Some are probably wanting to know why I share my joy and efforts when it comes Sudoku and other number puzzles. Well when my thirteen years of mandatory education came to an end, I felt this insane pressure suddenly disappear from my mind. The stress and anxiety from my need to meet with the demands of the school curriculum began to ease away, allowing me the time and space to recollect myself and prepare for my next move. However, I also began to realise that my knowledge was fading away at the same time. Each and every moment I revisited my school work, I would see the things I've learnt but struggle to comprehend the process. I was asking why something was the case. I lost touch with the basis and couldn't seem to connect things together the way I used to. It was as if I was becoming more stupid every day. My brain wasn't getting active. There wasn't any motivation to think. So I looked to try and change that. It was then I found a growing inclination to Sudoku. Had a book and a deck of cards with puzzles on it, so decided I would get the practice in and put my brain to the test.
It was only early this year, when I was introduced to six other Japanese logic puzzles after opening up an entire book of them (a book I was given as a Christmas present after my parents were aware of my interest in Sudoku). Some of these puzzles were of unusual design to me and while others came to put my mathematical abilities to the test. My interest in them grew much to the point where I felt the need to introduce others to them with my results along with a few live tutorials. It's not so much an obsession. It's rather a hobby. An activity always there for me whenever I am in the mood. An activity there to get me thinking. Here I share with you about each of the seven puzzles including the main one in Sudoku. I even provide some tips and approaches on how to go about solving some of them and show some supporting pictures. But be wary of my explanations as they might not sound as clear. Hopefully you will get an idea of the terminology. If not, then feel free to ask questions in the comment section and I'll be sure to answer them as soon as possible. Let's get stuck into this shall we?
Sudoku
Sudoku can come in many shapes and sizes, but the most common is the classic 9 x 9 version. A large grid of 81 cells divided evenly into 9 squares. You basically start off with a small amount of numbers, which are often set out in a pattern. The objective is then to work out from these where the rest of the numbers go so that the solution has each number from 1 to 9 on every row, column and 3 x 3 square. There are known to be four difficulties of standard Sudoku puzzles: easy, medium, hard and diabolical. The difficulties primarily depend on the amount of numbers given to work with, the way they are set out and which numbers appear more frequently. Regardless of the difficulty there is usually no need for guesswork, as there will always be at least one certain answer in front of the solver. This is where Sudoku also tests vision out and whether or not one can see the answers (I myself struggle to most times). In terms of process and challenge, I find these puzzles to be wedged in-between the two common literal games of word searches and crosswords.
Below are a few techniques I tend to follow when it comes to solving Sudoku puzzles.
The first of these is the mini-rows and mini-columns technique, where I separate rows and columns based on the 3 x 3 squares. This I like to consider a two-step process. In the first step, I look at the numbers and see whether they are positioned in the top, middle or bottom row of the particular square. From there I work my way across, eliminating certain rows where the same number can't be placed. The second step is going down and doing the same thing with separated columns to see if any more cells can be crossed out for a particular number. This is my go-to method when starting off any Sudoku puzzle, and I always find myself starting off with either the most common number or going in ascending order from one. The next couple of techniques will often depend on this one, as they require more numbers to work with (unless they are puzzles of very easy difficulty).
The second technique involves whole rows and columns. It's where I look at an entire row or column (irrespective of the 3 x 3 square) and focus on whichever numbers are missing. This method sees both rows and columns dependent of one another. If I were to pick one particular row, I'd then be focusing on a set of columns and vice versa. Take 4 for instance. If I were to find 4 missing in a selected row, I'd focus on the empty cells and seeing whether the columns they're in are already occupied by a 4. Sometimes different numbers can help determine where your focused number goes in this technique. Finally there's the single-cell technique. The attention is placed on just one cell and looks at each of the row, column and 3 x 3 square it sits in. It's another process of elimination, with any number inside these areas being ruled out. The technique only works when there is one option remaining for the empty cell, so the chances of success are very small.
The first of these is the mini-rows and mini-columns technique, where I separate rows and columns based on the 3 x 3 squares. This I like to consider a two-step process. In the first step, I look at the numbers and see whether they are positioned in the top, middle or bottom row of the particular square. From there I work my way across, eliminating certain rows where the same number can't be placed. The second step is going down and doing the same thing with separated columns to see if any more cells can be crossed out for a particular number. This is my go-to method when starting off any Sudoku puzzle, and I always find myself starting off with either the most common number or going in ascending order from one. The next couple of techniques will often depend on this one, as they require more numbers to work with (unless they are puzzles of very easy difficulty).
The second technique involves whole rows and columns. It's where I look at an entire row or column (irrespective of the 3 x 3 square) and focus on whichever numbers are missing. This method sees both rows and columns dependent of one another. If I were to pick one particular row, I'd then be focusing on a set of columns and vice versa. Take 4 for instance. If I were to find 4 missing in a selected row, I'd focus on the empty cells and seeing whether the columns they're in are already occupied by a 4. Sometimes different numbers can help determine where your focused number goes in this technique. Finally there's the single-cell technique. The attention is placed on just one cell and looks at each of the row, column and 3 x 3 square it sits in. It's another process of elimination, with any number inside these areas being ruled out. The technique only works when there is one option remaining for the empty cell, so the chances of success are very small.
Mini Rows & Mini Columns Technique (Step 1)
Take the small picture on the left for example. To start off with, I am given two fives in adjacent squares. I then look at which rows they occupy within their square. First one sees the 5 in the top row and the second sees the 5 in the bottom row. From this, I already know that the top right square will have the 5 somewhere in the middle row.
Sometimes I don't require the same digit to appear in two adjacent squares. In the picture on the right, we only have one 5 in the top left square. No other 5 is given, however there is a 4, 9 and 3 occupying the entire bottom row of the top centre square. This tells us that a 5 has to be in the middle row in this square and the bottom row of the top right square.
Mini Rows & Mini Columns Technique (Step 2)
After determining which row 5 goes in, you can check to see whether you can narrow it down further. This is when you look at the columns. In the following example on the left, we have a 5 in the middle column of the middle right square and a 5 in the right column of the bottom right square. This goes on to narrow it down to one cell (in green) where five can go in the top right square. Most times, we won't be as lucky as to have the same number given that many times. In the following example on the right, we don't have a 5 in the middle right square. However we have a 2 occupying the centre cell in the top right square and there is still the 5 in the bottom right square. This can still help determine where the 5 goes.
Whole Rows & Whole Columns Technique
This technique is one selecting any particular row or column (preferably one with most of the numbers filled) and figuring out which missing numbers go where. This is mainly determined by looking at the empty cells and going the opposite way to see if a missing number of the row appears (e.g. if selecting a row, one will focus on a few columns and vice versa). This ensures whether or not the missing number of the selected row can be placed in a certain cell.
- The first picture shows a particular row with numbers at focus. By determining where the 2 goes, one places their attention towards the first two empty cells and sees whether or not a 2 already occupies both columns the empty cells are a part of. Since there is a 2 in both columns, the empty cells of the selected row can't be a 2. This leaves one option in the centre-right square, so that is where the 2 goes.
- This process can even involve 3 x 3 squares as shown in the second picture. There is already a 7 in the centre-right square so the two empty cells of the focused column within that square can't be a 7. This adds to a 7 not being in the first empty cell as a 7 already occupies the row. So the 7 can only go at the bottom cell of the focused column.
- One can even determine where a number goes based on whether or not other missing numbers can go there. Take the third picture for instance. This focused row was missing a 1, 5 and 6. The column of the first empty cell (where the 5 is), already has two of the missing numbers in 1 and 6. This means neither can go in that cell. So the only other number that can go in the cell is 5.
Single Cell Technique
In the midst of solving a puzzle, I may now and then use this technique when the others don't seem to be working for me. This technique involves choosing a particular empty cell and looking at what is around it. We take a look at each number and see whether they are in the same row, column or 3 x 3 square as the empty cell. This helps eliminate certain numbers. Sometimes it can even take us straight to an answer. Look at the example below. Pretend that the nine isn't there and see whether any other number can go in that particular cell. In this situation, no other number can:
- 1 can't go in the green cell because a 1 is already in the column.
- 2 can't go in the green cell because a 2 is already in the row.
- 3 can't go in the green cell because a 3 is already in the row, column and 3 x 3 square.
- 4 can't go in the green cell because a 4 is already in the row.
- 5 can't go in the green cell because a 5 is already in the 3 x 3 square.
- 6 can't go in the green cell because a 6 is already in both the row and the column.
- 7 can't go in the green cell because a 7 is already in the column.
- 8 can't go in the green cell because an 8 is already in the row.
All of this means that only one number can go in the green cell; 9.
Jigsaw Sudoku
Jigsaw Sudoku is pretty much like regular Sudoku. We're to place each number once in every row column and group of nine cells. However, there is one significant difference. Instead of 3 x 3 squares, we are dealing with irregular jigsaw-like shapes (hence the name 'Jigsaw Sudoku'). Another minor difference about these puzzles compared to standard Sudoku, is that the shapes are symmetrical rather than the given numbers. These puzzles are slightly more complicated due to the design and the only techniques (from above) we can really use for them are the second and the third. They require a set of strong eyes and they get one thinking the tiniest bit harder. But I find them to be a lot more fun to solve. Jigsaw Sudoku is not a common puzzle and one won't stumble across it that regularly. But the design is nothing short of fascinating. From a personal perspective, I would highly recommend that people try it out for themselves.
Jigsaw Sudoku is pretty much like regular Sudoku. We're to place each number once in every row column and group of nine cells. However, there is one significant difference. Instead of 3 x 3 squares, we are dealing with irregular jigsaw-like shapes (hence the name 'Jigsaw Sudoku'). Another minor difference about these puzzles compared to standard Sudoku, is that the shapes are symmetrical rather than the given numbers. These puzzles are slightly more complicated due to the design and the only techniques (from above) we can really use for them are the second and the third. They require a set of strong eyes and they get one thinking the tiniest bit harder. But I find them to be a lot more fun to solve. Jigsaw Sudoku is not a common puzzle and one won't stumble across it that regularly. But the design is nothing short of fascinating. From a personal perspective, I would highly recommend that people try it out for themselves.
Killer Sudoku (Addoku)
Killer Sudoku is a hybrid puzzle that combines elements of regular Sudoku and another Japanese puzzle called Kakuro. It requires the solver to use their head a little more, as it involves calculating and mental arithmetic. We start out with the usual 9 x 9 Sudoku grid. But this time instead of given numbers, we have smaller subsections divided by either dotted lines or different colours. These are called cages. Each cage is represented by a number in the top corner of the first cell in that cage going across. This number is the total of all the numbers in a particular cage. The rules remain as they are. One is to place the numbers 1 through to 9 once on each row, column and 3 x 3 square, while also ensuring that all the numbers in the cages add up to the given total. One must also NOT repeat any numbers within a cage, even if they don't occupy the same row, column or 3 x 3 square. This type of puzzle can seem hard when you first look at it and it may remain that way. But it primarily depends on the difficulty and whether we have the right level of skill.
Allow me to share with you a very handy tip that is key to solving these puzzles both quickly and efficiently. With this puzzle you will always be focusing on the totals. Always look at the both the lowest totals and highest totals first. Knowing that numbers cannot be duplicated in cages, it makes room for obvious answers:
- If a cage covers two cells and the total is a 3, the only numbers that can occupy the two cells in that cage are 1 and 2 (as they both add up to make 3). A cage with 3 is only ever going to cover two cells.
- If a cage covers two cells and the total is a 4, the only numbers that can occupy the two cells in that cage are 1 and 3 (since 2 and 2 is repeating the same number). A cage with 4 is only ever going to cover two cells.
- If a cage covers two cells and the total is 17, the only numbers that can occupy the two cells in that cage are 8 and 9 (as they are the only two numbers that can reach the total).
- If a cage covers two cells and the total is 16, the only numbers that can occupy the two cells in that cage are 7 and 9 (since 8 and 8 is repeating the same number).
- Here are a few extra tips, though these may rarely occur:
1. A three-cell cage with a total of 6 = 1, 2 and 3
2. A three-cell cage with a total of 7 = 1, 2 and 4
3. A four-cell cage with a total of 10 = 1, 2, 3 and 4
4. A four-cell cage with a total of 11 = 1, 2, 3 and 5
5. A three-cell cage with a total of 24 = 7, 8 and 9
6. A three-cell cage with a total of 23 = 6, 7 and 9
7. A four-cell cage with a total of 30 = 6, 7, 8 and 9
8. A four-cell cage with a total of 29 = 5, 7, 8 and 9
These tips help narrow down options and provide answers. Once an answer is found, the rest of the answers will follow. Sort of like a chain reaction. No guesswork is required.
Killer Sudoku is a hybrid puzzle that combines elements of regular Sudoku and another Japanese puzzle called Kakuro. It requires the solver to use their head a little more, as it involves calculating and mental arithmetic. We start out with the usual 9 x 9 Sudoku grid. But this time instead of given numbers, we have smaller subsections divided by either dotted lines or different colours. These are called cages. Each cage is represented by a number in the top corner of the first cell in that cage going across. This number is the total of all the numbers in a particular cage. The rules remain as they are. One is to place the numbers 1 through to 9 once on each row, column and 3 x 3 square, while also ensuring that all the numbers in the cages add up to the given total. One must also NOT repeat any numbers within a cage, even if they don't occupy the same row, column or 3 x 3 square. This type of puzzle can seem hard when you first look at it and it may remain that way. But it primarily depends on the difficulty and whether we have the right level of skill.
- If a cage covers two cells and the total is a 3, the only numbers that can occupy the two cells in that cage are 1 and 2 (as they both add up to make 3). A cage with 3 is only ever going to cover two cells.
- If a cage covers two cells and the total is a 4, the only numbers that can occupy the two cells in that cage are 1 and 3 (since 2 and 2 is repeating the same number). A cage with 4 is only ever going to cover two cells.
- If a cage covers two cells and the total is 17, the only numbers that can occupy the two cells in that cage are 8 and 9 (as they are the only two numbers that can reach the total).
- If a cage covers two cells and the total is 16, the only numbers that can occupy the two cells in that cage are 7 and 9 (since 8 and 8 is repeating the same number).
- Here are a few extra tips, though these may rarely occur:
1. A three-cell cage with a total of 6 = 1, 2 and 3
2. A three-cell cage with a total of 7 = 1, 2 and 4
3. A four-cell cage with a total of 10 = 1, 2, 3 and 4
4. A four-cell cage with a total of 11 = 1, 2, 3 and 5
5. A three-cell cage with a total of 24 = 7, 8 and 9
6. A three-cell cage with a total of 23 = 6, 7 and 9
7. A four-cell cage with a total of 30 = 6, 7, 8 and 9
8. A four-cell cage with a total of 29 = 5, 7, 8 and 9
These tips help narrow down options and provide answers. Once an answer is found, the rest of the answers will follow. Sort of like a chain reaction. No guesswork is required.
Arrow Sudoku
Arrow Sudoku is much like normal Sudoku and has a couple of similarities to Killer Sudoku. As usual, each number from 1 to 9 goes in every row, column and 3 x 3 square. This time however, many starting numbers are replaced with a set of arrows crossing over multiple cells. The arrows are joint by a circle that occupies a single cell. The number in this circle is the sum of each number on the arrow line. Numbers can be repeated on an arrow line so long as they are on a different row, column and 3 x 3 square. Arrow Sudoku is extremely challenging and requires one to have a strong level of concentration. They can take as long as a couple of hours to completely solve and answers are not as easy to find here compared to the other puzzles. I've often found myself having problems whilst attempting to solve these ones. I consider them to be my least favourite as a result of this. But then again, it depends on the solver having the mind for this type of game.
- A cell with a circle is never going to be a 1, as that is the smallest number in the puzzle. Nothing can add up to make 1 here.
- In most cases, one can rule out a 2 being in a circle as this is extremely rare. The only time this can happen is if the arrow crosses over two cells that do not share the same row, column or 3 x 3 square.
- The same goes for every even number. Even totals can have two of the same number placed along an arrow line, but only when that arrow crosses over two cells which do not share the same row, column or 3 x 3 square.
- The number 9 can never be placed on an arrow as it is the highest number one can use.
Calcudoku
Calcudoku is one of the more recent creations in Japanese logic puzzles. It usually involves a 6 x 6 grid with all 36 cells separated into cages of 2, 3 or 4. These cages are too accompanied by a number along with an operation sign right next to it. This number is a target and the operation sign represents the process we have to use in order to reach the target. For example, a cage of two cells with a target number of 10 and a + sign means the two numbers within the cage must add up to 10 (which in this case, can only be 4 and 6). To solve this puzzle, one must follow the target numbers and processes while still following the common rule of each number from 1 to 6 appearing once in every row and column. It's much like Killer Sudoku, except it is often smaller, excludes 3 x 3 squares and uses all four basic mathematical operations. This puzzle primarily tests our ability to calculate in our heads. It's not overly difficult, but does require some great focus.
It is not only design that sees Calcudoku being similar to Killer Sudoku. Both of the logic puzzles share a significant solving method. That being the need to focus on the smallest target numbers along with the largest. This method in Calcudoku however can be affected by the shape of any particular cage. Unlike Killer Sudoku, this puzzle allows for the same digit to appear in a cage twice (so long as the digit doesn't meet in both the same row and same column). Completing these ones generally require us to play around with the numbers and see which ones reach certain targets. The process of elimination for cells can play a role here as well, though there is always one obvious answer in front of us from start to finish. Here are a couple of tips one can take into a Calcudoku puzzle:
1. Focus on the cages first and try to see which group of numbers go where. Then look at the cells.
2. Here are a few obvious answers I gathered for a standard 6 x 6 Calcudoku puzzle:
2-Cell Cages
3+, 2× = 1 and 2
4÷ = 1 and 4
5÷ = 1 and 5
5-, 6÷ = 1 and 6
10× = 2 and 5
15× = 3 and 5
18× = 3 and 6
10+ = 4 and 6
20× = 4 and 5
11+, 30× = 5 and 6
3-Cell Cages
6+ = 1, 2 and 3 (1, 1 and 4 possible for a non-straight cage)
2- = 1, 3 and 6 (2, 2 and 6 possible for a non-straight cage)
4- = 1, 1 and 6 (only for a non-straight cage)
90× = 3, 5 and 6
108× = 3, 6 and 6 (only for a non-straight cage)
96× = 4, 4 and 6 (only for a non-straight cage)
120× = 4, 5 and 6
144× = 4, 6 and 6 (only for a non-straight cage)
150× = 5, 5 and 6 (only for a non-straight cage)
17+, 180× = 5, 6 and 6 (only for a non-straight cage)
NOTE: A straight 3-Cell cage is an 'I' shape. A non-straight 3-cell cage is an 'L' shape
Takuzu (Binary)
This is generally one of the much easier puzzles to solve. Takuzu only ever involves using two symbols. Most of the time it's the two digits that represent the common binary number system; zero ('0') and one ('1'). The aim is to simply have an even number of zeros and ones in every column and row while avoiding the same number appearing next to each other more than twice. Each row and column must also be different from one another. However, this won't be as much of a worry when following the small steps. Binary puzzles with the standard 10 x 10 grid usually take around approximately 10 to 20 minutes for me to solve depending on the day. Despite how easy they are to solve though, I personally find Takuzu a tad boring. There is minimal detail and it's nothing but two digits being used. That being said, I wouldn't stop others from solving these if they wanted to. There is every chance that they will enjoy them more than I do.
NOTE: Each method is matched to a picture underneath by the number. These examples are also based on a standard 10 x 10 grid.
1. The cell in-between two adjacent ones with the same number will always be the other number, since we cannot have three of the same number next to each other.
(e.g. 1-?-1 = 1-0-1, 0-?-0 = 0-1-0)
2. When two of the same number appear in adjacent cells, the cells on either end will always be the other number. (e.g. ?-0-0-? = 1-0-0-1, ?-1-1-? = 0-1-1-0)
3. Each row or column must have an equal number of the two digits. With the matching picture below, there are five ones and four zeros. This means the last cell in the row has to be a zero. (e.g. 1-0-0-1-0-1-1-0-0-1)
4. Most times in any row or column with three empty cells next to each other remaining, an answer can already be given. With the matching picture below, there are three zeros and fours ones given. This leaves two zeros and a single one left. If the very bottom cell of the column was a one, then that would leave the two zeros to fit the two cells directly above it. This would make for three zeros next to each other. Since this cannot happen, the very bottom cell has to be a zero.
Futoshiki
The final Japanese logic puzzle I am discussing here is none other than Futoshiki. This one is arguably the easiest of them all and only takes around 5 minutes to complete (referring to the standard 5 x 5 puzzle). All one has to do is list the numbers from 1 to 5 at least once in each column and each row so that all the sequences are different. But also coming with this puzzle are inequality constraints in the form of arrows representing both 'greater than' and 'less than'. These arrows must be followed in order to solve the puzzle correctly. Sounds like quite bit to take in, but it isn't. There are only 25 boxes with some already occupied by a few given numbers to work with. Plus the inequality arrows are more of an assistance than a constraint. With plenty of these arrows and a couple of tips in hand, one should really have no problem trying to solve a standard Futoshiki puzzle. It's a nice game to kill a few minutes off, whenever one is left waiting.
For anybody interested in Futoshiki, I'd recommended you take into account these four easy tips:
1. Since 5 is the greatest number in a standard 5 x 5 grid and there is no larger number, it won't ever be seen in a box with an arrow pointing to it.
2. Since 1 is the lowest number in any Futoshiki puzzle and there is no smaller number, it won't ever be seen in a box with an arrow pointing away from it.
3. Any box with a 2 and an arrow pointing away from it, tells us the box that same arrow is pointing to is a 1.
4. Any box with a 4 and an arrow pointing to it, tells us the box that same arrow is pointing away from is a 5.
My message to people is simple. Give these puzzles a go (or any puzzles for that matter). They are not school or work papers. They are actually a game that helps keep our brain active through concentration and logical thinking. They keep us occupied at times of boredom or when patience is required. These Japanese number puzzles are not as hard as they look and the rules are not as complicated as some make them out to be. Take it from someone who has done so many of them in the past few months. Despite my struggles with the occasional one or two over time, I generally have very little problem with most that I solve. And believe it or not, I managed to pick up a few new strategies when working on these puzzles (and numbers in general). Others will most likely learn a thing or two as well. Just try Sudoku out. Try all of these out. I don't expect them to be everybody's cup of tea, but it's worth the time. Consider this as me caring for the health everyone's mind.
The final Japanese logic puzzle I am discussing here is none other than Futoshiki. This one is arguably the easiest of them all and only takes around 5 minutes to complete (referring to the standard 5 x 5 puzzle). All one has to do is list the numbers from 1 to 5 at least once in each column and each row so that all the sequences are different. But also coming with this puzzle are inequality constraints in the form of arrows representing both 'greater than' and 'less than'. These arrows must be followed in order to solve the puzzle correctly. Sounds like quite bit to take in, but it isn't. There are only 25 boxes with some already occupied by a few given numbers to work with. Plus the inequality arrows are more of an assistance than a constraint. With plenty of these arrows and a couple of tips in hand, one should really have no problem trying to solve a standard Futoshiki puzzle. It's a nice game to kill a few minutes off, whenever one is left waiting.
1. Since 5 is the greatest number in a standard 5 x 5 grid and there is no larger number, it won't ever be seen in a box with an arrow pointing to it.
2. Since 1 is the lowest number in any Futoshiki puzzle and there is no smaller number, it won't ever be seen in a box with an arrow pointing away from it.
3. Any box with a 2 and an arrow pointing away from it, tells us the box that same arrow is pointing to is a 1.
4. Any box with a 4 and an arrow pointing to it, tells us the box that same arrow is pointing away from is a 5.
My message to people is simple. Give these puzzles a go (or any puzzles for that matter). They are not school or work papers. They are actually a game that helps keep our brain active through concentration and logical thinking. They keep us occupied at times of boredom or when patience is required. These Japanese number puzzles are not as hard as they look and the rules are not as complicated as some make them out to be. Take it from someone who has done so many of them in the past few months. Despite my struggles with the occasional one or two over time, I generally have very little problem with most that I solve. And believe it or not, I managed to pick up a few new strategies when working on these puzzles (and numbers in general). Others will most likely learn a thing or two as well. Just try Sudoku out. Try all of these out. I don't expect them to be everybody's cup of tea, but it's worth the time. Consider this as me caring for the health everyone's mind.
Travis "TJ" James
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