the brain responsible for learning, storing information, and preventing memory loss.

The group then performed a test similar to the one they had trained with, along with the second group of people who did not exercise at all.

High scores in the training game did not lead to higher scores in the final test, and people who “trained” their minds did not do better.

“Brain training does not lead to IQ improvements”

Study author Bobby Stojanoski said: “We [thought] that if you get really good at one test and you train really long, maybe you’ll get better at similar tests.

“Unfortunately, there is simply no evidence to support this claim.

“In fact, we show that this doesn’t even lead to improvements in things similar to the training test, let alone things in a more general sense like IQ.”

‘Sleep better. Do sports regularly. Eat better. Education is great. That’s what we should focus on.

“If you want to improve your brain, go running”

“If you want to improve your cognitive self, instead of playing a video game or taking a brain training test for an hour, go for a walk, run, meet up with a friend. These are much better things for you.”

The results of dr. Stojanoski complements previous research showing that brain training has no effects that translate to general brain performance tests.

He added: “As a scientist, I would never say that anything permanently closes the door on anything.

“To say that, one would have to test every single possible condition and every possible combination of conditions to be really sure.

“Don’t expect to get better at anything else because you’re good at one thing”

“But I think this study provides compelling evidence that brain training doesn’t lead to improvements beyond the task you trained for.

“If you want to get really, really good at something, just keep going and you’ll get better at it.

“But you shouldn’t expect to get better at another task just because you’ve gotten better at that one thing.”

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Most published Sudoku puzzles have only one solution. If there is more than one solution, it is probably an error. However, puzzles with incomplete clues can have multiple solutions. At the extreme, a puzzle with no clues has 6,670,903,752,021,072,936,960 solutions according to Wikipedia.

I don’t know if it’s possible to have exactly 3 solutions, but boards with 2 and 4 (and more) solutions are easy to find. In general, I think boards with an even number of solutions are easier to build.

Finding the number of solutions is a generalization of Sudoku solving algorithms, and there are Sudoku algorithms that perform significantly better than brute force. Once the board is filled as much as possible, it can be brutally pushed the rest of the way.

Sudoku rules are simple and straightforward. It is precisely their simplicity that makes finding the solution and solving these puzzles a real challenge.

In order to be able to play Sudoku, the player only needs to know the numbers from 1 to 9 and be able to think logically. The goal of this game is clear: to fill the grid with the numbers from 1 to 9 and complete it. The challenging part lies in the restrictions placed on the player to be able to fill the grid.

Rule 1 – Each row must contain the numbers from 1 to 9, with no repetitions

The player must concentrate on filling each row of the grid while making sure there are no duplicate numbers. The order of placement of the digits is irrelevant.

Each puzzle, regardless of difficulty, begins with assigned numbers on the grid. The player should use these numbers as clues to find out which digits are missing from each row.

Rule 2 – Each column must contain the numbers from 1 to 9, without repetitions

The Sudoku rules for the columns in the grid are exactly the same as for the rows. The player must also fill these with the numbers from 1 to 9, making sure that each digit only occurs once in each column.

The numbers given at the beginning of the puzzle serve as a guide to find out which digits are missing in each column and in which position.

Rule 3 – The digits may only occur once per block (Nonet)

A regular 9 x 9 grid is divided into 9 smaller blocks of 3 x 3, also known as nonets. The numbers 1 through 9 can occur only once per nonet.

In practice, this means that the process of filling the rows and columns without duplicate digits within each block finds another constraint on the positioning of the numbers.

Rule 4 – The sum of each individual row, column and nonet must equal 45

To find out which numbers are missing from each row, column or block or if there are any duplicates, the player can simply count or use their math skills and add the numbers. If the digits appear only once, the sum of each row, column, and group must be 45.

1+2+3+4+5+6+7+8+9= 45

Other details to consider

1. Each puzzle has a unique solution

Each Sudoku puzzle has only one possible solution, which can only be achieved by correctly following the Sudoku rules.

Multiple solutions only occur when the puzzle is poorly designed or, most commonly, when the player makes a mistake in the solution and a duplicate is hidden somewhere in the grid.

2. Guessing is not allowed

Trying to guess the solution to each cell is not allowed under Sudoku rules. These are logical number puzzles.

The numbers assigned at the start of the game are the only clues the player needs to solve the grid.

3. Notes and Techniques

Writing down the numbers that are candidates for each cell is allowed, and even encouraged, by Sudoku rules. These help the player to track their progress and keep their reasoning organized and clear.

As these puzzles increase in difficulty, these notes also become essential in order to be able to use the advanced solving techniques needed to complete the grid.

New research shows that solving puzzles can help you stay sharp. Sharing on Pinterest, people who were in the habit of doing a crossword or Sudoku seemed to have sharper perceptions. Getty Images A new study adds more evidence that puzzles can be effective for brain health. However, it is still unclear how they can help us in the long term or if they can help prevent cognitive decline. According to a study recently published in the International Journal of Geriatric Psychiatry, the more engaged in games like Sudoku and crossword puzzles, the better the brains of people over 50 work. The researchers looked at data from about 19,100 participants in the PROTECT study to determine how often they solved word and number puzzles. Then they used a series of tests to measure attention, memory and reasoning. In short, the more people engaged in puzzles, the better they did on tests. According to the study tests, people who do jigsaw puzzles have brain function 10 years younger than their age. On short-term memory tests, the puzzle participants had a brain function that was eight years younger. The cross-sectional data analysis evaluated tests on approximately 19,000 people. The data were self-reported and participants completed cognitive tests online. “The improvements are particularly evident in the speed and accuracy of their performance. In some areas the improvement has been quite dramatic,” said Dr. Anne Corbett, lead author and Lecturer in Dementia at the University of Exeter Medical School. “We can’t say that playing these puzzles will necessarily reduce the risk of dementia later in life,” Corbett said. “But this research supports previous findings suggesting that regular use of word and number puzzles helps our brains work better for longer.” The researchers plan to follow participants over time. They also want to assess the impact of puzzle intensity, taking into account how long people have been puzzled.

Real results? dr Jerri D. Edwards, a professor at the University of South Florida at Tampa who studies brain games and cognition, said that because the study is correlative — not randomized — it doesn’t mean that playing games leads to better cognition. “It’s likely that people with better cognition like these activities and tend to engage in them,” she told Healthline. “Even people without cognitive decline engage in these activities, but when they experience cognitive decline, they probably stop doing it because it becomes frustrating or challenging,” she said. She pointed to research that found cognitive engagement in old age can be a buffer against decline. She also cited research that found poorer cognitive function can lead to a reduction in lifestyle including social activities. According to a large randomized clinical trial, computer-assisted cognitive training that targeted processing speed was better at protecting against deterioration over time than crossword puzzles in older adults, Edwards found. “Given that verbal skills improve with age, in normal aging we tend to get better at word-based games,” Edwards said. “On the other hand, some cognitive abilities tend to decline with age, such as mental quickness, divided attention, ignoring distractions, and redirecting our attention. It’s important to challenge our brains with these types of tasks as we age.” She encourages cognitive stimulation, but said she’s not aware of any evidence from randomized controlled trials that confirms it improves cognitive performance or reduces the risk of cognitive impairment decline or dementia in the long term. dr Jessica Langbaum, an Arizona Alzheimer’s researcher and associate director of the Alzheimer’s Prevention Initiative, said there’s evidence that cognitively stimulating activities like jigsaw puzzles can improve our skills like thinking, paying attention and reasoning. “However, what we don’t know is whether this is a direct causal link,” Langbaum told Healthline. “Nor do we know whether participation in these activities delays or prevents the onset of cognitive impairment such as dementia or dementia due to Alzheimer’s disease.” She said the study results are interesting, but noted that the data were self-reported, which may not be entirely reliable.

Brain balance A key concept in both normal brain aging and dementia (including Alzheimer’s disease) is that our ability to function is a balance between brain pathology and the brain’s cognitive strength, explained Dr. Gayatri Devi, a memory neurologist at Lenox Hill Hospital in New York City. “When the pathology is overwhelming, which occurs in aggressive dementia, no amount of brain power can help slow the progression,” she said. “Fortunately, most types of dementia and Alzheimer’s progress slowly, and we can boost our brain’s power, or cognitive reserve, to either delay or prevent the onset of dementia altogether.” Using crossword puzzles and other mental exercises to strengthen our Brain networks is a way to strengthen the brain, as is physical exercise. “The trick is to challenge and engage the brain as we age,” Devi said. You don’t have to be a puzzle fanatic to boost your brain, but you can also learn a new language or pick up a new hobby. “Regardless of the task, if the problem is challenging enough, all areas of the brain are more or less involved in finding a solution — which is good for overall strengthening of brain networks and improving cognitive reserve,” she said.

The singles rule requires that if a candidate k is only possible in a single cell of a row, column, or block, then that cell must be k. This situation can occur for one of two reasons. A naked single arises when there is only one possible candidate for a cell; a hidden single arises when there is only one possible cell for a candidate. Despite the name, hidden singles are much easier to find than naked singles. You should always start a sudoku by finding all the hidden singles. No grades required!

The dots in the cell in row 3, column 5 indicate that the numbers 4, 5, 6, and 8 are all possible in that cell. But in that upper middle block, only one cell can contain a 5. This is a hidden single. This is the most basic technique. Because a number can appear only once in each column or row, and must appear exactly once in each 3×3 block, it’s easiest to first look for cells that must contain a value, since no other cell in a 3×3 block does can this number. In this case, for example, the number 5 is excluded from all but one cell in the top-middle 3×3 block. The 5 in this cell is called a “hidden single” because it can only be in this one spot and this fact is “hidden” by the presence of the other markings. This process, called cross-hatching, is repeated for each row and column. Crosshatch scanning is generally all that is necessary for “simple” puzzles. Most people do this step without actually making any marks.

First off, if the rules discussed below sound pretty much the same, it’s because they’re all just permutations of the same idea in different dimensions. Here, if we use “A” and “B” for a given number of rows, columns, cells, or blocks, then we have: If a candidate k is possible at the intersection of A and B but not elsewhere in A, then it is it is also not possible elsewhere in B.

This idea is discussed in more mathematical detail in The 12 Sudoku Rules.

The blocked candidate rule, form 1: If a candidate is possible in a given block and row/column and it is not possible anywhere else in the same row/column, then it is also not possible anywhere else in the same block. The locked candidate rule, form 2: If a candidate is possible in a given block and row/column and is not possible anywhere else in the same block, then it is also not possible anywhere else in the same row/column. When all the singles are found, I usually start tagging. Then the task is to eliminate markers until only one remains in a cell so that we know the value of that cell. In this next technique, we take advantage of the fact that once a number can be assigned to a specific row or column of a specific block (even if its exact location is still unknown), no other block can have the same number in the same row or column.

In the example shown on the right, the position of the number 1 in both the top row and the leftmost column is already known. Also, the second row of the top left 3×3 block is already filled. The only possible spot for a 1 in the top left 3×3 block is then the third row. This means, however, that in the third row one of the boxes circled in red must contain the number 1 – not only for this 3×3 block, but in general. We can optionally remove it from the other cells in this row (uncheck it). These values ​​then lock that row and require that the 1 in the top right block be in one of the two positions shown in blue.

The same idea eliminates the possibility of 9s in the top row of the center block because in this case the only possible place for a 9 in the left center block is the top row.

There are several other blocked candidates on this board. They include 1’s, 4’s and 7’s. can you find her

By the way, can you find the hidden single 5? It is here.

There are two subset rules for locked cells. The naked subset rule states that if n candidates in a given set of n cells are all possible in the same block, row, or column, and no other candidates are possible in those cells, then those n candidates are not possible elsewhere in the same block , row or column. The hidden subset rule states that if n candidates in a given set of n cells are all possible in the same block, row, or column, and those n candidates are not possible elsewhere in the same block, row, or column, then none other candidates are possible in these cells. Thus, if a “subset” of three cells can be identified for which the only possibilities are exactly three numbers, then while we don’t know which of those cells contains which number, we still know that there are no other cells in the row are , column or 3×3 block containing this “subset” of cells can have any of these numbers.

Since in this example there are already exactly two locations in the top row that must contain “3” and “7” together, the fourth cell from the left cannot be 7; It must be 6. Subset elimination, along with cross-hatch scanning and block area testing, can solve “moderately difficult” puzzles in general. (This technique is also known as searching for bare or hidden pairs, triples, and quads.) Examples of lattice analysis

Other advanced techniques are required to solve more difficult Sudoku puzzles like this one. These techniques are based on either of the following two rules: if a candidate is possible in a given set of cells that form the intersection of a set of n rows and n columns, but is not possible elsewhere in that same set of rows, then you are also not possible elsewhere in the same column group. and its downside: if a candidate is possible in a given set of cells forming the intersection of a set of n rows and n columns, but not possible elsewhere in the same set of columns, then they are also elsewhere in it not possible same series of lines. With the grid technique, a specific number is selected, e.g. 5, and searched for known feasible possibility patterns such as X-Wings and Swordfish.

An X-wing pattern. Within the two marked columns, there must be a 4 in the top and bottom rows. The possibility of 4 in all five circled cells can be eliminated.

Two intertwined “Swordfish” patterns. The possibility of 5 in the circled cell can be ruled out.

Note that grid analysis is essentially a form of two-dimensional subset elimination, now correlating the subsets themselves over multiple rows or columns. So, for example, if you look at this blue pattern, there is a set of three columns (the third, sixth, and seventh) that 5 can represent only in rows 2, 5, and 7. Three Columns. .three rows. Just like “three numbers…three cells”. In exact analogy to subset elimination, this means that for these three rows, the five must be in one of these three columns. It’s tough! In fact, however, X-Wings and Swordfish are simply two simple species of a much broader category of beasts.

A 5×5 circuit containing the candidate number “1” hiding in a simple Sudoku. There are five columns and five rows that make up the sentence. All possibilities for 1 in these five rows and columns that are not on the grid itself can be eliminated. Note that if we list the line numbers where candidate 1 appears, we get {379 469 14679 46 139 1(not shown)}. So in this case the two marks shown can be eliminated either due to the 5×5 “hidden quintuplet” of 34679 or the “naked singlet” of 1.

Demonstrating this equivalence of all such “circuits” is how the Sudoku wizard treats all circuit analysis without distinction: look for areas of the board where a given candidate’s possibilities lie on a grid of n columns by n rows . Outside of this set of grid lines, additional possibilities may exist in either rows or columns, but not both. Eliminate all possibilities of that number in any row or column defined by this grid, but not including the grid points themselves. If you look at the JavaScript code for the Sudoku Wizard, you’ll see that it uses the same function (analyzeX, only 16 lines) used to find all possible X-wings, Swordfish, hidden sets and “naked” sets. The fact is that all of this is the same “beast”, only seen from different perspectives. (More on that later!)

A 4×4 grid based on

the contestant number 4 The point is simply that finding x-wings and swordfish and the like is not a daunting task. One only has to recognize the n-by-n lattice containing the set of candidate squares. You don’t really have to discover the circuit. The principle that works here is very simple: once a specific row or column can be assigned a number (even if its exact location is still unknown), no other cells in the same rows and columns are allowed to contain that specific number.

Since an n by n set of cells defines exactly where n numbers can go, the principle is the same regardless of the number n. So all we do in this type of analysis is identify a set of n rows and n columns that contain the possibility of a particular candidate at their intersections. All other possibilities, all those that are not at these grid points, can safely be eliminated.

Consider a simple naked triple:

12 * 13 23

What prevents the cell with * from being a 2? Well if there was a 2 then we would eliminate both possibilities of 2 in the same row and specifically in our naked triple right? Again, these cells would have to be 1 and 3, and that would eliminate the only possibilities for the cell that only contains 13. This is how all naked sets work. You have N cells with N possible candidates underneath. If an option removes one of these possibilities from the set, there are not enough candidates to bypass it and that option can be eliminated.

Now imagine what is called an XY wing:

* 12 23 13

Notice that the three cells that make up the XY wing form a bare triple. It’s just that this triple is “bent” – the three cells aren’t in the same row, column, or block. What are the implications of this? First, unlike a standard naked triple, it is possible for both the 12 and 23 cells to be 2s. So this is no help. But here’s the interesting thing: It’s still not possible that NONE of them are 2. Because they still have exactly the same relationship to the 13 cell. If neither is a 2, then they are 1 and 3, and then the 13-cell is left without a possibility. The cell marked with * cannot be 2. It’s that simple. XY-Wings are just curved naked triples.

You know how easy it is to find naked triplets. Well, finding XY wings is that easy. Just look for curved nude threesomes. For example, consider the panel on the right. The 2 in row 8, column 1 can be excluded by the “bent triple” in rows 7 and 8.

Note that of 1, 2 and 3 the 2 can be excluded because (a) of 1, 2 and 3 only the 2 is common at both ends of the triple and (b) the cell we are deleting the 2 of ” sees” all cells of our curved naked triplet that contain 2.

But beware! If the middle cell was 123 instead of 13, this strategy wouldn’t work because the 2 we hit wouldn’t be able to see all the cells of the subset containing 2 and if one of the ends was 123 instead of just 12 or 13, then the strategy wouldn’t work because two values ​​would be common at the two ends, not just one. (And it wouldn’t be an XY wing.)

The idea of ​​a “bent” bare subset is that the subset includes two domains – a row and a column, a row and a block, a column and a block, or even two rows or two columns. The key are the cells that are NOT at the intersection of these two domains. If there is a possible value k in these two regions outside of this intersection, then we have the simple rule: if a curved naked subset contains one and only one candidate k present in both of its non-intersection subregions, k can be defined as a candidate in any cell that sees all the possibilities for k in the subset is eliminated. For example, this is a block and column containing 1, 4, and 5. Again, 4 can be excluded. It is a so-called XYZ wing. Of course, if we choose 4 in the {14} cell, the 4 in the {1245} cell in the 6th row of its column can be excluded. And if we choose “not 4” in that {14} cell, that is, if we choose 1, then that excludes the 1 in the {145} cell below, which becomes {45} {145} in that block to {45 } {45}, and the 4 in cell {1245} is still excluded. Regardless of what happens in the {14} cell, the 4 in {1245} can be ruled out. This happens with all bare subsets – bent or not.

Why exactly does this work? Let’s use our general overlap idea to prove that if there is only one common value in the non-intersecting region of the subset, then that candidate must not be eliminated. The proof goes something like this: Suppose you have a curved naked subset that contains exactly one common candidate k, as shown on the right. We don’t care what’s in the crossroads. (This is what makes bent naked subsets so practical and fundamentally different from almost locked sets.) What we do know is that there are N candidates for N cells. Now remove the common candidate k from all places in A and B. We now have N-1 candidates in N cells. That would be fine if we could duplicate any of the other values, but we can’t because a, b, c, … are all inside A so can’t be duplicated, and d, e, f, . .., are all in B, so they can’t be duplicated either. Therefore none of the remaining candidate numbers can be duplicated and it is not possible to eliminate the common candidate k. Any candidate k elsewhere that would do so can be eliminated.

A final example that illustrates another rule of curved naked subsets. Look at the configuration on the right. The five squares in green and blue form a curved bare set consisting of the five numbers {15789}. Interestingly, there is no common candidate outside the intersection – {589} for the block and {17} for the row. In this case, the displayed 9 can be eliminated since it sees all cells of the subset that contain 9. (See for yourself that this is true. If this cell is 9, then the top row of this block must be 7 and 1, and that rules out any value for cell {17}.)

This points to another rule for curved bare subsets: if there is no common value k in the two non-intersecting regions of a curved bare subset, the subset behaves like a standard bare subset. That is, candidate k can be eliminated from any cell that can “see” all cells of the subset containing k. Let’s use our general intersection again to prove this rule. Suppose you have a curved bare subset that contains no common candidates, as shown on the right. Again, we don’t care what’s in the intersection. What we do know is that there are N candidates for N cells. Remove one of the candidates – say “a”. We now have N-1 candidates in N cells. Again, that would be fine if we could duplicate any of the other values, but we can’t for exactly the same reason as the previous proof. The candidates b, c, …, are all in A, so they cannot be duplicated, and the candidates d, e, f, …, are all in B, so they cannot be duplicated. Therefore, none of the remaining candidates can be duplicated and there is no way to fill the N cells. We cannot eliminate any of the candidates in the curved naked subset. Any possibility that would do that can be ruled out.

I propose that curved bare subsets should be fairly easy to see in a puzzle. The Sudoku Wizard can find some, but not all, curved bare subsets.

Closely related to curved bare subsets is an idea that has been called almost locked sets. An “almost closed” set is a set consisting of n cells containing n+1 values. Removing any value from this set “locks” the set as a bare subset. It turns out that a certain combination of almost locked sets represents a kind of curved naked subsets, so there is some overlap in this analysis. Let’s look again at the last example of a curved bare subset. The cell outlined in green forms a single-cell, almost interlocked set “A” = {17}. Here we have a cell with two candidates. Similarly, the four cells highlighted in blue form another nearly closed set, “B” = {789 1789 59 58}, which includes four cells and five candidates: 1, 5, 7, 8, and 9. Together they form the curved nudity Subset {17 789 178 59 58}.

We could represent the situation as follows:

5 8 |/ A–7…7–B | |\ 1………1 9

The almost blocked set A contains two candidates {17}; nearly barred set B has five candidates {15789}. Candidates 1 and 7 are special because if one set has one of these, the other set doesn’t have the other and must have it. For example, if we remove the 7 from both of these sets, we run into a problem because then they both need the 1. But only one of them can have the 1 because all 1s are in row 4. The same is true for the 7. So we can eliminate any 7 or 1 that “sees” all the 7s and 1s in our pair of almost closed sets. But it’s better than that. We can also eliminate any 5’s, 8’s or 9’s that all see their kind in pairs. That’s because eliminating one of those candidates will force either set A or B to need both the 7 and the 1, leaving the other without enough candidates. We can eliminate all candidates marked in red. So we have a new rule: if two nearly locked sets are mutually doubly connected, any candidate k elsewhere that “sees” all candidates k in the two sets can be eliminated. Similarly, in this penultimate curved bare subset example, where we have the curved bare subset {14}{145}{45}, you can split the two nearly locked sets in two different ways. If we use A={14 145} and B={45} then we can write:

14 145 45 4…

Here we have 5…5 / \ B A–1 \ / 4 4

Note that these sets are not doubly linked. However, they have the common value 4. Removing 4 from these two sets would result in both requiring 5, but they cannot both require 5, so we can remove 4 from any cell that can see all three of those cells. The same goes for the 1. More precisely, in this case we have to eliminate 4 from the cell colored red. So we have a rule: if two nearly locked sets are linked by candidate i and have another candidate k in common, then any candidate k elsewhere that “sees” all candidates k in the two sets can be eliminated. This turns out to be a far more common situation than the doubly linked pairs deal described above.

There are many things that can be done with almost locked sets. Basically, they can latch directly into chain sets because they can transmit a weak link. Because if one of the candidates in the set is set FALSE, all other candidates are set TRUE (and the set is locked). This is the essence of a weak link. One problem, however, is that they can be plentiful. It’s not uncommon for a Sudoku board to have 100 or more nearly closed sentences. As such, finding one or the other that leads to an elimination can be difficult and time-consuming. Still, you might only find one.

A little-heralded property of almost locked sets is that they also hold for monovalued lattices. The key here is that a value can turn off two possibilities of an almost locked set. You can’t do this with rank-based fast locked sets – a 7 anywhere knocks out a 7, not a 7 and an 8 – but it can for a grid-based fast locked set. Let’s see how this works.

Look at the group of 8 on the right. What now? Remember that a 8-grid is actually the same as a subset. (An X-wing is really a naked pair; a swordfish is a naked triple, etc.) All we have to do is look at the columns as a set and see which rows the 8’s are in. In this case we have: { 146 126 8 237 16 1467 1235 25 9 }. Take a good look there. The {126 16} form an almost closed set. These 8s are circled in blue. The 8 in r1c1 is doubly linked to this set—it would remove both the 1 and the 2, reducing the set to {6 6} (“two 8s in row 6”), which is impossible. r1c1#8 can be removed. Notice how EASY it is to see the almost complete set – it’s just an X-Wing with an extra cell.

Here’s another one. In this case we have {3 5 79 12489 24789 278 12789 279}. The key is the nearly locked set {79 278 279}. The 8 in row 7, column 5 is doubly linked to this set. If you assign 8 there, all possibilities for 8 in rows 7 and 8 are removed. But we cannot reduce the number of possibilities by two in an almost closed sentence. So r7c5#8 can be eliminated. The same applies to r9c5#8 and r9c4#8. The Sudoku Wizard analyzes this board for near-closed sentences and classifies it as sashimi.

Many methods have been discovered that go by other special names, but really only look for this kind of grid-based almost locked sets. See for example A1s. For example, it turns out that the technique called sashimi can be easily described in terms of almost closed sets. In the example on the right, the cells highlighted in green are a column-based, almost locked, set for 6s, comprising the following three sets of four rows: {16 17 367}. The red 6 is a weak link to this almost locked crowd because it is in row 6, which would reduce – lock – this crowd as {1 17 37}. Normally this is not a problem. But in the case of a sashimi, this would also mean that column 7 would now only be {3}. But the lock {1 17 37} bans any other 6 in row 3, allowing 6 to be eliminated. Nothing more than that. Four other examples of sashimi analyzed in terms of near-closed sentences are shown below.

impossible520 number 273 step 10

r4c3#4 is eliminated by an almost locked set in 4s in columns 6 and 9 which, when reduced, causes all 4s in column 1 to be disallowed.

impossible520 puzzles 202 step 22

r5c5 cannot be 4 because of the almost blocked set of 4s in columns 7 and 9: {58 38}. If r5c5 was 4, then this set would be reduced to {8 38}, and all 4s in column 4 would be excluded.

impossible520 puzzles 121 step 9

r2c3 cannot be 4. If this were the case, then the almost locked set {25 59} of 4s in columns 4 and 7 would be reduced to {5 59} and all 4s in column 1 would be excluded.

top1465 Puzzle 89 Step 31

r1c1#6 is eliminated by a simple line-based almost locked set in 6s on line 9: {14}. If r1c1 was 6, then the set would be locked as {4}. But the 6’s in row 2 would also be reduced to rows {4}, forcing two 6’s in column 4.

What’s interesting is that the Sudoku wizard originally only found column-based sashimi. This wasn’t a mistake. This was because the Sudoku wizard checked column-based subsets first and did not take into account that large column-based sashimi can alternatively be represented as smaller row-based sashimi. That is, the same sort of row/column relationship with grids (that an n x n grid in rows is always also an m x m grid in columns) has an analogy with sashimi. (If you want to play with it, add the ALSLARGE option when listing the A1s examples and look specifically for row sashimi. Then the Sudoku Wizard will find both large and small sashimi.) So we have the following additional rule: all sashimi in rows (columns) have complementary sashimi in columns (rows). These alternate eliminations always involve the complementary set of columns and rows of the original sashimi and involve the same elimination. Here is an example:

top1465 Puzzle 89 step 31 (row based sashimi)

top1465 Puzzle 89 Step 31 (column based sashimi)

Subsets aren’t the only grouping that can be almost locked. For example, consider the panel on the right. Here we have a simple X-type strong chain with 6s. The 6 highlighted in red is a weak link to the yellow polarity of the chain. The lower right block is interesting here. Note that the 6’s are positioned at a row/column intersection. I will refer to this as an almost closed area. The idea is similar to an almost locked sentence (and the Sudoku wizard treats it as such, and it’s actually very similar to how I solve Sudoku puzzles by hand). The 6 highlighted in red locks the 6s in the lower right block with the first row, forcing the opposite polarity in the chain. And any time a candidate forces both polarities of a strong chain, we have a problem. r6c9 cannot be 6.

That interested me when I read about “Franken” Swordfish. I don’t claim to understand the analysis presented there with all the Xs, defining and secondary sets, constraints and the like. To me it’s a simple case of a conjugate pair working with two nearly locked regions. It really doesn’t matter what’s in the top right box. Starting with candidate options highlighted in red and blue, you can track eliminations in red and blue lines. Where the two overlap, you can eliminate similar candidates. It works due to the pairing of the two almost locked areas. It’s actually a very clever little idea, one of hundreds I’m sure that could almost involve off-limits areas. The decisive factor is not where the candidates are, but where they are not.

There are basically 15 varieties of almost closed assortments. They are shown on the right. There must be at least one candidate on each of the red boxes and at least one on each of the blue boxes. The presence of a candidate in the yellow squares is optional. (The Sudoku wizard ignores them unless the block contains at least three candidates; otherwise, if there are only two, they reduce to a simple strong “conjugate pair”.) Importantly, there is no candidate in any of the gray cells.

In fact, these almost blocked areas shift the direction of a weak link. The top nine types rotate the direction of a weak link 90 degrees and switch the link from a row-based link to a column-based link or vice versa; the bottom six amplify the signal along a parallel row or column. Coupled with a strong chain, nearly closed areas can have the interesting consequence noted by the discoverer of the Franconian fish.

I don’t know if naming combinations of almost closed domains and conjugate pairs and such makes much sense. These things are everywhere; When I solve Sudoku puzzles, I just follow them and see where they lead. In combination with strong chains, they can easily lead to elimination.

By the way, it’s this cell highlighted in yellow that makes fast-locked ranges a different kind of idea than fast-locked sets. Unlike fast-locked sets, here we have a cell where both options disappear when a candidate is present. This yellow cell is actually a weak link to both the red and blue subsets, producing a FALSE in all four directions. Das ist wirklich ganz anders als bei fast gesperrten Sätzen, bei denen N-1-Kandidaten letztendlich irgendwo innerhalb des Satzes vorhanden sein müssen.

Vier einfache Beispiele reichen aus, um nahezu geschlossene Bereiche zu veranschaulichen. Durchsuchen Sie die Beispiele nach über 100 weiteren.

top1465 #250

Hier ist eine einfache. Die 1 in r1c3 ist eine schwache Verbindung sowohl zum {15 15}-Paar als auch zum fast gesperrten Bereich in Block 2, wodurch die Verbindung zurück zur anderen Polarität des {15 15}-Paars umgeleitet wird. r1c3#1 kann eliminiert werden.

top95 #20

Auch hier bewirkt die Umleitung der schwachen Verbindung durch einen fast gesperrten Bereich, dass ein Kandidat beide Paritäten eines konjugierten Paares erzwingt. r3c3#5 kann eliminiert werden.

unmöglich520 #228

Ein fast gesperrter Bereich vom Spaltentyp dupliziert das schwache Glied von der 9 in r3c9 in Spalte 8. Aber das erzwingt dann beide Paritäten der starken XY-Kette mit 7 und 9 – eine Unmöglichkeit.

unmöglich520 #517

Hier ist eine großartige Eliminierung – volle 9 Kandidaten in dieser ziemlich komplexen starken Kette werden auf einmal eliminiert, weil die 8 in r8c2 „nicht 8“ in r6c7 erzwingt, aber diese beiden 8en haben die gleiche Polarität. Das ist nicht erlaubt.

The possibilities are endless. Man kann mit Sicherheit sagen, dass Fast-Locked-Sets überall zu finden sind, und sie können bei der Generierung von Eliminierungen mächtig sein. Die folgende Tabelle fasst das Lösen des unmöglichen520-Sets durch den Sudoku-Assistenten zusammen. “A” ist hier eine einfache, fast gesperrte Mengenanalyse; “a” ist das in Kombination mit schwachen Gliedern zu starken Ketten. “Ms” und “Mw” sind Medusa-Analysen starker bzw. schwacher Ketten, bei denen keine fast-locked-Sätze involviert waren. Wenn eine Fast-Locked-Analyse vorhanden ist, verwendet der Sudoku-Assistent diese eindeutig ausgiebig.

Rätsel #Hinweise #Leckerbissen Methoden #B #A #W #M #H #P #Schritte —————————— ————————————– 00001~ 17 411 A 3 18 00002~ 17 408 A 2 24 00003 ~ 17 405 AA 20 32 00004 ~ 17 410 A 4 23 00005 ~ 17 402 AA MS 18 1 39 00006 ~ 17 398 A 8 20 00007 ~ 17 401 A 16 32 00008 ~ 17 412 A 12 19 00009 ~ 17 403 Aa Ms 5 1 21 00010~ 17 407 A 5 25 00011~ 17 409 A 5 25 00012~ 17 404 Aa 6 27 00013~ 17 405 Aa 7 30 00014~ 17 412 A 8 20 00015~ 17 10 2 A 17 10 2 A 17 410, 411 A 10 21 00017~ 17 409 A W3 5 2 29 00018~ 17 412 Aa Ms 5 1 33 00019~ 17 410 A 2 22 00020~ 17 411 A 3 21 00021~ 17 410 A 2 26 000 2 73 000 2~ 00023 ~ 17 405 AA 12 26 00024 ~ 17 411 A 9 19 00025 ~ 17 411 A 9 19 00026 ~ 17 411 A 3 20 00027 ~ 17 411 A 3 20 00028 ~ 17 406 A 10 25 00029 ~ 17 411 a 4 19 0003030 ~ 17 400 Aa 16 40 00031~ 17 407 A 4 20 00032~ 17 408 A 5 21 00033~ 17 400 Aa 5 29 00034~ 17 405 Aa 5 31 00035~ 17 404 Aa 5 31 00035~ 17 404 Aa 5 31 00 7 40 00 7 4~ 17 408 A 5 31 00036~ 17 404 Aa 6 26 00038~ 17 403 Aa 5 24 00039~ 17 404 Aa Ms 6 1 23 00040~ 17 405 A 7 28 00041~ 17 402 A 8 23 00042~ 17 411 A 9 24 00043~ 17 409 A 13 21 00044~ 17 406 Aa Mw 9 9 41 00045~ 17 407 A 5 26 00045~ 2a 2 0046 A,6 A,6 45 00047 ~ 17 399 A 9 29 00048 ~ 17 402 AA MW 15 17 43 00049 ~ 17 404 A MW 10 3 36 00050 ~ 17 397 AA 11 20 00051 ~ 17 404 AA 4 27 00052 ~ 17 409 AA 16 34 00053 ~ 17 403 A 4 25 00054~ 17 404 A 2 23 00055~ 17 409 A 2 25 00056~ 17 399 A 2 22 00057~ 17 408 A 13 26 00058~ 17 399 A 11 26 00059~ 17 403 A 11 26 00059~ 17 403 A Aa 4 28 00061~ 17 404 A 2 23 00062~ 17 411 A 2 22 00063~ 17 412 Aa Ms 14 1 28 00064~ 17 400 Aa 8 27 00065~ 17 401 Aa 4 21 00065~ 17 401 Aa 4 21 00066~ 17 00069~ 9 17,3 407 Aa 9 34 00068~ 17 406 Aa 8 23 00069~ 17 411 Aa 10 24 00070~ 17 407 Aa 8 21 00071~ 17 404 A 13 25 00072~ 17 409 A 4 22 00073~ 07 3 40 A Aa 12 31 00075~ 17 409 Aa Ms 13 1 34 00076~ 17 401 Aa 21 35 00077~ 17 409 Aa 11 22 00078~ 17 404 Aa 6 28 00079~ 17 409 Aa 6 30 0079~ 17 409 Aa 6 30 0079~ 3 P0 1w 6+ 29 00081~ 17 406 A 3 16 00082~ 17 402 A 2 24 00083~ 17 399 A 2 19 000 84~ 17 403 A 2 17 00085~ 17 410 A 2 17 00086~ 17 403 Aa 15 28 00087~ 17 403 Aa Ms 15 1 29 00088~ 17 404 Aa Ms 3 1 25 00089~ 0 17 402 Aa Ms 3 02~ 0 17 402 Aa Ms 17 402 A 5 31 00091~ 17 406 A 1 23 00092~ 17 404 A 20 26 00093~ 17 400 A 2 23 00094~ 17 405 A 3 22 00095~ 17 405 A 10 27 00961~ 2 17 0961~ 2 17,0 399 Aa 6 20 00098~ 17 403 A 9 27 00099~ 17 405 A 3 23 00100~ 17 407 A 13 29 00101~ 17 404 a 5 31 00102~ 17 406 Aa 14 30 0028~ 07 406 Aa 14 30 0028~ 07 4 AA 11 25 00105 ~ 17 399 A 7 36 00106 ~ 17 413 A 6 32 00107 ~ 17 411 AAW3 14 1 32 00108 ~ 17 416 A 15 26 00109 ~ 17 414 A 9 29 00110 ~ 17 399 AA 7 25 00111 ~ 17 403 414 5 21 00112~ 17 401 9 24 00113~ 17 402 4 20 00114~ 17 402 4 20 00115~ 17 406 2 23 00116~ 17 403 12 30 00117~ 17 409 0 7 21 10 23 00119~ 17 410 A 8 21 00120~ 17 403 A 7 17 00121~ 17 402 A 9 19 00122~ 17 401 A 8 32 00123~ 17 402 A 2 21 00124~ 17 410 A 0 4 3~ 17 410 A 20 00126~ 17 408 A 2 18 00127~ 17 411 A 1 23 00128~ 17 409 A 1 25 00129~ 17 408 A 3 28 00130~ 17 408 Aa Ms 13 1 36 00131~ 17 406 A 2 27 00132~ 17 405 A 4 26 00133~ 17 398 Aa 5 20 00134~ 17 408 Aa 15 38 00135~ 17 408 Aa 15 38 00135~ 0 17 29 Aa 17 08 Aa 8 31 00137~ 17 400 A 5 24 00138~ 17 410 Aa 4 28 00139~ 17 406 A 1 24 00140~ 17 408 Aa 13 30 00141~ 17 398 Aa 10 31 00142~ 17 399 Aa 10 31 00143~ 17 397 Aa P+ 7 1 24 00144~ 17 404 A 4 27 00145~ 17 408 A 2 22 00146~ 17 410 A 2 20 00147~ 17 407 A 3 22 00148~ 17 398 A 1 18 00149~ 17 409 Aa 11 34 00150~ 17 406 Aa 8 34 00151~ 17 404 A 5 27 00152~ 17 400 A 3 18 00153~ 17 404 A 5 27 00154~ 17 402 A 3 18 00155~ 17 405 Aa P++ 6 1 31 00156~ 17 410 AaW2 14 1 31 00157~ 17 411 Aa 6 27 00158~ 17 401 A Ms 4 1 29 00159~ 17 402 Aa 5 25 00160~ 17 404 Aa 4 25 00161~ 17 409 A 3 21 00162~ 17 400 A 2 21 00163~ 17 400 A 3 18 00164~ 17 402 A 3 18 00165~ 17 412 Aa 7 32 00166~ 17 414 A 6 22 00167~ 17 410 A 3 20 00168~ 17 409 A 3 22 00169~ 17 403 A 2 21 00170~ 17 404 A 9 22 00171~ 17 405 A 6 29 00172~ 17 403 Aa 5 24 00173~ 17 404 A 4 25 00174~ 17 407 a 1 19 0 0175~ 17 405 A 5 28 00176~ 17 410 Aa 9 26 00177~ 17 406 A 2 22 00178~ 17 401 AaW2 10 1 35 00179~ 17 401 Aa Mw P+ 13 6 1 34 00180~ 17 405 A 3 22 00181~ 17 405 Aa 5 25 00182~ 17 400 A 3 27 00183~ 17 398 Aa P+ 10 1 31 00184~ 17 398 Aa Ms 5 1 30 00185~ 17 403 Aa Ms 9 1 32 00186~ 17 400 A 8 25 00187~ 17 413 A 2 22 00188~ 17 409 A 2 21 00189~ 17 408 A 2 21 00190~ 17 414 A 2 19 00191~ 17 413 A 2 21 00192~ 17 401 A W2 9 1 25 00193~ 17 400 Aa 9 27 00194~ 17 406 A 4 28 00195~ 17 398 A 3 24 00196~ 17 395 A 3 24 00197~ 17 397 A 3 26 00198~ 17 411 A 4 28 00199~ 17 406 Aa 4 31 00200~ 17 409 A 18 21 00201~ 17 408 A 3 18 00202~ 17 411 Aa P+ 14 1 34 00203~ 17 411 Aa 14 29 00204~ 17 411 Aa 10 26 00205~ 17 397 A 2 22 00206~ 17 406 Aa 11 29 00207~ 17 404 A 9 21 00208~ 17 409 A 10 20 00209~ 17 410 A 4 29 00210~ 17 410 A 9 24 00211~ 17 403 Aa 7 24 00212~ 17 403 Aa 7 30 00213~ 17 400 Aa 7 25 00214~ 17 400 Aa 7 26 00215~ 17 406 A 15 30 00216~ 17 401 A 6 24 00217~ 17 411 A 2 21 00218~ 17 402 A 7 29 00219~ 17 3 99 Aa Mw 6 10 26 00220~ 17 393 A 2 22 00221~ 17 403 A P 1 1 27 00222~ 17 414 A 5 26 00223~ 17 405 AaW3Ms 12 1 1 38 00224~ 17 405 Aa 15 26 00225~ 17 405 Aa 15 26 00226~ 17 409 A 4 28 00227~ 17 408 A 3 25 00228~ 17 404 A W2Ms 12 1 1 38 00229~ 17 400 Aa 10 24 00230~ 17 400 A 2 19 00231~ 17 410 A W2 11 1 27 00232~ 17 407 A 6 25 00233~ 17 402 A 5 24 00234~ 17 408 A 4 20 00235~ 17 408 A 4 20 00236~ 17 408 A 4 23 00237~ 17 398 Aa 11 24 00238~ 17 402 AaW2 4 1 23 00239~ 17 400 A 6 30 00240~ 17 406 Aa 8 30 00241~ 17 409 A 3 22 00242~ 17 411 A 3 25 00243~ 17 402 A W3 11 1 37 00244~ 17 412 A 12 29 00245~ 17 408 Aa 25 44 00246~ 17 404 AaW2 8 1 26 00247~ 17 405 AaW3 17 1 32 00248~ 17 407 A 7 20 00249~ 17 404 A 5 20 00250~ 17 410 Aa 5 23 00251~ 17 405 AaW2Mw 28 1 19 45 00252~ 17 403 A 13 18 00253~ 17 411 A 11 22 00254~ 17 406 A 7 26 00255~ 17 407 A 12 24 00256~ 17 407 Aa 13 36 00257~ 17 411 Aa Mw 15 1 37 00258~ 17 409 A 6 22 00259~ 17 407 A 1 20 00260~ 17 406 A 3 28 00261~ 17 404 AaW2 10 1 30 00262~ 17 404 AaW2 P+ 7 1 1 37 00263~ 17 398 A 1 21 00264~ 17 399 A 4 28 00265~ 17 410 Aa 10 24 00266~ 17 410 Aa 15 35 00267~ 17 399 A 14 32 00268~ 17 400 A 9 25 00269~ 17 406 Aa 11 33 00270~ 17 408 Aa 16 29 00271~ 17 403 A 11 26 00272~ 17 407 A 9 22 00273~ 17 409 A 10 23 00274~ 17 406 A 15 25 00275~ 17 406 A 14 24 00276~ 17 411 A 8 28 00277~ 17 403 A 3 18 00278~ 17 406 Aa 13 30 00279~ 17 397 Aa 7 22 00280~ 17 409 A 6 25 00281~ 17 403 Aa 12 25 00282~ 17 403 Aa 12 25 00283~ 17 403 A W3 5 2 27 00284~ 17 404 A W3 5 2 27 00285~ 17 410 A 10 27 00286~ 17 410 A 2 25 00287~ 17 408 A 4 22 00288~ 17 406 Aa 6 20 00289~ 17 403 A 11 24 00290~ 17 410 A 21 29 00291~ 17 399 A 5 27 00292~ 17 405 Aa 13 34 00293~ 17 405 Aa 14 37 00294~ 17 409 A 12 19 00295~ 17 411 A 5 20 00296~ 17 405 A 8 21 00297~ 17 407 A 8 20 00298~ 17 402 Aa Mw 14 2 38 00299~ 17 405 A 13 26 00300~ 17 407 A 2 19 00301~ 17 404 A 7 27 00302~ 17 404 A 7 19 00303~ 17 406 A 6 22 00304~ 17 409 A 2 27 00305~ 17 413 Aa Ms 13 1 35 00306~ 17 408 A 5 2 0 00307~ 17 408 A 5 20 00308~ 17 410 Aa 14 31 00309~ 17 400 A 9 30 00310~ 17 399 A 3 15 00311~ 17 409 A 2 23 00312~ 17 398 Aa 8 28 00313~ 17 401 A 2 23 00314~ 17 399 Aa 9 28 00315~ 17 402 A 7 24 00316~ 17 405 A 2 21 00317~ 17 401 A 2 21 00318~ 17 408 A 4 21 00319~ 17 409 A 5 22 00320~ 17 402 A 4 25 00321~ 17 409 A 4 20 00322~ 17 416 Aa 10 32 00323~ 17 406 A W2 6 1 27 00324~ 17 406 a 1 20 00325~ 17 410 A 11 28 00326~ 17 407 A 6 25 00327~ 17 407 A 3 20 00328~ 17 409 Aa 5 24 00329~ 17 397 Aa 4 31 00330~ 17 401 Aa 6 29 00331~ 17 407 A 6 22 00332~ 17 408 A W3 5 2 27 00333~ 17 407 A 8 30 00334~ 17 407 A 7 20 00335~ 17 409 A W3 5 2 27 00336~ 17 403 A 13 30 00337~ 17 410 Aa 14 36 00338~ 17 403 A 11 24 00339~ 17 410 A 10 35 00340~ 17 405 A 3 27 00341~ 17 408 Aa 19 34 00342~ 17 412 A 5 21 00343~ 17 402 A 4 20 00344~ 17 400 A 4 26 00345~ 17 408 A 4 19 00346~ 17 408 A 4 23 00347~ 17 408 Aa 5 24 00348~ 17 409 A 1 17 00349~ 17 407 A 3 24 00350~ 17 409 A 3 24 00351~ 17 403 A 4 23 00352~ 17 401 AaW4Ms 15 1 2 38 00353~ 17 406 A 3 29 00354~ 17 410 AaW4Ms 19 1 1 35 00355~ 17 410 A 5 22 00356~ 17 406 A 4 20 00357~ 17 406 A 6 26 00358~ 17 409 Aa P+ 7 1 30 00359~ 17 410 Aa P+ 5 1 32 00360~ 17 410 A 3 27 00361~ 17 406 A 3 25 00362~ 17 408 A 8 22 00363~ 17 408 Aa 8 29 00364~ 17 403 Aa 16 35 00365~ 17 410 A 24 22 00366~ 17 410 A 12 28 00367~ 17 412 Aa Ms 12 2 42 00368~ 17 398 A 6 27 00369~ 17 405 A 3 30 00370~ 17 405 A 5 24 00371~ 17 396 Aa 6 31 00372~ 17 408 A 15 18 00373~ 17 403 A 3 24 00374~ 17 404 A 9 26 00375~ 17 404 Aa 18 44 00376~ 17 404 Aa 10 26 00377~ 17 408 A 7 26 00378~ 17 407 AaW2Mw 31 1 42 46 00379~ 17 407 AaW3 17 1 34 00380~ 17 412 A 18 31 00381~ 17 402 Aa Ms 12 1 29 00382~ 17 413 A 12 26 00383~ 17 405 A 6 30 00384~ 17 413 A 4 31 00385~ 17 399 A 4 32 00386~ 17 402 A 4 26 00387~ 17 399 A 10 20 00388~ 17 399 A 11 23 00389~ 17 401 Aa 6 30 00390~ 17 411 A 3 19 00391~ 17 406 Aa 6 28 00392~ 17 403 A 5 25 00393~ 17 401 Aa 7 27 00394~ 17 406 Aa 10 33 00395~ 17 399 A 4 27 00396~ 17 409 AaW3 10 1 34 00397~ 17 403 A 5 26 00398~ 17 404 Aa 8 40 00399~ 17 402 A 6 28 00400~ 17 401 Aa 7 27 00401~ 17 396 Aa 9 27 00402~ 17 408 A 4 25 00403~ 17 410 A 4 27 00404~ 17 409 A 4 27 00405~ 17 406 Aa Mw 3 1 27 00406~ 17 408 A 4 19 00407~ 17 413 A 15 22 00408~ 17 403 A W3 5 1 26 00409~ 17 406 AaW2 11 1 36 00410~ 17 404 A W2 3 1 18 00411~ 17 403 A 4 21 00412~ 17 411 A 2 24 00413~ 17 402 Aa 9 35 00414~ 17 408 Aa 9 36 00415~ 17 407 A 9 22 00416~ 17 399 A 3 20 00417~ 17 408 A 4 21 00418~ 17 406 Aa 5 32 00419~ 17 408 A 8 20 00420~ 17 397 A 5 21 00421~ 17 410 A W2 5 1 27 00422~ 17 409 A 4 25 00423~ 17 411 A 4 27 00424~ 17 410 A 4 27 00425~ 17 405 A 7 18 00426~ 17 405 A 16 27 00427~ 17 410 Aa 16 22 00428~ 17 404 A 5 25 00429~ 17 404 A Ms 3 1 30 00430~ 17 403 A Ms 2 1 29 00431~ 17 399 A W2 11 1 26 00432~ 17 403 A 6 26 00433~ 17 411 Aa 7 33 00434~ 17 397 Aa 7 28 00435~ 17 393 Aa 8 29 00436~ 17 407 Aa 2 26 00437~ 17 413 Aa 8 27 00438~ 17 401 Aa 6 36 00439~ 17 405 A 9 24 00440~ 17 409 Aa Mw 14 2 8 39 00441~ 17 404 Aa Ms 19 1 37 00442~ 17 405 A 9 32 00443~ 17 402 Aa 11 30 00444~ 17 406 A 5 16 00445~ 17 401 A 7 26 00446~ 17 396 A 18 22 00447~ 17 404 A 9 25 00448~ 17 407 A 5 24 00449~ 17 409 A 7 24 00450~ 17 413 A 7 23 00451~ 17 403 A 9 27 00452~ 17 417 AaW2 12 1 26 00453~ 17 406 Aa Ms 27 1 47 00454~ 17 403 A 10 22 00455~ 17 409 A 10 19 00456~ 17 404 A 6 27 00457~ 17 403 Aa 13 31 00458~ 17 401 A 4 24 00459~ 17 399 A 6 23 00460~ 17 400 Aa P 6 1 26 00461~ 17 403 A 1 24 00462~ 17 402 Aa 6 24 00463~ 17 402 Aa 3 26 00464~ 17 402 A 3 25 00465~ 17 405 A 7 22 00466~ 17 400 A 7 23 00467~ 17 404 AaW2 4 1 22 00468~ 17 408 Aa 8 23 00469~ 17 408 Aa 8 23 00470~ 17 402 Aa 16 35 00471~ 17 404 a 8 35 00472~ 17 401 AaW3 20 1 33 00473~ 17 401 A 6 28 00474~ 17 403 Aa 23 36 00475~ 17 408 A 11 19 00476~ 17 403 A 9 30 00477~ 17 406 Aa P++ 16 1 40 00478~ 17 407 A P++ 2 1 28 00479~ 17 406 A P++ 3 1 26 00480~ 17 409 A P++ 3 1 29 00481~ 17 408 A 5 22 00482~ 17 408 Aa 12 37 00483~ 17 411 Aa Mw P++ 16 37 1 4 5 00484~ 17 411 Aa Mw P+ 9 2 1 40 00485~ 17 400 A 12 28 00486~ 17 409 A 18 24 00487~ 17 401 Aa 6 24 00488~ 17 405 Aa 9 31 00489~ 17 405 Aa 4 32 00490~ 17 392 Aa 3 23 00491~ 17 406 Aa 6 27 00492~ 17 401 Aa 6 23 00493~ 17 411 Aa 10 39 00494~ 17 407 A 12 32 00495~ 17 402 A 3 20 00496~ 17 403 Aa 4 26 00497~ 17 404 A 6 29 00498~ 17 405 A 3 25 00499~ 17 408 Aa Ms 8 2 29 00500~ 17 409 A 22 28 00501~ 17 404 Aa 3 21 00502~ 17 401 A 3 26 00503~ 17 409 A 7 28 00504~ 17 407 A 2 24 00505~ 17 410 A 9 20 00506~ 17 406 Aa 12 32 00507~ 17 407 Aa 21 43 00508~ 17 405 Aa 8 26 00509~ 17 400 A 3 20 00510~ 17 405 A 2 21 00511~ 17 410 A 4 23 00512~ 17 408 A 14 23 00513~ 17 409 Aa 2 25 00514~ 17 408 Aa 2 25 00515~ 17 407 A 13 26 00516~ 17 415 Aa 26 40 00517~ 17 410 Aa Mw P+ 9 4 1 35 00518~ 17 406 AaW2 10 1 36 00519~ 17 408 Aa 10 30 00520~ 17 406 Aa 6 28

It’s interesting to compare these results to when almost-locked set analysis is turned off (in the table below). The number of puzzles where Sudoku Assistant resorted to pure hypothesis goes up from 18 (about 4%) when using almost-locked set analysis to 95 (about 18%) when not using it.

puzzle #clues #tidbits methods #B #A #W #M #H #P #steps ——————————————————————– 00001~ 17 411 W3Mw 1 33 24 00002~ 17 408 Mw 1 26 00003~ 17 405 Mw P+ 8 1 34 00004~ 17 410 Mw 1 24 00005~ 17 402 Mw 40 32 00006~ 17 398 Mw 1 19 00007~ 17 401 Mw 2 25 00008~ 17 412 Mw 13 22 00009~ 17 403 Mw 2 22 00010~ 17 407 Mw 2 24 00011~ 17 409 Mw 2 24 00012~ 17 404 Mw 3 23 00013~ 17 405 Mw 3 24 00014~ 17 412 Mw 12 20 00015~ 17 410 Mw 1 21 00016~ 17 411 Mw 1 21 00017~ 17 409 W3Mw 2 4 28 00018~ 17 412 Mw P+ 1 1 30 00019~ 17 410 Mw 1 21 00020~ 17 411 Mw 1 20 00021~ 17 410 Mw 1 22 00022~ 17 406 Mw 5 32 00023~ 17 405 Mw 2 21 00024~ 17 411 Mw 1 19 00025~ 17 411 Mw 1 18 00026~ 17 411 Mw 1 19 00027~ 17 411 Mw 1 19 00028~ 17 406 Mw 1 22 00029~ 17 411 Mw 1 17 00030~ 17 400 W3Mw P+ 1 7 2 32 00031~ 17 407 Mw 1 19 00032~ 17 408 Mw 1 19 00033~ 17 400 W3Mw 1 6 30 00034~ 17 405 W3Mw 1 6 32 00035~ 17 404 W3Mw 1 6 32 00036~ 17 400 W3Mw 1 6 31 00037~ 17 404 Mw 9 23 00038~ 17 403 Mw 5 20 00039~ 17 404 Mw 14 22 00040~ 17 405 Mw 7 22 00041~ 17 402 Mw 13 31 00042~ 17 411 Mw 3 22 00043~ 17 409 Mw 17 28 00044~ 17 406 Mw P+ 11 2 32 00045~ 17 407 Mw 3 27 00046~ 17 406 Mw P+ 30 1 38 00047~ 17 399 Mw 8 39 00048~ 17 402 Mw P+ 5 1 30 00049~ 17 404 Mw P+ 10 3 36 00050~ 17 397 Mw 11 18 00051~ 17 404 Mw 3 23 00052~ 17 409 Ms 9 34 00053~ 17 403 Mw 1 24 00054~ 17 404 Mw 1 23 00055~ 17 409 Mw 1 25 00056~ 17 399 Mw 1 21 00057~ 17 408 Mw 5 25 00058~ 17 399 Mw 4 27 00059~ 17 403 W2Ms 1 3 21 00060~ 17 407 Mw 6 25 00061~ 17 404 Mw 1 22 00062~ 17 411 Ms 1 21 00063~ 17 412 Mw 3 20 00064~ 17 400 Ms 3 28 00065~ 17 401 Mw 1 19 00066~ 17 399 Mw P+ 7 1 19 00067~ 17 407 Mw 3 29 00068~ 17 406 Mw 1 20 00069~ 17 411 Mw 1 20 00070~ 17 407 Mw 1 19 00071~ 17 404 Mw 1 27 00072~ 17 409 Mw 1 20 00073~ 17 408 Mw 1 23 00074~ 17 410 Mw 12 27 00075~ 17 409 W3Mw 1 3 24 00076~ 17 401 W3Mw 1 22 23 00077~ 17 409 Mw 1 19 00078~ 17 404 Mw 3 23 00079~ 17 409 Mw 17 26 00080~ 17 406 Mw P+ 2 1 27 00081~ 17 406 Mw 1 19 00082~ 17 402 Mw P+ 10 1 25 00083~ 17 399 Mw 1 19 00084~ 17 403 Mw 1 17 00085~ 17 410 Mw 1 17 00086~ 17 403 W2Mw 1 8 18 00087~ 17 403 W2Mw 1 9 21 00088~ 17 404 Ms 3 24 00089~ 17 402 W3Mw 1 22 23 00090~ 17 402 Mw 15 27 00091~ 17 406 Ms 1 23 00092~ 17 404 Ms 2 19 00093~ 17 400 Mw 1 21 00094~ 17 405 Ms 1 22 00095~ 17 405 Mw 1 23 00096~ 17 410 Mw 1 21 00097~ 17 399 Mw 12 22 00098~ 17 403 Mw 3 21 00099~ 17 405 Ms 3 28 00100~ 17 407 Mw 1 23 00101~ 17 404 Mw 15 27 00102~ 17 406 Mw P++ 2 2 23 00103~ 17 403 Ms P+ 1 2 29 00104~ 17 407 Mw 10 30 00105~ 17 399 P+ 1 30 00106~ 17 413 Mw 22 24 00107~ 17 411 W3Mw P++ 1 19 3 36 00108~ 17 416 Mw 15 31 00109~ 17 414 Mw 6 34 00110~ 17 399 W2Mw 1 3 20 00111~ 17 403 Mw 12 17 00112~ 17 401 Mw 18 21 00113~ 17 402 Mw 17 22 00114~ 17 402 Mw 17 22 00115~ 17 406 Mw 3 25 00116~ 17 403 Mw 11 24 00117~ 17 409 Ms 3 24 00118~ 17 409 Mw 1 20 00119~ 17 410 Mw 1 20 00120~ 17 403 Mw 1 17 00121~ 17 402 Mw 1 17 00122~ 17 401 Mw 2 29 00123~ 17 402 Mw 1 20 00124~ 17 410 Mw 1 27 00125~ 17 409 Mw 1 20 00126~ 17 408 Mw 1 18 00127~ 17 411 Ms 2 22 00128~ 17 409 Ms 2 24 00129~ 17 408 Mw 9 24 00130~ 17 408 Mw 2 32 00131~ 17 406 Ms 2 26 00132~ 17 405 Mw 2 27 00133~ 17 398 Mw P+ 21 1 21 00134~ 17 408 Mw 8 35 00135~ 17 407 Mw P+ 13 1 32 00136~ 17 400 Mw 13 26 00137~ 17 400 Mw 1 23 00138~ 17 410 Mw 12 26 00139~ 17 406 Mw 1 23 00140~ 17 408 Mw P+ 2 1 26 00141~ 17 398 Mw P+ 9 1 28 00142~ 17 399 Mw P+ 9 1 28 00143~ 17 397 Mw P+ 14 1 20 00144~ 17 404 Ms 1 30 00145~ 17 408 Mw 1 22 00146~ 17 410 Mw 18 23 00147~ 17 407 Mw 1 21 00148~ 17 398 Mw 18 19 00149~ 17 409 W2Mw 1 17 34 00150~ 17 406 Mw 6 36 00151~ 17 404 P+ 1 21 00152~ 17 400 Ms 2 19 00153~ 17 404 P+ 1 21 00154~ 17 402 Ms 2 19 00155~ 17 405 Mw P++ 2 1 31 00156~ 17 410 Ms 6 26 00157~ 17 411 Mw 1 22 00158~ 17 401 W3Mw 2 3 24 00159~ 17 402 Mw 1 24 00160~ 17 404 Mw 9 21 00161~ 17 409 Mw 1 20 00162~ 17 400 Mw P+ 2 1 17 00163~ 17 400 Ms 2 19 00164~ 17 402 Ms 2 19 00165~ 17 412 Mw P+ 6 2 29 00166~ 17 414 Mw 1 21 00167~ 17 410 Mw 1 19 00168~ 17 409 Mw 1 21 00169~ 17 403 Mw 1 21 00170~ 17 404 Ms 2 21 00171~ 17 405 Mw 16 26 00172~ 17 403 Mw 1 23 00173~ 17 404 Mw 3 26 00174~ 17 407 Mw 1 19 00175~ 17 405 W4Mw 1 4 28 00176~ 17 410 Mw P+ 7 1 23 00177~ 17 406 Mw 4 20 00178~ 17 401 Mw P+ 7 1 23 00179~ 17 401 Mw P+ 10 1 31 00180~ 17 405 Mw 1 23 00181~ 17 405 Mw P+ 2 1 22 00182~ 17 400 Mw 1 25 00183~ 17 398 Mw P+ 1 3 24 00184~ 17 398 Mw 5 28 00185~ 17 403 Mw 15 29 00186~ 17 400 Mw 11 21 00187~ 17 413 Mw 1 23 00188~ 17 409 Mw 1 22 00189~ 17 408 Mw 1 21 00190~ 17 414 Mw 1 20 00191~ 17 413 Mw 1 22 00192~ 17 401 Mw 2 22 00193~ 17 400 Mw 1 22 00194~ 17 406 Mw 1 24 00195~ 17 398 Ms P+ 3 1 23 00196~ 17 395 Mw 1 22 00197~ 17 397 Mw 2 24 00198~ 17 411 Mw P++ 21 2 33 00199~ 17 406 Mw P+ 1 1 28 00200~ 17 409 Mw 2 20 00201~ 17 408 Mw 1 18 00202~ 17 411 W2Mw P+ 1 10 1 33 00203~ 17 411 Mw 6 29 00204~ 17 411 W4Mw 1 4 26 00205~ 17 397 Ms 2 26 00206~ 17 406 Mw 4 25 00207~ 17 404 Mw 1 20 00208~ 17 409 Mw 1 20 00209~ 17 410 Mw 1 33 00210~ 17 410 Mw 1 24 00211~ 17 403 W2Mw 1 4 23 00212~ 17 403 W2Mw 1 4 29 00213~ 17 400 Mw 32 24 00214~ 17 400 Mw 32 25 00215~ 17 406 Mw 2 24 00216~ 17 401 W2Mw 1 2 24 00217~ 17 411 Mw 1 21 00218~ 17 402 Mw 3 28 00219~ 17 399 Mw 17 20 00220~ 17 393 Ms 1 24 00221~ 17 403 Ms P+ 1 1 27 00222~ 17 414 Mw P+ 4 1 26 00223~ 17 405 W3Mw 1 13 32 00224~ 17 405 Mw P+ 19 2 33 00225~ 17 405 Mw P+ 16 2 35 00226~ 17 409 Mw 4 28 00227~ 17 408 Mw 2 25 00228~ 17 404 W3Ms 1 7 38 00229~ 17 400 Mw P+ 6 1 24 00230~ 17 400 Mw 1 19 00231~ 17 410 W2Mw 1 2 26 00232~ 17 407 Mw P+ 10 1 31 00233~ 17 402 Mw 10 23 00234~ 17 408 Mw 1 19 00235~ 17 408 Mw 1 19 00236~ 17 408 Mw 4 27 00237~ 17 398 Mw 48 18 00238~ 17 402 W2Mw 1 1 21 00239~ 17 400 W3Ms 1 2 30 00240~ 17 406 Mw 27 26 00241~ 17 409 Mw 13 23 00242~ 17 411 Mw 15 27 00243~ 17 402 W3Mw 1 17 37 00244~ 17 412 Mw 15 30 00245~ 17 408 Mw 14 33 00246~ 17 404 W2Mw 1 5 28 00247~ 17 405 W3Mw 1 15 25 00248~ 17 407 W2Ms 1 3 20 00249~ 17 404 Ms 2 21 00250~ 17 410 Mw 2 20 00251~ 17 405 Mw P+ 8 1 27 00252~ 17 403 W2Mw 1 18 21 00253~ 17 411 W2Mw 1 22 32 00254~ 17 406 P+ 1 23 00255~ 17 407 Mw 5 31 00256~ 17 407 Mw 11 26 00257~ 17 411 Mw 9 34 00258~ 17 409 Mw 17 24 00259~ 17 407 P+ 1 19 00260~ 17 406 Mw 8 25 00261~ 17 404 W2Mw 1 2 27 00262~ 17 404 W2Mw P+ 1 8 1 28 00263~ 17 398 Ms 1 21 00264~ 17 399 W2Ms 1 5 29 00265~ 17 410 Mw 13 24 00266~ 17 410 Mw 4 29 00267~ 17 399 Mw P+ 4 1 33 00268~ 17 400 Mw P+ 3 1 33 00269~ 17 406 W2Mw P+ 1 15 1 27 00270~ 17 408 Mw 7 34 00271~ 17 403 Mw 1 21 00272~ 17 407 Mw 1 19 00273~ 17 409 Mw 1 19 00274~ 17 406 Mw 1 20 00275~ 17 406 Mw 1 23 00276~ 17 411 Ms 3 24 00277~ 17 403 Mw 1 17 00278~ 17 406 Mw 5 29 00279~ 17 397 W2Mw 1 4 21 00280~ 17 409 W2Mw 1 11 24 00281~ 17 403 Mw 9 23 00282~ 17 403 Mw 9 23 00283~ 17 403 W3Mw 2 4 28 00284~ 17 404 W3Mw 2 4 29 00285~ 17 410 Mw 15 28 00286~ 17 410 Mw 11 26 00287~ 17 408 Mw 1 21 00288~ 17 406 Mw 5 21 00289~ 17 403 Mw 14 23 00290~ 17 410 Mw 5 35 00291~ 17 399 Mw 5 28 00292~ 17 405 Mw 15 35 00293~ 17 405 Mw 15 35 00294~ 17 409 Mw P+ 16 1 27 00295~ 17 411 Mw 9 20 00296~ 17 405 Mw 1 19 00297~ 17 407 Mw 1 18 00298~ 17 402 Mw 7 32 00299~ 17 405 Mw 2 23 00300~ 17 407 Mw 13 21 00301~ 17 404 Mw 2 27 00302~ 17 404 Mw 1 20 00303~ 17 406 W2Mw 1 15 26 00304~ 17 409 W3Mw 1 4 28 00305~ 17 413 Mw 2 26 00306~ 17 408 Mw 1 18 00307~ 17 408 Mw 1 18 00308~ 17 410 W2Mw 1 12 26 00309~ 17 400 Mw 7 41 00310~ 17 399 Ms 1 16 00311~ 17 409 Mw 1 24 00312~ 17 398 W2Mw P+ 1 4 1 27 00313~ 17 401 W2Mw 1 4 22 00314~ 17 399 W2Mw P+ 1 33 1 24 00315~ 17 402 W2Mw 2 16 22 00316~ 17 405 Mw 1 21 00317~ 17 401 Mw 1 21 00318~ 17 408 W2Mw 1 2 23 00319~ 17 409 W2Mw 1 1 24 00320~ 17 402 P++ 1 26 00321~ 17 409 Mw 3 21 00322~ 17 416 Mw 12 25 00323~ 17 406 W3Mw 2 5 30 00324~ 17 406 Mw 2 21 00325~ 17 410 Mw 3 24 00326~ 17 407 Mw 23 19 00327~ 17 407 Mw 1 19 00328~ 17 409 Mw 15 18 00329~ 17 397 Mw 5 30 00330~ 17 401 Mw 4 26 00331~ 17 407 Mw 2 24 00332~ 17 408 W3Mw 2 4 28 00333~ 17 407 Mw P+ 1 1 25 00334~ 17 407 Mw 4 23 00335~ 17 409 W3Mw 2 4 29 00336~ 17 403 Mw 4 28 00337~ 17 410 Mw 7 37 00338~ 17 403 Mw 1 25 00339~ 17 410 Mw 6 34 00340~ 17 405 Mw 1 26 00341~ 17 408 Mw 4 27 00342~ 17 412 Mw 2 26 00343~ 17 402 W2Mw 1 1 22 00344~ 17 400 W2Mw 1 18 32 00345~ 17 408 Mw 1 18 00346~ 17 408 Mw 12 20 00347~ 17 408 Mw 15 18 00348~ 17 409 Ms 1 17 00349~ 17 407 Mw 3 26 00350~ 17 409 Mw 2 25 00351~ 17 403 W3Mw 1 1 22 00352~ 17 401 Mw P+ 13 2 29 00353~ 17 406 Mw 1 27 00354~ 17 410 Mw 12 25 00355~ 17 410 W2Mw 1 2 25 00356~ 17 406 W2Mw 1 1 22 00357~ 17 406 Ms 4 26 00358~ 17 409 Mw P++ 8 2 26 00359~ 17 410 Mw P++ 7 2 30 00360~ 17 410 Mw 3 29 00361~ 17 406 Mw 3 27 00362~ 17 408 Mw 1 23 00363~ 17 408 Mw 24 33 00364~ 17 403 W2Mw 1 15 32 00365~ 17 410 Mw P++ 5 1 27 00366~ 17 410 Mw 2 27 00367~ 17 412 Mw 7 36 00368~ 17 398 Mw 13 26 00369~ 17 405 Mw 1 31 00370~ 17 405 Mw 3 24 00371~ 17 396 Mw 15 25 00372~ 17 408 Ms 3 20 00373~ 17 403 W3Mw 1 1 22 00374~ 17 404 Mw 1 24 00375~ 17 404 P+ 1 26 00376~ 17 404 Mw P+ 3 1 26 00377~ 17 408 Mw P+ 1 1 25 00378~ 17 407 Mw P+ 26 2 34 00379~ 17 407 Mw 16 27 00380~ 17 412 Mw 13 31 00381~ 17 402 W2Mw P+ 1 11 1 24 00382~ 17 413 Mw 17 25 00383~ 17 405 Mw 5 31 00384~ 17 413 Ms 2 32 00385~ 17 399 W3Mw 1 6 28 00386~ 17 402 Mw 1 25 00387~ 17 399 Mw P++ 48 1 19 00388~ 17 399 W2Mw P++ 1 10 2 26 00389~ 17 401 Mw 16 30 00390~ 17 411 Mw 3 22 00391~ 17 406 Ms 5 34 00392~ 17 403 Mw 1 24 00393~ 17 401 Mw 20 27 00394~ 17 406 Mw 1 29 00395~ 17 399 Ms 2 24 00396~ 17 409 W3Mw 2 3 27 00397~ 17 403 W2Mw 1 25 26 00398~ 17 404 Mw 11 33 00399~ 17 402 Mw 2 25 00400~ 17 401 Mw P+ 4 1 25 00401~ 17 396 Mw 3 22 00402~ 17 408 Mw 23 27 00403~ 17 410 Mw 23 29 00404~ 17 409 Mw 23 29 00405~ 17 406 Mw 1 26 00406~ 17 408 Mw 1 18 00407~ 17 413 Mw P+ 10 1 18 00408~ 17 403 W3Mw P+ 2 10 2 32 00409~ 17 406 W2Mw 1 3 30 00410~ 17 404 Mw 9 17 00411~ 17 403 Ms 3 20 00412~ 17 411 Mw 1 24 00413~ 17 402 Mw P+ 11 1 27 00414~ 17 408 Mw 1 33 00415~ 17 407 Mw 2 21 00416~ 17 399 Ms 2 19 00417~ 17 408 Mw 14 19 00418~ 17 406 Mw 7 29 00419~ 17 408 Mw 1 19 00420~ 17 397 Mw 16 18 00421~ 17 410 W2Ms 1 2 26 00422~ 17 409 Mw 15 27 00423~ 17 411 Mw 15 29 00424~ 17 410 Mw 15 29 00425~ 17 405 Mw 15 17 00426~ 17 405 Mw P+ 9 1 34 00427~ 17 410 Mw 4 21 00428~ 17 404 Ms 3 29 00429~ 17 404 Mw 3 29 00430~ 17 403 Mw 3 29 00431~ 17 399 W2Mw 1 19 27 00432~ 17 403 Mw 1 25 00433~ 17 411 Mw P+ 1 1 30 00434~ 17 397 Mw 7 30 00435~ 17 393 W3Mw 1 5 27 00436~ 17 407 Mw 6 24 00437~ 17 413 Ms 3 30 00438~ 17 401 Mw 17 33 00439~ 17 405 Mw 2 21 00440~ 17 409 Mw P+ 3 2 32 00441~ 17 404 Mw 65 32 00442~ 17 405 Mw 5 26 00443~ 17 402 Mw 3 24 00444~ 17 406 Mw 5 26 00445~ 17 401 Mw 6 23 00446~ 17 396 Mw 1 23 00447~ 17 404 Mw 6 26 00448~ 17 407 Ms 1 23 00449~ 17 409 Ms 1 23 00450~ 17 413 Ms 1 23 00451~ 17 403 Mw P+ 2 1 23 00452~ 17 417 W2Mw 1 20 26 00453~ 17 406 Ms P+ 6 2 31 00454~ 17 403 Mw P++ 41 1 21 00455~ 17 409 Mw P++ 44 1 19 00456~ 17 404 Mw 3 26 00457~ 17 403 Mw P+ 9 1 27 00458~ 17 401 P+ 1 23 00459~ 17 399 Mw 1 24 00460~ 17 400 Mw P+ 6 1 28 00461~ 17 403 Ms 1 25 00462~ 17 402 Mw 1 24 00463~ 17 402 Mw 2 22 00464~ 17 402 Mw 5 25 00465~ 17 405 Mw 16 22 00466~ 17 400 Mw 2 20 00467~ 17 404 W2Mw 1 4 21 00468~ 17 408 W2Mw 1 10 21 00469~ 17 408 W2Mw 1 10 18 00470~ 17 402 Mw 1 25 00471~ 17 404 Mw P+ 15 1 33 00472~ 17 401 W3Mw 1 9 29 00473~ 17 401 Mw 3 28 00474~ 17 403 Mw P+ 2 2 31 00475~ 17 408 Mw P++ 14 1 26 00476~ 17 403 Mw 5 29 00477~ 17 406 Mw P++ 21 2 29 00478~ 17 407 Mw P++ 2 1 26 00479~ 17 406 Mw P++ 4 1 24 00480~ 17 409 Mw P++ 2 1 27 00481~ 17 408 Ms 3 21 00482~ 17 408 Mw 14 28 00483~ 17 411 Mw P++ 6 2 28 00484~ 17 411 Mw P++ 8 3 32 00485~ 17 400 W2Mw P++ 1 5 3 35 00486~ 17 409 Mw P+ 15 2 32 00487~ 17 401 Mw 21 17 00488~ 17 405 W2Mw 1 20 30 00489~ 17 405 Ms 2 33 00490~ 17 392 Mw 2 19 00491~ 17 406 Mw 3 25 00492~ 17 401 W2Mw 1 4 28 00493~ 17 411 Mw P+ 7 2 31 00494~ 17 407 Mw 5 31 00495~ 17 402 Mw 1 19 00496~ 17 403 Mw 1 25 00497~ 17 404 Mw 16 27 00498~ 17 405 Mw 2 25 00499~ 17 408 W2Mw 1 23 31 00500~ 17 409 Mw 3 30 00501~ 17 404 Mw 3 18 00502~ 17 401 Mw P 4 1 27 00503~ 17 409 Mw P+ 8 1 34 00504~ 17 407 Mw 1 23 00505~ 17 410 Mw 5 18 00506~ 17 406 Mw 20 29 00507~ 17 407 Mw 1 26 00508~ 17 405 Ms 3 22 00509~ 17 400 Ms 2 19 00510~ 17 405 Mw P+ 20 2 25 00511~ 17 410 Mw 7 30 00512~ 17 408 Mw P+ 13 1 25 00513~ 17 409 Mw 1 24 00514~ 17 408 Mw 1 24 00515~ 17 407 Mw 23 27 00516~ 17 415 Mw 3 30 00517~ 17 410 Ms P+ 1 3 28 00518~ 17 406 W2Mw 1 3 31 00519~ 17 408 Mw 1 22 00520~ 17 406 Mw 2 22

The Sudoku Assistant by default does not look for almost-locked sets because it slows down the processing to do that, and they have not proven to be generally decisive in solving puzzles. But when those checkboxes are checked, it currently does the following: Finds all possible almost-locked sets. Checks to see if there are any mutually-linked pairs of almost-locked sets (bent naked subsets). If so, it checks for weak links and eliminates them. 8 1 2…2* \| / A–3…3–B /| | 9 7………7…7* Here almost-locked set A has five possibilities in four cells, and almost-locked set B has three possibilities in two cells. Together there are six possibilities in six cells — a bent naked subset. Looks for nodes that are weak links to two already-linked almost-locked sets and eliminates them. (7* in this case). 2…2—B / | 3–A | \ | 7..7*…7 Looks for strong chain nodes that are themselves weakly associated with an almost-locked chain. If a chain or set of chains is connected in an impossible way, such that it would force two of the N candidates of the almost-locked set to be FALSE, that chain can be eliminated. 2…2*–2 / 7 3*…3–A . \ . 7…7*—7- Due to the chain, the 2* and 7* have to be such that either both are TRUE or both are FALSE. But they can’t both be TRUE, because that would not leave enough candidates for almost-locked set A. They can be eliminated.

If a grid of cells can be made for which the value in one cell directly forces the value in another cell, we can sometimes follow that grid to see if it leads to a logical inconsistency. This can be done even though the exact contents of a cell are unknown. Some algorithms look for “strong cycles” with “weak edges,” but there is another way to think about this. What is significant is that in a “binary grid” every other cell along the grid either has one value or another — only two possible values. (The grid is made by specifically selecting only cells that have exactly two possibilities.) If any two such cells are in a single row, column, or 3×3 block, it is as if they were the ONLY possibilities for their cells.

a complicated X cycle. Only the values of 5 are shown. Lines connect mutually exclusive possibilities. Alternating nodes on the chain are sprayed red. The circled value is excluded by both of the chain options, so it can be eliminated. Thus, if we can say, because of a connecting chain of grid points, that two cells “either must both by 5 or neither is 5” then if they are in the same block, row, or column, neither can be 5. In the case on the left, we have a grid with two endpoints in the same 3×3 block. They are opposite “parity” — we don’t know which value is where, but regardless of that, whenever one is not 5, the other is 5. We can eliminate the circled possibility. That particular cell cannot be a 5, because it is in a block with two other cells, one of which has to be a 5. Since cells with only two possible solutions are relatively easy to spot, this technique lends itself reasonably well to human solving of Sudoku puzzles.

In fact, we can generalize the idea of strong binary grids to three dimensions. If we allow the grid to not just involve a specific candidate number, a 3D grid is generated. The Sudoku Assistant uses the Jmol applet to help visualize this. (The different colors here represent different candidate numbers, with red, at the top being 1 and white, at the bottom, being 9.)

A full 3D Medusa analysis involves extending these idea to weakly linked chains (encompassing X cycle Type 2 as well as Y cycles). The idea here is that one may have a sequence of three “nodes” (row, column, and candidate value, in 3D) A — b — C, for which A and C are members of strong chains (their cells have only two possible values). We call this a “weak link” or “weak edge” if A, b, and C all are in the same row, column, or block — or, in a 3D sense, the same cell. In that case, the “strongness” of one chain can be transferred to the other chain, provided the right condition is applied. (In other words, we can assert “if A is 5…then C is NOT 5”, but we cannot assert that “if A is NOT 5, then C is 5.”) The Sudoku Assistant can follow these weak links to any number of associated chains in order to eliminate possibilities.

The classic example of weakly linked chains in the form of a Y-cycle is an XY-wing, shown below.

* XZ X… YZ Y… XY

X… and Y… here are weak links. The first links the strong chains XZ and XY; the second links the strong chains XY and YZ. It doesn’t really matter if those links are there or not. (If they are not there, then we just have one strong chain.) The effect is the same: “NOT Z” in the lower left cell gets transmitted to the top right cell through the chain:

Z–X–(weak X)–X–Y–(weak Y)–Y–Z – + – + – + ——direction of transmission—->

as “Z” at the other end. Or, if the other Z is false, then we have:

Z–X–(weak X)–X–Y–(weak Y)–Y–Z + – + – + –

So one or the other of the Zs on the ends must be TRUE, and the * cannot be Z.

The term I use for * in the above situation is weak corner. Here we have again A — b — C, for which A and C are members of strong chains. To be a weak node, A and b must be in the same row, column, block, or cell, and the same is true for for b and C. But if these aren’t the same row, column, block, or cell, then we don’t have a weak link. Instead, we have a weak corner. The “strongness” of one chain cannot be transferred to the other chain via a weak corner, With the proper connecting of strong chains by weak links, one can infer valuable information about such a node. In particular, the elimination of the corner node, b, can be forced by the right sort of chain connection.

If we know that two cells of a strong chain are positioned such that both must be X or both must not be X, and together they exclude X from a block, then neither can be X. On the right, for example, the two red Xs are linked because they are part of the same strong chain. But if they are both TRUE, then all the Xs in the top right block are disallowed. There must be some X in that box, so their being TRUE leads to a contradiction. The XA cell must be A, the XC cell must be C, and the XB cell must be X. This technique has been referred to as a strong block hinge. Medusa variants also target cells in the same row or column, or target values in the same cell. X A X… X… X… X… X B X… X C X… A strong block hinge.

Cells XA and XC cannot be X,

and the cell labeled “XB” must be X.

The chain connecting XA and XC need not be strong. If one of the ends is a weak node, then the result is that we have a weak hinge, and we can eliminate that weak node. In the example on the right, if XA is X, we cannot say anything about XC because the X in XC is only weakly linked to the X of XB. (Only when XB is X can we say anything about XC; when XB is not X, we can’t conclude anything about XC.) But the weak link does work the other way. So if XC were X, then XB would have to be “not X” (B), and XA would have to be X. But then the two ends of the hinge would be X, and that would remove all possibilities for X in Row 2. So XC must be C, not X.

The Sudoku Assistant will find these sorts of hinges if the “hinge” box is checked. XA X… X… X… X… XB X… X… X C X… A weak hinge targeting Row 2.

The cell labeled “XC” must NOT be X.

The addition of weakly associated nodes to the mix of strong 3D chains allows for a method of solving Sudoku puzzles that transcends the idea of “forcing chains” or “trial and error” to allow for the formation of hypotheses regarding possible candidate elimination and their direct proof. Using 3D Medusa chains, we can follow the implications of eliminating or not eliminating a mark and see what comes of it. There are two strategies for this analysis, which I am calling proof or disproof.

The disproof strategy goes like this:

Consider a certain chain, 3, that has two possible states, 3C and 3c. Hypothesize that a weakly associated node, say one of the 3c-associated nodes, is TRUE — that it cannot be eliminated.

Mark all Chain 3c nodes with the lowercase letter “a” meaning FALSE — “If the hypothesis is true, then this node must be FALSE.

Mark all Chain 3C nodes with an uppercase “A” meaning “if the hypothesis is TRUE, then these nodes must also be TRUE (not eliminated).” This works because Chain 3 is a strong chain.

Mark all associated weak nodes of 3C with a capital letter “B” meaning “if the hypothesis is TRUE, then these nodes will be false (that is, they can be eliminated).”

Recursively add any chains directly involving any of these associated weak nodes with alternating capital letters/lowercase letters.

Check to see if we have either of the following two forcing conditions within any subset (all of a specific candidate in a row, column, or block, or all the candidates of a given cell):

All members are FALSE. Then, since one HAS to be true, the hypothesis is disproved.

Two members are TRUE. Then, since one MUST be false, the hypothesis is disproved.

If either of these is the case, we are done. The hypothesis is disproved and the node can be eliminated.

The proof strategy is essentially the same, but the objective is to prove a hypothesis rather than disprove it, starting with the proposition that a certain mark can be eliminated, and seeing what would have to be true to have that happen. There used to be a section here about this strategy, but I’ve removed it, because it adds nothing new and isn’t very easy for me (at least) to use.

An example of a pattern that is handled easily by hypothesis and proof is what has been referred to as an XYZ-wing. Both the * and Z here are weak corners between the two strong chains XZ and YZ. Let’s hypothesize that the * is Z. Any Z shown, then, would be FALSE. But the X of XZ and the Y of YZ both must be TRUE, because they are parts of the strong chains associated with that * node. OK, so if they are TRUE then they each force the X and Y of XY Z to be FALSE. But then all of X, Y, and Z have to be FALSE. That leaves nothing at that position, and the hypothesis is disproved. Therefore, * is not Z. QED.

* XY Z XZ YZ

Note that in practice you don’t really have to think about chains at all if you don’t want to. Just follow the logic. One node TRUE means all those “related to it” are FALSE. From there, look for strong connections — a cell with only two values in it, or a row, column, or block with only two of the TRUE node’s value. Those are the strong chains that can be set TRUE. Go from there to find more FALSE nodes, and on and on. Periodically look to see if there is any row, column, block, or cell that satisfies any of those last three conditions. It’s really no more than that.

A good trick Gail Nelson taught me is that you can to this hypothesizing starting with a cell with only two values in it and go both ways by placing a circle around the implications for one option and a square around the implications for the other option. You often don’t have to go far in either direction to come to a satisfactory conclusion:

The two logical chains converge to a particular value in a specific cell. Then that cell is that value. The two logical chains converge to two different values in the same cell, row, column, or block. Then every other possibility in that cell, row, column, or block can be eliminated.

This, I find, is a very powerful technique. Except for the Easter Monster puzzle, this technique has solved every Sudoku puzzle I have thrown at it. It is definitely my choice of last resort when solving puzzles by hand. I should probably get Sudoku Assistant doing this, but I haven’t done that yet.

Full trial-and-error analysis involves adding some element of depth. The idea here is that if a hypothesis gets to a “dead end” — no more conclusions to make, but still no solution — then all we really have to do is start the sequence over from that configuration. We can look for singles, locked candidates, subsets, grids, chains, and even do more “subordinate” hypothesizing. If this process is taken to the extreme, then you basically have a depth-first backtracking solver. The Sudoku Assistant doesn’t go that far. Instead, it takes a middle ground and considers two optional added elements of depth:

+depth just adding analysis involving single cells, rows, columns, and blocks (singles and locked candidates), and

++depth also adding analysis involving multiple cells, rows, and columns (n-tuples, grids). Note that using ++depth amounts to more than just one level of what people usually refer to as “depth” because these methods can’t be replaced by simple “If X then Y” type logic. They require “If X and Y then Z” logic, and that is the logic of depth. (“OK, let’s hypothesize that X is true, and then let’s also hypothesize that Y is true, then….”)

Further discussion of dead ends and depth can be found here.

Beyond These Techniques

The mathematics behind Sudoku

Some more interesting facts

A well-formed Sudoku puzzle has a unique solution. A Sudoku puzzle can have more than one solution, but in this case the kind of logical reasoning that we described when discussing solving strategies may fall short. There are examples of rank 3 Sudoku puzzles with 17 well-formed defaults. However, it is not known for how many given values ​​a rank 3 Sudoku can be well-formed.

Practice: Can you come up with a Sudoku puzzle that isn’t well-formed?

Another interesting question (which you may have asked yourself while solving the exercise above) is how many different symbols must be used under the circumstances for a puzzle to be well-formed. It turns out that for a rank n Sudoku, at least n2-1 different symbols must be used for the puzzle to have a unique solution. Because if we had an n-rank puzzle using only n2-2 symbols and we found a solution, then swapping the places of the two missing symbols would lead to a different, different solution. In particular, for the usual rank 3 Sudoku, at least 32-1=8 different digits must be used in the defaults for the puzzle to be well-formed; Otherwise the puzzle has more than one solution.

The fact discussed above can be rephrased as follows: if a rank n Sudoku is well-formed, then it must have n2-1 distinct digits under the circumstances. It is important to note that this is not the same as stating that a rank n Sudoku is well-formed if it has n2-1 distinct digits in the given values. Remember that the converse of a true statement is not necessarily true. The next exercise illustrates this with a concrete example.

Exercise: The following rank 2 Sudoku has 22-1=3 different digits among the given values. Find two different solutions.

For 4×4 Sudoku, a case-by-case analysis using the two substantially different grids proves that a well-formed puzzle must have at least four distinct digits in the numbers given.

Finally, it is fascinating to note that although there are computer programs that can solve rank 3 sudokus quickly and easily using a backtracking method, solving a sudoku of any rank n is a much more difficult problem. As the rank of a Sudoku increases from n to n+1, the additional computation time needed to find a solution increases fairly rapidly. This places the game of solving rank-n Sudoku puzzles in a class of problems that computer scientists have called NP-complete. An NP-complete problem satisfies the following two properties:

Any solution to the problem can be found relatively quickly, i. H. in polynomial time, can be checked. If the problem can be solved relatively quickly, then any problem that satisfies property (1) can, too.

Although checking a solution to an NP-complete problem can be done relatively quickly and easily, there is no known algorithm to find a solution because the computation time increases so rapidly compared to the size of the problem.

Solving Sudoku puzzles involves a series of logical deductions to arrive at a unique solution.

But after playing Sudoku puzzles, you’ve probably found yourself in situations where you feel like you have no choice but to guess.

However, guessing is hardly a logical solving technique. Therefore, it may seem strange that in a puzzle involving logical deduction, you have to make assumptions.

So let’s see if you ever need to guess to solve a Sudoku puzzle.

Can all sudokus be solved without guessing?

The short answer is yes. Every real Sudoku puzzle can be solved without ever having to guess.

Another way of thinking is that every Sudoku puzzle can be solved logically. Although it may require highly complicated solving techniques that you are unfamiliar with.

Where the answer starts to get complicated is when you factor in the fork or use brute force to solve a puzzle.

Sudoku guessing vs branching or brute force

There will be times with very challenging puzzles where there is only one remaining pair of numbers that are candidates for a particular cell.

While it may be theoretically possible to reduce the two candidates to a single number through a sophisticated (or even undiscovered) solving technique, the most advanced Sudoku solvers turn to branching instead.

In the bifurcation, one of these two remaining candidates is chosen for the given cell and it is checked whether it is possible to complete the sudoku without encountering a problem.

If they run into a problem (like the same number appearing twice in the same row or column), they know they must have picked the wrong number. And therefore the correct solution would be the other candidate.

Because this process can be considered “guessing,” it can get a bad rap in the Sudoku community. Instead, many solvers try to find a logical process to find the answer so they don’t have to rely on branching.

Whether you consider branch rates or not, it’s not the same as randomly choosing numbers to place in cells.

Rather, it is a brute-force method, systematically playing through all the remaining possibilities to find the answer.

No Sudoku puzzle, properly constructed, requires you to simply guess by inserting random numbers into cells.

All sudokus can be solved without guessing.

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Solving various Sudoku puzzles requires learning three or four elementary steps and practicing how to use them more efficiently. Sudoku puzzles are among the easiest to create. There is software that can generate these puzzles without human intervention. All they need is a difficult input and they will create a brand new Sudoku puzzle with different numbers filled in the squares.

Sudoku is like those factory made hamburgers that people love to hate. With that in mind, I’ve compiled the following list of fourteen reasons why I personally find Sudoku a waste of time. The good news is that all of this is subjective. You may find Sudoku to be a valuable addition to your lifestyle, despite its numerous downsides.

1. It has no soul

As I mentioned above, Sudoku has been created by computer programs for quite some time. What puts me in a jigsaw puzzle is knowing that behind this is an ingenious and creative person who has sat down for hours, maybe even days, to create something that will inspire others to boggle their brains a bit.

When playing a man-made puzzle, don’t just try to figure it out based on general logical principles. You also learn about its creator and how it thinks because it imprints in its creation the knowledge you need to solve it. Sudoku has none of that; Software can “hand out” hundreds of nearly identical puzzles in an afternoon.

2. It repeats itself

While there are probably no two identical Sudoku puzzles in the world in terms of number placement, the principles by which the game is played are so limited and straightforward that you’ll soon find that there’s almost no real variety . Play a few dozen games and you’ve played them all. Once you get pretty good at solving harder Sudoku puzzles, you probably won’t come across one that you can’t solve.

3. It doesn’t offer a real sense of accomplishment

When you encounter Sudoku for the first time, it sure can be challenging, and solving the puzzle can give you a small performance boost. This doesn’t last long though, and the moment you get really good at the game and start solving the puzzles consistently and with ease, all that sense of accomplishment is gone.

You just keep doing the same thing over and over with no real sense of adversity. You can counteract this by downloading one of the many Sudoku apps with timers and challenging yourself to beat your own time count at various difficulty levels, but even this has a cap you cannot exceed.

4. One mistake can cost you the whole puzzle

Whether you’re a new Sudoku player still struggling to get the hang of the game, or a seasoned Sudoku veteran who just didn’t have enough coffee that day, it’s really easy to make a mistake. You can completely miss it until you’re almost done with the puzzle and realize it’s all wrong. It’s almost impossible to trace your incorrect entries back to the original one, meaning you can’t fix what you broke and have to admit defeat just because you wrote an incorrect number somewhere fifteen minutes ago.

5. It’s frustrating

If you’re a Sudoku beginner, you’ll probably find even the simplest of puzzles quite challenging. Even experienced players occasionally get stuck with a puzzle and become increasingly frustrated at not being able to solve it. When you can’t solve a sudoku, it’s easy to get stuck, overthinking, and stressing out. Rather than being a fun pastime, it can distract you from other more important and enjoyable things in life.

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6. It’s addictive

Sudoku addiction is a real thing. Most people assume it’s harmless because it’s just a numbers game, but they tend to forget that an item doesn’t have to be a chemical to create addiction. Sudoku certainly contains conceptual elements that can create this effect.

All addictions are unhealthy, even playing Sudoku. However, don’t confuse enthusiasm with addiction. The easiest way to tell them apart is to remember that enthusiastic activity adds value to a person’s life, while addiction takes it away. An item becomes an addiction when the person feels they cannot go a day without it and begin to neglect other aspects of their life in favor of this activity.

7. It requires a lot of patience

It takes the average person more than a few minutes to solve a sudoku. It requires patience, time and attention. If you are not very patient by nature, you will probably be very disappointed and upset with the game.

8. It’s a distraction

People love a good distraction. Most of our day-to-day lives have become quite stressful over the last few decades, and we all like to indulge in a bit of escapism from time to time, be it through TV, books or Sudoku puzzles. While this can be a helpful coping mechanism when used diligently and carefully, escapism can quickly overtake us and snap us out of reality, causing our life to fall apart even more severely around us.

9. It can make you feel incomplete

There are thousands if not millions of Sudoku puzzles in the world. If you’re an avid Sudoku player, you might know that feeling that you want to try to solve them all, but know you’ll never succeed. This can leave you feeling a little incomplete and empty.

10. It can cause an unbalanced lifestyle

While Sudoku can be an excellent activity for your brain, some studies have shown that it can help keep your mental health in check; it doesn’t work wonders for your physical health. If you’re not careful, you can immediately abandon your workout routines to just sit in a comfortable chair and puzzle for a few hours. Depending on your schedule, finding a balance in this regard can be quite a challenge, so watch your Sudoku intake.

11. It can lead to diminishing responsibility

Because of its addictive nature, Sudoku can quickly make people neglect other activities or people around them. A parent may be more concerned with a Sudoku puzzle than what their child is doing. A person may not listen to their partner while attempting to complete a puzzle, or you may choose to play a puzzle or two while waiting for your bath to be drawn and become distracted until your bath is crowded. It’s important to keep an eye on life and the world around you, no matter how much you enjoy a game.

12. It can cause anxiety

Playing Sudoku can be a relaxing activity after a long day, but it can just as easily cause even more stress. Getting stuck on a puzzle or anticipating the puzzle difficulties, layouts and solutions to come can be a source of anxiety for many people. If you find yourself one of them, you should probably avoid Sudoku.

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13. It may cause you to give up your mental health treatment

It is well known that mental health is essential. We’re also familiar with the fact that Sudoku is often promoted as an activity that can help keep your mind sharp and active. For people suffering from depression and/or anxiety, Sudoku can be quite helpful at times. But in some cases, people may choose to abandon their mental health treatment plans and rely only on Sudoku to get them through; but while it is helpful, Sudoku is not enough.

14. It’s a crossword threat

It is common for avid puzzle game players to stumble upon Sudoku and abandon crosswords entirely. The crossword community is quite concerned about the threat Sudoku poses to their favorite pastime. They often criticize Sudoku players, whether they have done crosswords before or not, for being too lazy to learn abbreviations, inversions, anagrams, homophones, etc.

The downside is that while Sudoku is very engaging mentally, it just requires you to train your brain to work logically within a very narrow system. In contrast, crossword puzzles require you to constantly expand your general knowledge.

Everything considered

There are many notable advantages to engaging in Sudoku, but the disadvantages are also numerous and can be quite damaging. In order to get as many benefits as possible from the Sudoku game while minimizing the disadvantages, you must be aware of your behavioral and instinctual tendencies at all times.

While individual puzzles only take a few minutes to keep you engaged in Sudoku on a regular basis and keeping you in check for the rest of your life, you need to monitor yourself and your decisions around the clock. Most of the time, Sudoku takes away more than it gives.

Practice creates masters

Sudoku is a game that rewards many hours of repeated practice. The more you play, the better you become. If you play Sudoku every day, you’ll find that you’ve almost got a “sensual memory” of sorts for the puzzle grid – you’ll recognize patterns and develop the ability to seize opportunities more quickly.

The more Sudoku situations you see through many hours of Sudoku play every day, the more likely it is that you can solve the puzzle in every situation.

Train your brain

Just as we need to walk, run, lift weights and exercise to keep our muscles strong, playing Sudoku is a way to exercise our brain “muscles”. Some studies have shown that playing brain games like Sudoku can even improve people’s cognitive abilities. Playing Sudoku is not only fun and relaxing – it can also help you develop your attention and focus.

Improving problem-solving skills

By playing Sudoku, you learn to solve problems through deductive reasoning – and this kind of logical reasoning can be useful in other areas of life as well.

Deductive reasoning skills are important for success in life. You may find that playing Sudoku helps you improve your problem-solving skills in other areas of your life, whether it’s playing other puzzle games, or approaching complex problems at work, or finding new inspiration for ideas to help you to improve your home or life.

Clear your mind

Playing Sudoku every day can also be a calming, almost meditative way to pass time. You might find that playing Sudoku every day becomes your own little ritual – escape from the daily rush and focus on solving the puzzle, filling in the empty spaces in the grid and bringing a sense of order to the rows , columns and squares.

Playing Sudoku can give you that little mental break you need to return to your daily work and other life challenges with renewed energy.

While most people will never be as good at Sudoku as Thomas Snyder, it’s interesting to look at some of the key personality traits of the greatest Sudoku players to see if we can emulate them – maybe if we get more like Sudoku -Champions can think, We can improve our Sudoku skills the same way!

Here are a few typical personality traits of the best Sudoku players:

Orderly, methodical spirit

Thomas Snyder is a scientist by trade, so it’s no surprise that he brings a rigorous, scientific process to solving Sudoku puzzles. The best Sudoku players know how to break through the visual clutter of a Sudoku grid and instantly look for the most valuable clues and information they need – they know how to apply a process to the Sudoku grid where they can see, which positions are most promising. They have a good visual memory that helps them remember which numbers are which. They seem to have a supernatural ability to scan, sort and analyze their options in the sudoku grid at any given moment.

Focus, focus, focus

Some of the best Sudoku players seem to be exceptionally good at turning their minds off of the outside world and spending hours concentrating on solving Sudoku puzzles. Sudoku is a game that requires concentration.

If you love the feeling of immersing yourself in a Sudoku puzzle (or solving a series of Sudoku puzzles) to the point where you lose track of time, then perhaps you have in common the intense focus that excellent Sudoku players have in common is.

Cool under pressure

Sudoku competitions are tough – players have to compete against each other as well as against the clock. Solving the most demanding Sudoku players requires speed, quick decision-making, and a certain amount of confidence and “courage”. The best Sudoku players are resilient under pressure. Even when a riddle is devilishly challenging, they remain calm and continue to work on their analysis and do whatever it takes to solve the riddle.

show determination

Great Sudoku players are made, not born. Sure, it probably helps to have some level of logic and reasoning, and some people may have more innate skills in these areas than others – but if you want to be great at Sudoku, it’s not just a matter of talent, it’s a matter of time and effort to develop your skills.

The best Sudoku players have taken a fun game and taken their skills to a higher level. Whether we ever play at a Sudoku World Championship or not, that kind of determination and diligence is admirable and something we can try to emulate in other walks of life.

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Sudokus are phenomenal brain teasers, but when you find yourself cracking simpler puzzles in the blink of an eye, you may have moved on to more difficult puzzles only to find that they feel impossible! The great thing about Sudoku is that there is always something else to try when you get stuck, and if you work carefully you should be able to solve the puzzle using deductive reasoning and logic alone. If you’re looking for extra help with your next puzzle, or have come here from a looming sense of desperation that your last guess ruined your current game, we’ve got your back!

Sudoku is a logic based puzzle that will train your brain and be fun. However, the 81 squares in a simple 9 by 9 grid can understandably overwhelm a complete novice. While Pennydelipuzzles.com has many Sudoku puzzles suitable for all levels of solvers, there are also a few ways to simplify the mechanics of the puzzle so you can have an enjoyable time. Here are some tips and tricks to keep in mind when solving a Sudoku puzzle as a beginner.

Focus on a single row, column, or square

Focus on just part of a square, row, or column instead of tackling the entire grid at once. Slowly work your way up until you have filled in all 81 squares. You can start with a single square, then a row, then a column. Getting rid of all other distractions will help you solve the Sudoku grid much faster.

Use scanning techniques

Scanning techniques include scanning rows and columns within each triple box area. Then eliminate numbers or squares until you find cases where only a single number fits in a single square. You can scan from one direction, from two directions, or for a single candidate. If a given cell can only contain a single number, then that number is the only candidate.

Avoid guessing

Sudoku is a methodical process of elimination, so you’ll end up spending more time solving a puzzle if you resort to guessing your answers. There are many tricks and techniques to narrow down which number goes in a particular spot. If you’re not sure what goes in there, move on. If you’re stuck on a particular field, don’t focus on it too much. Shift your gaze to another area of ​​the grid and try to see what other solutions are possible.

Use techniques to narrow your solutions

Naked singles

There are several techniques you can use to narrow down your solutions. Singles are the easiest techniques that every beginner should know. Here are the most commonly used:

In the naked single, the candidate values ​​of an empty cell are determined by examining the values ​​of the filled cells in the field, row, and column to which the cell belongs. If the blank cell has only one candidate value, then that is most likely that cell’s value.

Hidden singles

A hidden single is similar to a naked single, but only affects the cells where the candidates are located. Regardless of what other candidates are in a cell, if the cell contains a candidate that appears only once in the grid, it must be assigned to that cell. This means there is only one cell left to place that digit for any given digit and house. The cell has more than one candidate left, but the correct digit is hidden among the others.

Draw in your grid

Pencil marking involves writing small numbers in a square. It is a method that allows you to keep track of the remaining candidates for the cells that you have not solved yet. Then you cross out or remove the candidates one by one. This will help you not to get confused with all the different possible solutions.

There’s no shortage of ways to tweak the puzzle to your liking. At the end of the day, what matters is that you have fun.