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Recreational Maths. Modeling Sudoku Puzzles with Python. Generating a random complete solution We show how to characterize puzzles by logic programs and how to use those characterizations to build puzzles automatically. We can control the difficulty level of the We begin by expressing the constraints in a logic puzzles by choosing how and when to invoke the logic formalism.

For this paper, we present constraints program. Introduction Given n as the width of a section for us, 3 , and m the width of the puzzle for us, 9 , the logic rules Cellular puzzles take the form of cells whose val- of Figure 2 represent the constraints. The puzzle starts with clues, which are cells Lines 1 and 2 introduce numsmall and num- that already possess their value. It is the pleasant large as predicates true over the range 1. Lines 3, 4, and 5 give the sig- by entering values in all the cells in a way that sat- nature of the ternary predicate place and the vari- isfies the constraints.

We use place I,J,N to exactly one solution. To assist the enthusiast, the represent the fact that the value N is placed in cell puzzle may contain a hint sequence, which is a se- I,J. Figure 1 shows a sample Su- umn J. Line 8 constrains columns in a similar way. We say a set Given a section numbered K,M and a value N, of 9 cells is complete if the numbers 1.

The constraints that link has that value. The first hint, bE, directs our attention When we run these rules as a logic program, we to a cell that must contain 2, because in its section, get a very large number of results.

We pick the first all other locations for 2 are excluded by being filled result as a desired solution, relying on the random- in with other numbers or because their row or col- ization performed by Aspps. We also randomize umn already has a 2. If our preloading is inconsistent with the to generate cellular puzzles of varying degrees of constraints, we choose a different random preload- difficulty. Algorithm to reduce the clue set Hints 3.

Eventually, all the origi- dH aC nal clues are either removed or preserved; the pre- served clues become the final set of clues. We try solving the puzzle with Aspps. The Aspps soft- ware has two stages, the grounder and the solver. The solver em- ploys full backtrack, using a heuristic to pick facts Fig. Time seconds and memory MB require- to assert and following the consequences.

The fol- ments for generating Sudoku puzzles lowing results can arise from applying Aspps. This situation should never happen, because each iteration 4. Generating a hint sequence tests a less-constrained puzzle than the pre- vious one.

In a cellular puzzle, most atoms turn out to be false. Therefore head. By ex- lookahead. In practice, propagation. That list begins with the clues them- unless we are trying to generate a hard puz- selves.

The rest of the positive instances of place zle, we tell the grounder not to use looka- in the log form the hint sequence. Performance to find a sole solution. Although the puzzle is well-formed, it is too hard for general en- The logic-program approach to generating cellu- thusiasts. The miss- Aspps code. The algorithm of Figure 3 is encoded ing clue is required to keep the puzzle well- in about lines of Perl [WS90], much of which is formed.

These last two situations require the solver to dis- Given the Aspps rules, generating a puzzle has tinguish. Either the grounder clues. On a 3GHz Pentium 4 run- let the grounder use one-step lookahead. We also ning Linux, we accomplish the first phase for Su- apply one more test after reaching the final set of doku puzzles of various sizes in time and memory clues: Can the grounder solve the final set of clues shown in Figure 4.

The time for the first phase has without using lookahead? If so, then the puzzle a large variance, especially for larger puzzles. In this case, we re- second phase. This value has much smaller vari- ject the puzzle completely and start afresh. Puzzle types We have investigated several cellular puzzle types. These puzzles are the same as Sudoku, but the two diagonals are also complete. Reading this part of the chapter is optional. Now, let’s continue solving the puzzle, by the process of Reduction.

That is reducing the Matrix, removing exclusions newly created, if any. Let’s repeat the process of Reductions, deleting these values from their respective rows, columns and major squares respectively, till no more reductions are possible. First, since cell 1,2 takes the value ‘3’, we can’t have any more ‘3’ in row 1, or column 2 or the top left major square consisting of cells 1,1 , 1,2 , 2,1 and 2,2. So, delete ‘3’ from all the above cells in the Possibility Matrix where the puzzle is yet to be solved.

We have: 2 3 1,3 1,3,4 4 1 2,3 2,3 3 2 4 1 1 3,4 1,2,3 1,2,3 29 Solving Sudoku SAP Now, since cell 3,4 takes the value T, we can’t have any more T in row 3, or column 4 or the bottom right major square consisting of cells 3,3 , 3,4 , 4,3 and 4,4. So, delete T from all the above cells in the Possibility Matrix where the puzzle is yet to be solved. We have no such T to be deleted, though, and this is what we now have: 2 3 13 13,4 4 1 2,3 2,3 3 2 4 1 1 3,4 1,2,3 1,2,3 30 Solving Sudoku SAP Removing the deleted numbers, the puzzle reduces to the following: 2 3 1 4 4 1 2,3 2,3 3 2 4 1 1 4 2,3 2,3 We see that Cell 1,3 now has a T, Cell 1,4 has ‘4’, and 4,2 has ‘4’.

Again, let’s repeat the process of deleting these values from their respective rows, columns and major squares respectively. Since cell 1,3 takes the value T, we can’t have any more T in row 1, or column 3 or the top right major square consisting of cells 1,3 , 1,4 , 2,3 and 2,4.

We have, again, no such T to be deleted, and this is what we continue to have: 2 3 1 4 4 1 2,3 2,3 3 2 4 1 1 4 2,3 2,3 31 Solving Sudoku SAP Since cell 1,4 takes the value ‘4’, we can’t have any more ‘4’ in row 1, or column 4 or the top right major square consisting of cells 1,3 , 1,4 , 2,3 and 2,4.

So, delete ‘4’ from all the above cells in the Possibility Matrix where the puzzle is yet to be solved. We have, yet again, no such ‘4’ to be deleted, and this is what we still continue to 2 3 1 4! We have, once again, no such ‘4’ to be deleted, and this is what we again continue to have: 2 3 1 4 4 1 2,3 2,3 3 2 4 1 1 4 2,3 2,3 32 Solving Sudoku SAP Since cell 2,3 takes the value ‘2’, we can’t have any more ‘2’ in row 2, or column 3 or the Top Right Major Square consisting of cells 1,3 , 1,4 , 2,3 and 2,4.

So, delete ‘2’ from all the above cells in the Possibility Matrix where the puzzle is yet to be solved. We have: 3 1 4 4 1 2,3 2,3 3 2 4 1 1 4 2,3 2,3 Let’s now remove the deleted nos.

Are you sure? We had made an assumption along the way, didn’t we? Do you remember that we deleted ‘3’ from Cell 2,3 and assumed the Value ‘2’ in 2,3? What if we had deleted the Value ‘2’ from Cell 2,3 and assumed the Value ‘3’ instead? So, let’s apply the Exhaustive Tie Breaker Rules.

We have: Let’s now remove the deleted nos. Isn’t that interesting? For all practical purposes, you should be satisfied if you get one of the final solutions. In fact, most Sudoku solvers wouldn’t even know that there may be more than one solution to a puzzle when the puzzle at hand has more than one solution , if they solve it by any intuitive methods. Hence, when you verify answers, don’t be surprised if you find that you have a right solution, but it is not the same as the one given by the Sudoku publisher.

Ideally, if a puzzle has multiple solutions, the publisher should give all the possible solutions. Most Sudoku publishers don’t give puzzles with multiple solutions, if the first place. But multiple solutions can be there, very rarely, though! We’ll see about this later. No, not quite fully. Not as yet. Let’s quickly construct the Possibility Matrix, as below: 2,4 2,4 1,2,3, 4 1,2,4 3 1 2,4 2,4 1 3 2,4 2,4 2,4 2,4 1,2,4 3 Now, how do we proceed?

Do we have to go in for the Tie Breaker? Not quite as yet! Column 3 has “2” and “4” as possible values in all the 4 Cells, but 2 of these cells can take only the values ‘2’ or ‘4’; they can’t take any other values. In such a case. Cell 1,3 and Cell 4,3 can’t take these values, ‘2’ and ‘4’. So, remove ‘2’ and ‘4’ from these Cells. We call it ‘Group Reduction’. We had initially made an assumption of the value ‘2’ for Cell 1,1 , when ‘4’ was also a possible value, right?

Let’s now assume the value of ‘4’ for Cell 1,1 , and get the 2 more Solutions, as below: So, this Sudoku SAP puzzle has 4 possible solutions. How is that possible? When you do not define a Sudoku or a Sudoku SAP puzzle completely, it can have more than one solution. The more ill-defined the puzzle i. Isn’t this an interesting observation? Now, you are completely equipped to solve a Sudoku puzzle proper, but before you do that, let’s make sure you have learnt to solve the Sudoku SAP fully.

Creating the Possibility Matrix 2. Reduction 3. Group Reduction 4. Create the Possibility Matrix. Perform Group Reductions. Resolve impasse thru Tie Breaker and arrive at one possible Solution. Repeat Tie Breakers with other possible values to arrive at all possible Solutions, if required. Declare the Problem as Unsolvable If Conflicts arise which are not resolvable. Very Simple and Simple Puzzles are solved by Stage 2. Moderately Difficult Puzzles are solved by Stage 3.

Difficult Puzzles are solved by Stage 4. Very Difficult Puzzles by whatever names different composers choose to call them are solved by Stage 5 and beyond. Why not try to solve the following Sudoku SAP Puzzles, and get your fundas right, so that you’ll find it a lot easier to solve the original Sudoku Puzzles a little later?

Try all of them, and you won’t regret it, as each has an interesting story to tell you. The 2 Solutions are: 5. What’s interesting about the Puzzle 1 above? This puzzle is the same as the one we have solved under 6. That is, in our solved puzzle, we came to a point of deadlock which we resolved thru Tie Breaker. Here, one of the Cells where we had impasse previously has been assigned a specific number, and so the impasse no longer exists. So, you can see that it gets solved more easily.

Puzzle 2 is quite simple. This is here for you to cut your teeth before going on to solve more difficult Sudoku SAP and original Sudoku Puzzles. What’s interesting about Puzzle 3 above? This puzzle is also the same as the one we have solved above, with the impasse in tact. Substitute All T by ‘2’, ‘2 7 by ‘3 7 , 7 3 7 by 7 4 7 and 7 4 7 by T in the puzzle we have solved under 6.

Effectively, you will get the answers by simply substituting all Ts by 7 2 7 s, 7 2 7 s by 7 3 7 s, 7 3 7 s by 7 4 7 s and 7 4 7 s by Ts.

This is just a repeat exercise to make sure you don’t go wrong anywhere, and if you do, for you to be able to recover easily, by making substitutions of the above numbers from the solved example.

Puzzle 2 was simple, but Puzzle 4 is more complex. Despite more cells being filled with numbers than in the case of Puzzle 2, Puzzle 4 leads to Tie Breaker. This shows that for a Puzzle to be simple, it is not adequate if more cells are filled with numbers, but the right cells must be filled with their numbers.

Puzzle 5 is a Diagonal puzzle, and so we find that in the first step, nothing gets resolved. But the moment you start applying the Tie Breaker Rule, the puzzle gets resolved easily, with multiple answers.

Puzzle 6 has multiple solutions, but we need to apply the Tie Breaker Rule twice, since there are 2 Cells which can take one of two values, and there are 2 more Cells which can take two more of other values. First you apply the Tie Breaker for one set of 2 Cells, and for each of these two values, you again apply the Tie Breaker for the other set of 2 Cells. So, you get 4 possible solutions, overall.

Puzzle 7 is wrongly composed. Such a puzzle is very rarely given, if at all. And such puzzles could be given by a mistake in transcription, or deliberately to see if you are able to realize the mistake in the puzzle early. Row I has a ‘2’ in Cell 1, 1. So, all other cells in Row 1, which are empty, can’t take the value ‘2’. Therefore, the Row Exclusions for Row 1 is just the value ‘2’.

Similarly, let’s write down the other Row Exclusions too, as below. Likewise, Column 1 has a ‘2’ in Cell 1,1 , and a ‘3’ in Cell 3,1.

So, all other cells in Column 1, which are empty, can’t take the values ‘2’ and ‘3’. Therefore, the Column Exclusions for Column 1 are the values ‘2’ and ‘3’.

Similarly, let’s write down the other Column Exclusions too, as below. So the other Cells in this major square, viz. Remember the possible values for the squares are only T, ‘2’, T, and ‘4’.

Take the case of Cell 1, 2 , for example. Let’s face it: Some people will always be faster than others; if you happen to solve them slower, don’t worry. Try to learn to do them as fast as you can, but once you’ve reached your best speed, don’t let the thought of your speed in comparison with someone haunt you. This is not only true of Sudoku, but of every other competitive activity in life. Do your best, and then, don’t be obsessed with the thought of having to match or beat someone else in speed.

After you have learnt to solve Sudoku SAP Puzzle, learn to increase your ability to solve them quicker. You hone it by practice. The more puzzles you solve, the faster you become. However, there are a few shortcuts that can help you solve them faster. Here are some tips: 1. Constructing the Possibility Matrix is a mechanical process; there’s no great intelligence involved. I feel that this is best got done with the help of the Computer.

However, if you want to do it manually for any reason, please feel free to do so. A kill joy? Not really. Let’s understand this one clearly. There are mainly 2 parts to solving Sudoku our way.

One: Constructing the Possibility Matrix – this is mechanical, with almost no scope for use of one’s intellect.

Two: Solving the puzzle after the Possibility Matrix has been constructed – this involves using the intellect. I do not recommend use of the Computer for the second part.

But I do recommend use of the Computer for the first part. There are the following added advantages in using the Computer to construct the Possibility Matrix: i We may easily go wrong when we do it manually, but the Computer won’t. When you solve your higher school math problems, do you miss the thrill just because you used a Calculator to do your arithmetic? However, I leave it to you to decide how you will solve your Sudokus. Whether you’re constructing a Possibility Matrix or applying the process of Reductions, you may solve the puzzles faster if you take up the most potential squares first.

The most potential squares for solution are the ones where there is the best probability of resolution. Let me explain, with the example of one Now, we need to construct the Possibility Matrix.

When the Computer creates a Possibility Matrix, it hardly matters in which sequence it creates. So, I had suggested that you may go from top to bottom, left to right. In the puzzle above, we had originally solved it mechanically, top to bottom, left to right, just as a computer program might.

Let’s now try to do it the smarter way. We had originally taken up Cell 1,2 first for finding the possibilities, and found it can take values 3,4. Let’s now try to see if any cell can take a unique value rather than 2 or more possible values. How do you know if a cell can take a unique value?

If 3 values are precluded, it can take a unique value. Let’s inspect and see if there is any cell where 3 values are precluded. Row 3 has 2 cells filled in, column 1 has 2 cells filled in, and top left major squares have 2 cells filled in already.

For example, take Cell 2,1. It can take only the value ‘4’, since Column 1 has ‘2’ and ‘3’ filled in already, and Row 2 has T filled in.

Let’s fill that in too. We see that we’re not able to fill in any more Cells likewise. Now, creating the Possibility is a lot simpler, isn’t it? From experience, you’ll see that this will not only make your task more interesting, but also it will be easier and take less time to solve the puzzle itself.

Smarter ways of taking up cells that can be resolved faster is a sure way to speed up solving Sudoku. And you’ll be able to do this with practice. Creating a new table at every stage, or preferably, periodically will help you retract in case you have made any silly mistakes along the way. You could always go back to the last stage’s results and continue from there rather than having to correct a mistake midway. Discover the 7 Secrets Solving Sudoku is Simple, if only you learn to do it right!

Learn it starting with Sudoku SAP! Focus initially on Accuracy than Speed. You’ll gain speed with practice. Possibility Matrix is better than Intuitive techniques. With it, You’ll surely solve any Sudoku!

Preferably use a Computer for it. Apply Reduction and Group Reduction, repetitively, as much as possible. Avoid mistakes as backtracking will be difficult. When you reach an impasse, apply Tie Breaker, starting with fewer option Cells. Even if you had missed out on some reductions.

He suggests that the two of them go see a movie together and offers to carry the Heroine’s purse. Ikki ends up doing so before the Heroine is able to him an actual response. At the movie theater, surrounded by crowds of people, Ikki asks the Heroine whether it bothers her. After hearing her feeble response, he then comments on her unusual behavior since that morning, though the Heroine does not comment on it.

Ikki then breaks away from the Heroine so that he can purchase the movie tickets. He ends up purchasing some for a romantic drama and the two watch it together.

As they watch the movie, the Heroine reflects on what she learned earlier that morning, initially wondering whether time has been reversed. After concluding that it is not the issue, she wonders what else it could be. She then notices Ikki caressing her hand and blushes, upon their two eyes locking contact. Ikki then lets go. Eventually, the movie ends and the two are left as the only people in the theater. Ikki faces the Heroine and expresses his feelings about how his “condition” affects their relationship in many ways.

As the Heroine wonders what Ikki’s condition could be, they are requested by a movie theater employee to leave so that the area could be cleaned up. The two leave to have drinks nearby. While seemingly alone, Ikki tries chatting with the Heroine about the movie, though she is mostly quiet and does not say much.

Ikki quickly notices a group of his fangirls waiting outside. Upon this revelation, he offers to take the Heroine home, adding that they both have to go to work in the morning; upon hearing this, the Heroine becomes worried about whether she works and what she will be doing. Ikki quickly picks up on her concerned expression and, after a light-hearted joke, asks what she would want him to do to make up for their lost time earlier. The Heroine considers the request before asking that Ikki take her to her work in the morning.

Ikki obliges, though he comments on how unusual it is that she would ask him something such as that. Eventually, the Heroine returns home, where she lies on her bed and wonders what her relationship is with Ikki. She then starts to lament on the fact that Orion has seemingly left her. As time passes, Orion somehow manages to manifest in the Heroine’s bedroom, surprising but also elating the Heroine.

She attempts to ask him a question though she ends up letting him speak first. Orion explains that her situation is much different than how it was before, and that it appears that she is in a different world, though it is likely that she could now be in the “real world”. The Heroine, however, cannot understand what Orion is saying. Before Orion can say anything more about the Heroine’s situation, suddenly, he disappears, leaving the Heroine alone in her room. She fruitless calls out for him. The Heroine’s cell phone suddenly starts to vibrate which indicates a new text message.

She checks it and sees that Ikki had just sent her a new text message, where he reveals his suspicions about her new behavior and asks whether she is feeling alright.

A new text message is sent moments after. This time, however, it is from an anonymous person asking about the Heroine’s “daily report”, which she assumes is work-related.

The Heroine drops down on her knees and sighs over how confusing things are becoming for her. The following morning, the Heroine answers her ringing cell phone, only to find Ikki on the other end telling her that he has already arrived at her apartment.

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The Crossword Solver finds answers to classic crosswords and cryptic crossword puzzles. Enter the length or pattern for better results. Click the answer to find. The following morning, the Heroine answers her ringing cell phone, He then hands over his sudoku puzzle to Kent before he gets them all. We show how to characterize puzzles by logic programs and how to use those characterizations to build puzzles automatically. We can control the difficulty.