Puzzle Genius

Sudoku is great fun whatever your age. But when it comes to younger people, this classic logic puzzle can be a fantastic educational tool. Like all the best learning tools, sudoku works well precisely because it is so much fun to play. Kids learn best when they are enjoying themselves, and sudoku has a lot to teach — and not just about numbers. Indeed sudoku, whilst traditionally using numbers, is not a math game. But it has a whole lot to teach.

At the bottom of this page you will find some free sudoku puzzles designed specifically for kids, that you can download and print out. But first, here are seven incredible benefits of using Sudoku as a learning tool.

1. It Reinforces Figure — And Letter — Recognition

Right from an early age, very simple kids sudoku puzzles are an excellent way to promote and reinforce the recognition of number forms. Even the simplest 4x4 puzzles are great at this.

By turning recognition into a game, the child is not only gently encouraged to differentiate between figures, but because they must find missing numbers, they will naturally create figures in their mind’s eye. This mental creation of numbers strongly reinforces the forms.

Of course, sudoku doesn’t just have to be played with numbers. Letters can be used instead, adding more learning opportunities. We’ve included both number and letter variants in our free downloadable kids sudoku puzzles at the bottom of the page.

2. It Teaches Problem Solving Skills

The aim of sudoku is to work out the missing numbers in a grid. The whole game is a puzzle that is crying out to be solved, so naturally playing it encourages and develops problem solving skills.

This can be done as gradually as necessary. A simple grid with a single missing number might seem to be so easy as to be pointless, but it’s like a gateway drug. When a child works out the missing figure, they experience a rush of excitement at having solved the problem; they are primed to solve more.

Building up the difficulty slowly and steadily maintains the challenge. The child is obliged to add a little more effort every time, and think up new ways of finding the answer — and being rewarded with the dopamine hit that comes with success.

As puzzles grow in size, complexity, or both, the child will have to find new ways to solve them. Thus what started as an easy game can soon become a fun and rewarding exercise in lateral thinking.

3. It Promotes Teamwork

Larger sudoku puzzles (typically full-size 9x9 and above) are a fantastic tool for encouraging working within a group. Puzzles can be split into racks and stacks, or columns, rows, and blocks, and each piece assigned to one or more children.

With a simple grid, the kids may initially solve the puzzle by working individually on their own portion. But ramp up the difficulty even just a little, and before long they will be obliged to co-operate and communicate to ensure their solutions do not ‘collide’ with those of the others in the group.

Take the difficulty up another notch, and the team will be encouraged to work together to come to a solution for the puzzle, pooling their techniques and knowledge.

For larger groups or older kids, try using 16x16 grids, or even better, Samurai Sudoku. The latter is a ‘multi-sudoku’ game with interlocking grids — perfect for splitting up and working on as a team.

4. It Encourages Attention to Detail

In a sudoku grid, a single mistake inevitably leads to disaster. Just one number out of place renders the entire puzzle unsolvable — not that it’s always immediately obvious!

It only takes a few failed solutions for most children to learn that they must check and double-check their answers before writing them in the grid, thus promoting careful attention to detail.

5. It Expands Memory

Logic is essential in solving sudoku, but so is memory. As they work through a grid, a child will be constantly putting numbers into very short-term memory, sometimes for just a few seconds at a time.

Sudoku is a great workout for the brain. Just as concentrated exercise can improve overall fitness, so working short-term memory improves overall memorisation and recall skills. Speaking of working out the brain…

6. It Improves Focus & Concentration

Sudoku demands a level of concentration that just isn’t necessary for most other kinds of puzzles. Simple math games, crosswords, word searches and so on, can all be done piecemeal by dipping in and out as and when. But to solve a sudoku grid effectively, it’s necessary to hold a lot of information in short term memory at once.

Losing focus, or lacking concentration, leads to mistakes or quite simply not being able to find a solution. Therefore the child is obliged to put all their attention into the job in hand. Studies show that concentration is like a muscle, and that repeated training leads to long-term improvement.

7. It Builds Confidence

If you’ve completed a sudoku puzzle then you know the rush of satisfaction that comes with putting that final number in the grid. One of the amazing things about sudoku is the range of difficulty that can be applied to a single concept. A child can learn the basics on a really easy 4x4 grid in a matter of minutes, yet be constantly challenged and stretched by the exact same set of rules right up to mind-bending super difficult 16x16 grids. Every win is an opportunity to boost their confidence and self-esteem, all whilst having lots of fun.

Download Free Sudoku Puzzles For Kids

Now you know why sudoku is such a great learning aid, as well as being a fun game, here are some grids that we have prepared especially for children.

We’ve included three grid sizes: 4x4, 6x6, and regular 9x9. There are eight 4x4 puzzles, and twelve of each of the larger sizes (which also include letter-based variants). Full solutions are of course also included.

Kids Sudoku Grids To Download & Print

Kids 4x4 Sudoku Grids

Kids 6x6 Sudoku Grids

Kids 9x9 Sudoku Grids

Right click or long-tap and Download Linked File or click or tap to open in a new window then choose Print from your browser.

The pages have been formatted so they will print on both American letter paper, as well as standard A4.

Looking For More Puzzles?

We’ve got you covered! We publish a brand new free puzzle every day. And of course, we have a large range of sudoku puzzle books of varying difficulty and size.

For kids sudoku, we highly recommend Amelia Baker’s range of books, which we collaborated on. You can find out more about those here. Amelia’s books include excellent tutorials written specifically for younger players, and lots of puzzles from 4x4 to 9x9.

Looking for a handy sudoku reference that you can print out and keep? Look no further! Our printable sudoku rules are just the ticket. Download the PDF below and print at home (or at work — we won’t tell your boss if you don’t!)

The PDF has been formatted to print nicely both on international A4 and US Letter paper.

Click or Tap Here For Printable Sudoku Rules

Sudoku For All Levels

Don’t forget that Puzzle Genius offers sudoku books for players of all levels. Be sure to check out the whole range here.

Five Methods For Finding Your Way Through Any Maze or Labyrinth

Mazes are a fascinating kind of puzzle. Completely unlike symbol-based teasers such as sudoku or suguru, they present a different kind of challenge to the brain. According to at least one neuropsychobiology study, solving mazes activates a network within the brain from the visual to parietal regions. Working these puzzles even activates subcortical and cortical motor areas — areas normally associated with movement and coordination. To the brain, solving a maze on paper is like walking through a real, physical labyrinth.

Looking for some good mazes that will challenge even the smartest brain? Look no further! Mazes for Smart People is our collection of 100 huge mazes. With five levels of difficulty and four maze types, it will keep you busy for hours.

How can we go about solving mazes? Are there tricks and techniques that make the process easier? Or must we resign ourselves to trying every path, every twist and turn, until we eventually emerge at the exit? The brute force approach will work, and for some people that’s enough. Smarter minds seek efficiency though.

Fortunately there are techniques we can employ to help us find the solution more quickly. However, these methods are only ever an aid to brute force. There is no single magical method that will always lead you to the correct solution first time. Mazes are not sudoku and cannot be solved first time with logic alone. A well-designed maze always requires a little bit of trial and error. It’s all part of the fun.

With that said, let’s dive in and look at five different methods you can use to solve almost any maze.

1. Start At The End

Let’s begin by saying right now that this method won’t work with all mazes. At Puzzle Genius we design our mazes in such a way to completely negate this method. Are we evil? No, we just want to make good mazes that present a real challenge! Not all maze-setters are so conscientious.

Here’s a simple maze, typical of the kind you might find in a kids activity book.

Simple maze

If we begin at the start of the maze, we are immediately faced with a choice — left or right? If we go right, we have another choice — down or straight on? And on it goes. The maze has been front-loaded with branches, designed to confuse you from the off.

But what happens if we start at the end? There’s only one possible path, and we can follow it for more than half the puzzle before we get to a branch — in the blue circle below:

Simple maze solution

After that there are only three more choices to make before we reach the goal. In each of those decisions it’s easy to see the correct path and where there is a dead end, because we are so close to the end of the maze.

Lots of mazes are designed this way — front-loaded with branches designed to confuse you at the start. This example is a very simple maze, but even more complex mazes can suffer from this ‘problem’ (in quotes because not everyone will see it as a problem — some may say it’s an opportunity).

Starting at the end then, is a technique that’s always worth a try.

2. Hand on The Wall

This is probably the most well-known method for maze-solving. It’s usually suggested for physical labyrinths like the corn mazes favoured by farmers around the world, or the box hedge mazes found in the gardens of stately homes.

The technique is simple: when you enter the maze, place your right hand on the wall to your right (or left hand on the wall to the left), and keep it there as you move through the maze. Eventually you should come to the exit.

We say should because this theory, while sound, does not work in all mazes. It will only get you through a maze that can be deconstructed into a single line. By which we mean a maze that you could draw in one go without taking your pen off the page.

The maze in the example above, for instance, is a single-line maze. Another way of thinking of this type of maze is to imagine that it is made from a giant piece of spaghetti — or a long rope if you prefer. You could lay down your pasta, shaping it and twisting it around the corners until you had created the maze.

Here’s a super-simple maze that you can try this technique on:

A very simple single line maze

The first thing to note is that this is a single-line maze. You can put your finger on the top right corner and trace around every line of the maze without lifting it off. You could, given a long enough piece of spaghetti, recreate this exact maze without breaking it (though you would need do make some tight folds as you doubled it back on itself).

The reason the hand on the wall technique works with a maze like this is because it is made from one path, you can effectively trace your way around the whole maze in one go.

Here’s how that looks on our example. If we pictured ourselves walking into this maze from the top and placing our right hand on the right-hand wall and tracing a line with it as we went, this is the line we would draw:

A very simple single line maze with a hand-on-wall line overlaid.

Is it efficient? Well, it’s clearly not the quickest route through the maze. But did it work? Hell yeah! Maze solved.

Had we started with the left-hand, we would have gone the quicker way — but hindsight is a wonderful thing and is of no help when starting our journey through a real, complex maze.

Remember, this technique only works on mazes that can be constructed from a single line. It won’t work on any that include islands, like this for example:

A maze with an island

Depending which side you started on, you could potentially find yourself going round and round the blue island forever! Islands like that are common in physical labyrinths, placed there purposely to defeat this simple but effective maze-solving technique.

3. Dead-End Pruning

This can be a time-consuming method, but it will always produce a clear path through the maze by the end of the process. The technique is simple enough — starting at the end of the maze, block off every dead end you find. Eventually only the one-true path will remain.

A visual example will make this clearer. Let’s begin with this simple maze:

A simple maze.

Starting from the bottom we can block off dead ends. We could do this two ways - either by simply barring them with a line, or by filling them in. We’ve done both ways here for demonstration purposes.

The same example maze with some dead ends pruned.

The method you use is a personal preference. Filling in dead ends makes it easier to see the remaining path, but it’s obviously more time-consuming and uses more ink or graphite!

As you get more practice with this technique you’ll find you can trace dead-ends back quite some way and block off several with a single stroke, rather than blocking every one individually. For example, these pink blocks are unnecessary - the green block takes care of all those little dead ends in one go.

Unnecessary dead ends.

As you can see, dead-end pruning is still a rather brute-force technique for finding your way through a maze. It’s fool-proof, but time consuming.

4. Sub-Mazing

This is essentially a variant of dead-end pruning. Consider the following maze:

A new example maze.

Notice anything in particular?

If you look carefully, you’ll find that almost half of the maze is a complete dead-end! We can trace a wall from one side of the maze to the other, effectively creating a sub-maze in which the correct path cannot possibly pass.

The same maze with a sub maze highlighted.

This is an exaggerated example to make a point — it’s rare to find such obvious sub-mazes. However, it is not uncommon to find large chunks of a maze that are one huge dead-end that can be cut out.

Here’s another thing to look out for:

A maze with something to look out for.

This time we don’t have a clean break between two parts of the maze, but we almost do, and that in itself is very helpful.

We can draw a line from the left to the right with just a single break in it. That means the path through the maze must go through that break. We can therefore split this maze into two sub-mazes.

The same maze with the single hole for the path highlighted.

5. Hansel and Gretel

If you ever read the story of Hansel and Gretel, this final method will make perfect sense to you. In the fairytale, our two heroes set out into the woods to escape the house of the evil witch armed only with some stale bread to help them find their way. By dropping a trail of crumbs as they went, they were able to see the paths they had already tried, and thus discount them every time they had to make a new choice about which direction to take.

We can use the same method to solve any maze. Instead of breadcrumbs we draw a line as we make our way through the maze, showing the path we have already taken.

At first glance this may sound like the hand on the wall method, but whereas that technique prescribes a very strict path through the maze (and can be confounded by islands, bridges and tunnels), the Hansel and Gretel method is more freeform and will always work, provided we follow one simple rule: never take a path we have already been down twice. The reason is simple: if we’ve been there and back, it must be a dead end.

This method leaves the choice of direction at every junction up to us. We can try to head in the general direction of the exit, rather than follow every twist and turn. And if we see that a particular path is a certain dead end, we aren’t obliged to trace our way around it anyway, the way we would with the hand on the wall approach.

Here’s an example of drawing a Hansel and Gretel path through a simple maze:

An example of a Hansel and Gretel breadcrumb trail through a maze.

Were we to use the hand on the wall method, we would have had to trace our way around the obvious dead-end number 1, and we would also have had to trace our line right to the end of the dead end number 2. With this approach we could just turn right around and retrace our steps. Similarly we could avoid dead ends 3, 4, and so on, continually working towards the exit.

If we come to a junction where we have to choose between a route that already has a breadcrumb line and one that doesn’t, we should choose the one that doesn’t.

Again, provided we never take a path with two breadcrumb lines, we will always find the exit, even in mazes with islands.

Which Method To Use

So there you have it — five different ways to find your way across a maze. Which is best? Which one should you use? Ultimately it’s down to you. If you want a guaranteed solution with maximum efficiency, then Hansel and Gretel is the way to go.

If, on the other hand, you enjoy the unknown and like working your way around blindly, but want to simplify the puzzle to make your life a little bit easier, then dead-ending or sub-mazing is your friend.

Ready For A Real Challenge?

All the example mazes used in this tutorial were, by necessity, very, very simple! If you’d like to try some techniques on a proper maze, then you can download and print a Puzzle Genius maze below. This is a level one maze, similar to those you’ll find in Mazes For Smart People. If you get stuck, we’ve also provided the solution to download in a separate PDF. Good luck!

Practice Maze To Print

Level 1 Practice Maze

Level 1 Practice Maze — Solution

Right click or long-tap and Download Linked File or click or tap to open in a new window then choose Print from your browser.

Knowing the combination of digits that can fit into a killer sudoku cage isn’t just a useful technique, sometimes it is absolutely necessary in order to solve or start a puzzle. Remembering common unique combinations is essential if you want to improve your time for solving killer sudoku puzzles. Unless you have a photographic memory though, you probably won’t memorise all of them, which is why this cheat sheet can be handy.

As well as cell cage combinations, we've included required digits further down. Some cells always require particular digits, regardless of the number combination that goes into them. Knowing these is a great way to eliminate candidate numbers from blocks, rows, and columns.

Is it cheating? We call it a cheat sheet, but is it really cheating? Only you can decide! Our view is that a reference like this is no more cheating than using a dictionary to check your spelling. For us, puzzles like killer sudoku are all about the logic and not an exercise in memory or recall.

New to killer sudoku? Be sure to check out our Killer Sudoku From Scratch tutorial.

Looking for some excellent killer sudoku puzzles? We have plenty of lovely killer sudoku books to keep you busy!

Cell Cage Combinations

These are all possible combinations of digits for a given cage size and sum. Bolded sums have only one combination.

2-Cell Cage Combinations

3 — 12

4 — 13

5 — 14 23

6 — 15 24

7 — 16 25 34

8 — 17 26 35

9 — 18 27 36 45

10 — 19 28 37 46

11 — 29 38 47 56

12 — 39 48 57

13 — 49 58 67

14 — 59 68

15 — 69 78

16 — 79

17 — 89

3-Cell Cage Combinations

6 — 123

7 — 124

8 — 125 134

9 — 126 135 234

10 — 127 136 145 235

11 — 128 137 146 236 245

12 — 129 138 147 156 237 246 345

13 — 139 148 157 238 247 256 346

14 — 149 158 167 239 248 257 347 356

15 — 159 168 249 258 267 348 357 456

16 — 169 178 259 268 349 358 367 457

17 — 179 269 278 359 368 458 467

18 — 189 279 369 378 459 468 567

19 — 289 379 469 478 568

20 — 389 479 569 578

21 — 489 579 678

22 — 589 679

23 — 689

24 — 789

4-Cell Cage Combinations

10 — 1234

11 — 1235

12 — 1236 1245

13 — 1237 1246 1345

14 — 1238 1247 1256 1346 2345

15 — 1239 1248 1257 1347 1356 2346

16 — 1249 1258 1267 1348 1357 1456 2347 2356

17 — 1259 1268 1349 1358 1367 1457 2348 2357 2456

18 — 1269 1278 1359 1368 1458 1467 2349 2358 2367 2457 3456

19 — 1279 1369 1378 1459 1468 1567 2359 2368 2458 2467 3457

20 — 1289 1379 1469 1478 1568 2369 2378 2459 2468 2567 3458 3467

21 — 1389 1479 1569 1578 2379 2469 2478 2568 3459 3468 3567

22 — 1489 1579 1678 2389 2479 2569 2578 3469 3478 3568 4567

23 — 1589 1679 2489 2579 2678 3479 3569 3578 4568

24 — 1689 2589 2679 3489 3579 3678 4569 4578

25 — 1789 2689 3589 3679 4579 4678

26 — 2789 3689 4589 4679 5678

27 — 3789 4689 5679

28 — 4789 5689

29 — 5789

30 — 6789

5-Cell Cage Combinations

15 — 12345

16 — 12346

17 — 12347 12356

18 — 12348 12357 12456

19 — 12349 12358 12367 12457 13456

20 — 12359 12368 12458 12467 13457 23456

21 — 12369 12378 12459 12468 12567 13458 13467 23457

22 — 12379 12469 12478 12568 13459 13468 13567 23458 23467

23 — 12389 12479 12569 12578 13469 13478 13568 14567 23459 23468 23567

24 — 12489 12579 12678 13479 13569 13578 14568 23469 23478 23568 24567

25 — 12589 12679 13489 13579 13678 14569 14578 23479 23569 23578 24568 34567

26 — 12689 13589 13679 14579 14678 23489 23579 23678 24569 24578 34568

27 — 12789 13689 14589 14679 15678 23589 23679 24579 24678 34569 34578

28 — 13789 14689 15679 23689 24589 24679 25678 34579 34678

29 — 14789 15689 23789 24689 25679 34589 34679 35678

30 — 15789 24789 25689 34689 35679 45678

31 — 16789 25789 34789 35689 45679

32 — 26789 35789 45689

33 — 36789 45789

34 — 46789

35 — 56789

6-Cell Cage Combinations

21 — 123456

22 — 123457

23 — 123458 123467

24 — 123459 123468 123567

25 — 123469 123478 123568 124567

26 — 123479 123569 123578 124568 134567

27 — 123489 123579 123678 124569 124578 134568 234567

28 — 123589 123679 124579 124678 134569 134578 234568

29 — 123689 124589 124679 125678 134579 134678 234569 234578

30 — 123789 124689 125679 134589 134679 135678 234579 234678

31 — 124789 125689 134689 135679 145678 234589 234679 235678

32 — 125789 134789 135689 145679 234689 235679 245678

33 — 126789 135789 145689 234789 235689 245679 345678

34 — 136789 145789 235789 245689 345679

35 — 146789 236789 245789 345689

36 — 156789 246789 345789

37 — 256789 346789

38 — 356789

39 — 456789

7-Cell Cage Combinations

28 — 1234567

29 — 1234568

30 — 1234569 1234578

31 — 1234579 1234678

32 — 1234589 1234679 1235678

33 — 1234689 1235679 1245678

34 — 1234789 1235689 1245679 1345678

35 — 1235789 1245689 1345679 2345678

36 — 1236789 1245789 1345689 2345679

37 — 1246789 1345789 2345689

38 — 1256789 1346789 2345789

39 — 1356789 2346789

40 — 1456789 2356789

41 — 2456789

42 — 3456789

8-Cell Cage Combinations

36 — 12345678

37 — 12345679

38 — 12345689

39 — 12345789

40 — 12346789

41 — 12356789

42 — 12456789

43 — 13456789

44 — 23456789

9-Cell Cage Combinations

45 — 123456789




Required Digits

These are digits that must be present somewhere within a cage for a given sum.

3-Cell Cage Required Digits

8 — 1

22 - 9

4-Cell Cage Required Digits

12 — 1,2

13 — 1

27 — 9

28 — 8,9

5-Cell Cage Required Digits

17 — 1,2,3

18 — 1,2

18 — 1,2

19 — 1,2

20 — 1,2

21 — 1

31 — 9

32 — 8,9

33 — 7,8,9

6-Cell Cage Required Digits

23 — 1,2,3,4

24 — 1,2,3

25 — 1,2

26 — 1

34 — 9

35 — 8,9

36 — 7,8,9

37 — 6,7,8,9

7-Cell Cage Required Digits

30 — 1,2,3,4,5

31 — 1,2,3,4

32 — 1,2,3

33 — 1,2,6

34 — 1

36 — 9

37 — 8,9

38 — 7,8,9

39 — 3,6,7,8,9

40 — 5,6,7,8,9

If you want to try your hand at suguru but don’t know where to start, you’ve come to the right place. Assuming no prior knowledge of this captivating puzzle, we’re going to look at how the grid works, cover the simple rules, then dive in to the techniques we can use to solve it. Let’s get started, by looking at…

The Suguru Grid

Suguru grids do not have to be square. It’s possible to make puzzles of various shapes and sizes. Here at Puzzle Genius we are purists at heart, and we prefer the perfection of the square form. But we do publish puzzles of differing sizes — and of course — different levels. To start us off, here is an easy 1-star grid, six cells across by six down. It’s taken from our popular Pocket Suguru series:

Suguru Grid Labelled

Like a sudoku grid, we have some numbers filled in to start us off (clues). But that’s where the similarity ends. You’ll see there are no columns, rows, or blocks here. Instead we have regions

There is no fixed number of regions in a suguru grid, and the regions are not of fixed sizes. Indeed those two variables, along with the number of pre-filled clue cells, contribute to the difficulty of the puzzle.

Suguru Rules

Suguru’s rules are very simple. In fact there are only two:

  • Every region must contain the digits from 1 to the number of cells within that region.
  • Neighbouring cells cannot contain the same digit.

Looking back at our example grid, we note that the green shaded region has five cells, so it must contain all the digits from 1 to 5.

When we talk about neighbouring cells in suguru, we mean any cell that touches another horizontally, vertically, or diagonally. Here’s our example grid again:

Excluded Cells

If we consider the green cell 5, all the cells shaded red are its neighbours, so none of them can contain the number 5.

Just as with sudoku, we can always solve a suguru puzzle with logic. Guessing is not a strategy.

Solving Suguru Puzzles

Just like sudoku, some cells in a suguru puzzle are very easy to solve right off the bat with nothing more than a simple application of the rules.

Our puzzle has a an obvious place to begin. The region in the bottom right has four cells and is missing a 2 and a 4:

A region missing 2 numbers

The presence of the 2 highlighted in red means we cannot put another 2 anywhere inside the red outline. That rules out one of our empty cells in the green region. Therefore the 2 has to go in the other one, in turn leaving only one place for the 4.

Filled in two cells

We can apply the same logic here:

Two more empty cells

Our green region is missing a 1 and a 4. The red highlighted 1 rules out the left-most free cell in our region, leaving only one place it can go. The 4 must therefore go in the other one.

Filled in cells

We can quickly fill in another couple of cells the same way:

Filled out more cells

The red highlighted 4 makes them easy to work out.

Elimination

Once we’ve filled in the most obvious cells, we can move on to using an elimination strategy. Consider the region and cell highlighted:

A highlighted region

At first glance this looks a little more complicated because now we have three cells within a region that we need to fill in. But actually we can apply the same logic to eliminate certain possibilities.

Highlighted 3 and 2

The red 3 and 2 prevent us from putting either of those numbers in our cell in question, which only leaves the 1.

Elimination By Implication

There’s another way we can use elimination, and it’s a bit more subtle. So far we’ve used pre-filled and solved cells to eliminate candidates. But in suguru we can also eliminate candidates based on implication. Consider the region with no numbers in it at all:

A completely empty region

We know we need to find all the digits from 1 to 5 to fill this region. We can use the neighbouring (red) region to help us get started. The red region needs a 2 and a 3 in order to be completed. Given how our green region is partially surrounded by the red one, we can automatically eliminate 1, 2 and 3 as candidates for the cell with a “?” inside, and of course we can eliminate the 5 as well. Even if we didn’t already have the 1 and the 5 filled in inside the red region, we could still eliminate 1, 2, 3 and 5 based purely on implication. Putting any one of them into our “?” cell would make the red region impossible to solve! Therefore we can deduce that the only possible digit that can go there is 4.

Filled in the 4

That’s a simple example of elimination by implication, but it’s a powerful technique that can be used to solve regions that at first glance appear to have no clues to go on. To see how powerful, here's an example from a different puzzle:

No clue numbers

It looks like we have no clues at all to help us here, but of course that’s not true. There are certain things we do know. The green region has five cells, so requires the digits 1-5. The red region has four cells so needs 1-4. Elimination by implication tells us what the “?” cell contains. Given its position surrounded by the red region, it cannot contain a 1, 2, 3, or 4. If it did, it would be impossible to put any of those digits into the red region. Therefore it must contain a 5 - it’s the only option.

Solved

Looking back at our example puzzle, it can easily be solved now with simple elimination.

The solved suguru puzzle

Note Taking

Just as with sudoku, writing in small numbers to show candidates for a cell can be a useful thing to do — especially on harder grids. It can ease cognitive load and show you opportunities that might not be obvious otherwise. Consider this puzzle:

A more complex suguru puzzle

We’ve already solved the 5 in the green region based on elimination by implication. Where do we put the 1 and the 3 though? For now we don’t know, so we can write them in as little numbers:

Little numbers in a suguru grid

Now when we come to consider the green cell, we can immediately see that it cannot be a 1, 2, 3 or 4, so must be a 5. Yes, we could have worked this out without writing in the small numbers, but when working bigger puzzles with larger regions, taking notes like this can make it easier to spot answers as your notes will ripple out across the puzzle. When that happens, some combinations of notes will eliminate candidates in others, solving cells that then ripple back through your earlier notes.

Wrapping Up

Suguru is all about elimination. Elimination by implication can give you a quick ‘in’ to fill out some extra clue cells, helping you get started. Taking notes on harder puzzles will uncover hidden elimination opportunities.

If you find you get stuck in a puzzle, work through noting all possible candidates in the empty cells, and you will find pairs of candidates that, when combined, eliminate candidates from pairs of neighbours. 

To help you practice, we’ve put together a set of easy level suguru puzzles you can print out, along with the solutions in a separate PDF. You can download both below.

If you enjoy suguru and would like enough puzzles to keep you going for an entire year, then our A Year of Suguru books are for you. Large grids, and a daily rotation through five levels of difficulty mean you’ll have a fresh challenge every day of the year. Choose from Easy-Intermediate and Intermediate-Hard editions.

Practice Grids To Print

Suguru Practice — Grids

Suguru Practice — Solutions

Right click or long-tap and Download Linked File or click or tap to open in a new window then choose Print from your browser.

If you’ve never played killer sudoku but want to get started, or if you’ve taken a look at a killer sudoku grid and wondered how on Earth you’re supposed to begin solving it, then you’ve come to the right place. 

Killer sudoku is like regular sudoku but with added mathematics. Unlike the classic game, there’s arithmetic involved. If that’s not your thing, killer sudoku probably isn’t for you!

A quick word of caution before we get started: killer sudoku is very much based on regular sudoku. You really need to know how to play the classic version before tackling this kind of puzzle. So if you’ve never solved a sudoku puzzle before, I would recommend reading through our Sudoku From Scratch guide, and doing some simple puzzles first (there are some free ones to download included in our guide).

With that said, let’s dive in and look at a killer sudoku grid, and the rules that govern the puzzle. Here’s an easy grid to start us off — it’s a 1-star grid from our Pocket Killer Sudoku range.

Killer sudoku grid

There are lots of things that are familiar about this grid if you are used to sudoku:

  • It’s nine cells across by nine cells down.
  • The grid can be split into nine rows and nine columns.
  • There are nine blocks - sometimes called boxes, regions, or nonets
  • Although we can’t see it here, the grid will ultimately be filled with the numbers 1-9.

There are also a couple of very obvious differences between a killer sudoku puzzle and the regular variety.

  • Killer sudoku has an extra type of region, which we call a cage (highlighted above in yellow). Cages are delimited by dashed lines. They are not restricted to columns, rows, or blocks - they can cross those boundaries.
  • None of the final digits are filled in. Instead we have a lot of little numbers to provide our starting clues.

The Rules

The first rule of killer sudoku is the same as sudoku:

  • Every column, row, and block must contain the digits from 1 to 9 once, and only once.

To this we add a new rule:

  • The sum of the digits within each cage must equal the clue number shown in that cage.

There’s the arithmetic; every killer sudoku puzzle involves a bit of adding up.

Looking back at our example grid above, we can see that the yellow cage has a little 8 written in the top cell. That means that the three numbers that go in that cage must add up to eight.

Whether you want to do your killer sudoku arithmatic in your head, or with the aid of a calculator, is up to you. But beware: any mistakes in adding up clue numbers will lead to errors that make the puzzle impossible to solve!

Solving Killer Sudoku

Solving killer sudoku puzzles requires a number of techniques. Some of those are the same as regular sudoku. Hidden and naked singles, single free cells, matching pairs and so on, all work exactly the same. But of course to use those we need to have some cells filled in, and when we begin a killer sudoku there aren’t any. So we can’t begin solving a killer sudoku puzzle the same way we would with a regular grid. We have to find another means of making headway. We need some specialist techniques, and that’s what we’re going to learn now.

Killer Sudoku Techniques

The 45 Rule

When we add up all the digits from 1 to 9, we get 45. We can therefore deduce that the sum of every row in a killer sudoku grid must be 45, that the sum of every column must be 45, and the sum of every block must also be 45. How does this help us solve a puzzle? Consider our example grid again:

the 45 rule

Looking at the top left block we notice that all but one of the cages are fully contained within it. The 9 cage spills over into the next block. We know that the sum of the block must be 45. We can add up the clue numbers (12+21+10+9) to get 52. That’s bigger than 45 - the maximum sum possible in a block. Therefore everything over and above 45 must fit into the single cell that spills out of the block. 52 minus 45 is 7, so that cell must contain a 7. Which in turn means the other cell in the cage has to be a 2:

Filling in two cells

Naturally we can use the same technique on rows and columns as well as blocks. 

Sometimes cages don’t spill out out an area, they spill into one. Here’s an example from another 1-star grid:

Spilling into a block

The principle is exactly the same. We add up the sum of the cages wholly contained within the block, to work out the value of the cell that has spilled into it.

6 + 8 + 8 + 14 = 36. We need all the cells in the block to add up to 45, so the cell that has spilled into it (the only one not shaded green), must contain a 9 (45 - 36 = 9):

Filling in the inside spill

Looking for places to apply this 45 rule is a good way to start a puzzle because it allows us to fill in some cells very quickly. Our example puzzles here are 1-star (easy) level. Harder puzzles won’t give up their secrets so easily. However, it’s worth keeping an eye out for places to apply this rule as you progress through a puzzle, because as cells get filled in, they can be used in the sum calculations, opening up more places to use this simple technique. 

Multiples of 45

We can take the 45-rule further by applying it to multiple columns, rows, or blocks simultaneously:

Two columns combined

Neither of these columns alone are candidates for the 45 rule, but by combining them we find we have a single cell spilling inside. Working out its value is simple. We know the sum of the highlighted columns must be 90 (2 columns x 45). If we add up the clue numbers we get 85. Therefore the cell spilling into the columns must contain a 5, because 90 - 85 is 5.

Filling in the empty cell

Here’s another example, this time combining three blocks together:

Combining three block

Adding all the clue numbers together gives us 137. As we have three blocks, we know the total should be 135 (3 x 45), so the extra 2 must go in the cell spilling out:

Filling in the spill out cell

That in turn tells us we can put a 1 in the second and last remaining cell in that cage:

Filling in the cage

Limited Sum Combinations

For any given cage, there are only a limited number of digits that can legitimately add up to its clue number. This block contains a cage with an 3 in it, and it’s made of two cells. For brevity, we notate this as a 3(2) cage.

A 3(2) cage

The only numbers that add up to 3 are 1+2. Therefore one of the cells must contain a 1 and the other must contain a 2. On its own this information doesn’t allow us to solve either cell definitively, but it narrows them down to two possible digits each, and we can write those in as little numbers, just like in sudoku:

Writing in little numbers

We know from sudoku that matching pairs like this are very useful. This one allows us to eliminate the numbers 1 and 2 from the rest of the row and and the rest of the block. 

Now let’s consider the example of an 8(2) cage (so that’s a two cell cage with an 8 written in it). Here are the possible combinations of numbers that could go in those cells:

  • 1 and 7
  • 2 and 6
  • 3 and 5

Why not 4 and 4? Because no two cell cage could possibly contain the same digit twice — it would mean putting the same digit twice into a column or row. 

Knowing that each cell in an 8(2) cage could contain a 1, 2, 3, 5, 6 or 7 isn’t very useful. But other cages are much more restricted - as was the case with our 3(2) example earlier. To save you working out other combinations yourself, here are all the two-cell cages that can only contain two digits:

  • 3(2) - 1 or 2
  • 4(2) - 1 or 3
  • 16(2) - 7 or 9
  • 17(2) - 8 or 9

If you want to see the limited sum combinations for larger cages, take a look at our killer sudoku cheat sheet.

Required Digit Sums

There are certain sum combinations that always contain certain digits. For example, our yellow highlighted 8(3) cage right at the top of the page could comprise any of the following:

  • 1 and 2 and 5
  • 1 and 3 and 4

Knowing that any of the three cells in the cage could contain a 1, 2, 3, 4 or 5 is not very handy. But knowing that one of them must contain a 1 is much more useful. It allows us to write a little 1 into each of those three cells, thus eliminating it from the rest of the block. 

Writing in little 1s

We’ve put the required digit sums into our cheat sheet too, to save you some more time. As with limited sum combinations, learning some of these by heart is a sure way to speed up your solving skills.

Combining & Reducing

We can push a puzzle a long way just by combining the techniques we have looked at so far, along with using what we already know about sudoku. Let’s work through a quick example. Consider the middle block of this right-hand stack:

Combining techniques

The first thing we can and should do, is work out the contents of the cell from the 15(3) cage that is spilling inside this block. Adding up the cages wholly contained in our block yields 41, so the intruding cage's cell must contain the missing 4 to bring us up to the requisite 45.

Filling in the spill in cell

Next, based on what we know about limited sum combinations, we deduce that the 17(2) cage can only contain an 8 and a 9. And we can also work out that the 3(2) cage can only contain a 1 and a 2. Look what happens when we write in those little numbers:

Combining little numbers

Knowing that this block already has the digits 1, 2, 4, 8 and 9 taken care of, reduces the possible candidates for the other cells. So whereas an 8(2) cage could normally take candidates 1,2,3,5,6 and 7, here we can reduce that list to 3,5,6 and 7. And in fact if we zoom out and look at the stack again, we notice that there’s already a 5 in the middle column, so we can reduce the options even further in half of the cage.

Zooming out

The same logic applies to the 13(2) cage. Normally it could hold a 4,5,6,7,8, or 9. Given what we know must already be in the 17(2) and 3(2) cages, and the cell that’s already filled in, we can reduce those candidates to 5,6, and 7. And again, the 5 already in the column reduces the options further in the first cell in that cage:

Candidate numbers for the 13(2) cage

Working across the puzzle in this way, combined with regular sudoku techniques, is enough to solve easy-intermediate puzzles like the examples presented here.

Wrapping Up

The key to solving easy killer sudoku puzzles is to use all the tools we’ve covered, to eliminate possible candidates. Often you will find that you can reduce and reduce the possibilities within a cage so far that you solve one cell in the cage. That in turn reduces the possible candidates for other cells within the same cage, and so on.

Remember to always look at the repercussions of each cell you solve — in killer sudoku even more so than in regular sudoku, solving a single cell can have huge repercussions that ripple out across the whole puzzle.

What starts off as a seemingly impenetrable grid, can very quickly evolve into something that becomes easy to solve as you combine the killer sudoku sum clues with regular sudoku techniques.

If you want to practice solving 1-star puzzles, we’ve put together a set you can download and print out below, as well as the solutions to check your answers. And of course, we have a full range of high-quality killer sudoku books, including our popular A Year of Killer Sudoku - with a new puzzle every day for a year.

Practice Grids To Print

Killer Sudoku Practice — Grids

Killer Sudoku Practice — Solutions

Right click or long-tap and Download Linked File or click or tap to open in a new window then choose Print from your browser.

Part Three - Speeding Up

In part one of this tutorial we began by looking at the basics of sudoku, using smaller than normal puzzles. Part two took us through using racks and stacks to solve 1-star level full size puzzles. Now it’s time to ramp things up to the next level - literally.

We’re going to look at a 2-star grid and learn a couple of techniques that will help you solve puzzles faster, and prepare you for even higher-level puzzles along the way.

If you’ve arrived here without any knowledge of sudoku, I’d highly recommend going back and covering those earlier sections first - everything here will make a lot more sense that way.

Levelling Up

Here’s a 2-star puzzle, taken from our Pocket Sudoku — Classic range:

2-star sudoku grid

If you want to follow along yourself, you can download this grid to print out at home here.

A quick glance at the puzzle tells us there aren’t any easy single free cells to fill in - every row, column, and block has at least two empty cells. So we begin by working the racks and stacks. This time I’m going to do the racks first, because why not? There’s no right or wrong way to attack a sudoku puzzle.

Working across the top rack, we can fill in a 2 and a 4. When we get to the 6, there are two possible cells it could go in:

Top rack

We could move on and look for the next number — it’s what we did back in part two of this tutorial in situations like this. But there’s a better way. Instead, we will write a tiny little number 6 in both the candidate cells, like this:

Two tiny 6s

You’re probably wondering how this will help. Don’t worry, we’ll come on to that in a bit. First though, here’s some sudoku etiquette about how to write in little numbers. It’s best to always write them in the ‘correct’ place within a cell, because as we work through the puzzle we may end up putting more than one little number in a cell. Just jotting them down willy nilly will make our puzzle messy and hard to read. It’s better to have a system. To figure out where to write little numbers, just imagine the cell is filled with all the possible values, like this:

all the little numbers

Sticking to this convention really makes working with the puzzle easier. By the way, it’s best to use pencil for solving sudoku puzzles because later you’ll want to erase the little numbers. It’s also good to have cells big enough to put several little numbers in without having to write microscopic digits. It’s why all Puzzle Genius sudoku books have generous sized grids - even our popular Pocket Sudoku range.

Back to our puzzle. There’s nothing more we can do with the top rack, so we move on to the middle one. Straight away we see that we have two candidate slots for the 1, so we can write in two little numbers:

Candidate cells

Working through the rack we can fill in the 3, 4, 7 and 8. This is what the grid looks like so far:

The grid so far

On to the third rack. The first opportunity we have to write anything in is the number 8. Again we have two candidate cells. We write in the little numbers, like this:

Candidate cells

Now look what happens when we reach the number 9:

The nines

There’s a 9 in the middle block and top row (of the third rack), and there’s a 9 in the third block and bottom row. Logic tells us we have to put a 9 in the first block and middle row of the rack. There are two empty cells to consider, but one of them is out of bounds because there’s already a 9 in the column that cell is in. That leaves just one candidate cell, and it already has a little number 8 inside it. Remember, those little 8s are there to tell us one of the two cells must contain a number 8. When we write our 9 into the green cell (the only place it can go), we are ruling out the possibility of an 8 going in there:

Ruling out the 8

Now the only place left to put our 8 is the one remaining cell with a little 8 in it. Writing in the little numbers earlier meant we got to solve two cells in one go! 

Solving two cells in one go

We would have found the 8 eventually, even without the little numbers. Our second tour through the racks — after we had worked through all the stacks — would have shown us where to put it. But doing it this way means we got to fill it in much earlier, saving us time and potentially making it easier to work through the stacks, because every completed cell makes the puzzle a little bit easier.

There are more benefits to using little numbers, and we’re going to cover one of those right now, as well as learning another technique that can speed up our sudoku solving powers!

Matching Pairs & Repercussions

Let’s continue with our example puzzle. I’ve gone ahead and worked through the first stack, and the middle stack as far as the number 8. This is what the grid looks like so far:

the grid so far

It’s starting to get a bit crowded - but don’t worry, all those little numbers are going to be a great help. Starting right now, when we add in the number 9 in the middle stack:

candidate cells

Looking back at the top left block, we notice the green cells both contain a little 7 and a little 9. When two cells in the same row, column, or block, contain two — and only two — little numbers, that tells us that they can only hold one of those two numbers. In other words one of those green cells must be a 7, and one must be a 9. There are no other numbers that could go there. And now that we’ve just filled in that 9 in the top row (middle stack), we’ve ruled out the possibility of the top green cell holding a 9!

ruling out a cell

The repercussion of filling in the 9 in the top block of the middle stack is that now we know which green cell holds the 7 (the top one) and which holds the 9 (the bottom one):

Filled in the 7 and 9

Repercussions are an amazing way to solve puzzles more quickly. The technique is simple: every time we fill in a cell definitively (that is to say with a full size number, not a little number), we look at the puzzle to see if that number helps us solve any other cells. The repercussion of filling in the 9 in the middle stack was that we could solve the 7 and 9 in the left stack.

Before we move on working to the last stack, we can see if the 7 and 9 we just solved have any repercussions themselves

The 7s in the left stack are already all solved, and we don't have enough information to solve the 7s in the rack. But guess what? The new 9 means we can solve the third 9 in the top rack:

Repercussion of a 9

Now we've solved all the 9s in the puzzle. But that’s not the only repercussion. The 7 and 9 we added earlier have given us a single free cell in a column:

A single free cell

Filling it in with the number 5 gives us three more repercussions:

  1. A single free cell in a block (green square, below).
  2. A missing 5 in the left stack (yellow square).
  3. A missing 5 in the bottom rack (purple square).

Repercussions of repercussions

Filling in those numbers in turn gives new repercussions, and so on as the empty squares fall like dominoes. In fact, just by following the repercussions, we can solve the whole puzzle! We don’t even have to work through the final stack, and we certainly don’t need to work through the racks a second time.

Why not try and complete the rest of the puzzle yourself? When you are ready, you can download the solution here.

Wrapping Up

You now have enough techniques under your belt to solve 1 and 2-star puzzles, and with a bit of effort some harder ones too. There are plenty more techniques to discover though, and filling in those little numbers is key to making many of them work.

If you want to practice solving 2 star puzzles, we’ve put together some 2-star puzzles you can download and print out below, as well as the solutions to check your answers. And of course, we have a full range of high-quality sudoku books, including our popular A Year of Sudoku - with a new puzzle every day for a year.

Practice Grids To Print

9x9 Sudoku 2-Star Practice — Grids

9x9 Sudoku 2-Star Practice — Solutions

Right click or long-tap and Download Linked File or click or tap to open in a new window then choose Print from your browser.

Part Two - Racks and Stacks

In part one of this tutorial we began looking at sudoku from the first principles. If you’ve never tried sudoku before, we’d recommend going back and reading that page because it covers the basic foundations. 

Here in part two we are going to move on to full-sized 9x9 grids - the kind you’ll find in most books, including ours (though we do have some bigger versions for the more adventurous).

The Sudoku Grid

Here’s an empty 9x9 sudoku grid:

lablelled 9x9 grid

It’s just like the little 4x4 grid we looked at before, only bigger. The grid uses all the single-digit numbers from 1-9. Each row, column, and block must have each number 1-9 once, and only once.

Solving Sudoku Puzzles

Now let’s look at a real puzzle. This is an easy one - a one-star puzzle from our Pocket Sudoku - Classic range.

Example 1

If you are new to sudoku, then at first glance a puzzle like this can look a bit intimidating. All those empty cells! What’s more, on closer inspection you might notice that unlike the little grids we looked at in part one, this one doesn’t have any rows, columns, or blocks with just a single empty cell.

Fear not. This really is an easy puzzle, and we’re going to deconstruct it to see why.

When we say deconstruct, we mean literally. We are going to break the puzzle apart and turn it into smaller bite-sized chunks that are much simpler to solve.

There are two ways we can break apart a sudoku grid. The first is by splitting it into three equal parts horizontally, along the thick lines, like this:

9x9 racks

We call each of these horizontal sections a rack. As with everything in sudoku, different puzzle designers have different names for things. Rack is pretty common, and it’s what we stick with here at Puzzle Genius, but you might see racks referred to as ranks or even (confusingly) rows, in other places.

Looking back at the original grid above, you’ll see there are thicker lines running top to bottom as well. Splitting the grid vertically along those gives us three stacks, like this:

9x9 stacks

Again, different folks may have different names. We stick with stacks. 

Now obviously we aren’t going to physically cut out a sudoku puzzle and chop it up like this. But thinking of a grid as smaller racks and stacks makes it easy to tackle the puzzle because we’re only looking at three rows, columns, or blocks, at a time

One Rack At a Time

To see how using racks and stacks is helpful, let’s work through our example puzzle, starting with the top rack. 

9x9 rack example 1

Step one of solving this grid is to look at where all the number 1s are in this first rack. There are two of them already filled in for us (highlighted here in blue). The first is in the first block and middle row. The second is in the second block and bottom row. We know every block and every row has to contain every number, once. So in this rack, logic tells us we need to put a number 1 somewhere in the third block, and somewhere in the top row. And guess what? There’s only one empty cell that falls in both the last block and top row. We have to put a 1 there. Like this:

9x9 rack example 2

We can repeat this step, this time looking for number 2s. But there aren’t any number 2s in this rack, you say. And you’re right! No need to worry, we can move on and look for number 3s.

9x9 rack example 3

There’s a number 3 in the first block and bottom row, and one in the last block and top row. We must be missing a number 3 in the middle block and middle row. And again, there’s only one empty cell in that block and row (the green one), so we have to put a 3 in it, thus ensuring each row and block has a single number 3.

As you’ve surely figured out by now, we look for number 4s next, but there aren’t any. The number 5 already appears in each row and block, so we move on to the 6s. And here we have a bit of a problem:

9x9 rack example 4

There are two 6s, and logic has narrowed down the possible homes for the third to two cells. Which one is correct? Right now we don’t know. We never guess in sudoku, because if we get it wrong we won’t know straight away and by the time we realise there’s an error in the grid, it will be far too late to work out where we made a mistake and we’ll have to start over. What then, are we to do? Simple: we carry on looking for the next numbers. We’ll come back to the 6s later. Right now we can check the 7s (they are all present already), and then the 8s:

9x9 rack example 5

There’s only one 8, so we can’t draw any conclusions about where to put the others yet. The same is true of the 9. 

We’ve done all we can with this top rack (for now), so we can move on to the second rack and repeat the process.

There’s a 1 we can fill in in the top row of the rack, but the 2s are inconclusive, and we don’t have enough 3s, 4s or 5s to work those out yet. Once again, the 6s present us with a minor dilemma:

9x9 rack example 6

With a 6 in the middle row and first block, and one in the top row and last block, we know we have to add one somewhere to the bottom row and middle block. There are two empty cells there, but unlike the first rack, this time we can work out which is the correct place. How? By looking at the puzzle as a whole again:

9x9 rack example 7

Looking at the entire grid allows us to consider not just the row and block, but also the column. And look - of the two cells we could consider for the 6, one of them is in a column that already contains a 6, so we know we can’t put it there. That only leaves one possible place for our missing 6:

9x9 rack example 8

The 7s are all present and correct, and we only have one 8 so can’t do anything with that. The 9 in the middle row and the 9 in the top row tell us we need to put a 9 in the bottom row (middle block), and now we've only one free place to put it, so we can fill that in:

9x9 rack example 9

We do the whole process again for the bottom rack, where we find a missing 3 and 8.

This is what the grid looks like after working through the three racks:

9x9 rack example 10

Working through the racks has allowed us to fill in enough empty cells that there’s now a column with just one number missing - a nice easy one to fill in. We also have an easy block to complete:

9x9 rack example 11

Here’s what the puzzle looks like so far:

9x9 rack example 12

Stacking Them High

With the racks done, we can turn our attention to the stacks. The principle is exactly the same - take each stack in turn and work through the numbers to see what we can fill in. First up, the 1s in the left-most stack:

9x9 rack example 13

With a 1 in the top block and right column, and another in the middle block and left column, we know we have to find a place to put a 1 in the bottom block and middle column. There are two free cells there, so which do we go with? Once again, we can look at the whole grid to tell us:

9x9 rack example 14

One of those rows already has a 1, so that only leaves one possible empty cell to put our missing 1.

After going through all the stacks using everything we’ve learned so far, our grid looks like this:

9x9 rack example 15

We’ve made some impressive progress, but obviously we’re not done yet. So how do we move forward? We could go back and go through the racks again. Having added in some extra numbers by working the stacks, we will almost certainly find that re-doing the racks will yield more answers. Indeed repeatedly working through racks and stacks might get us all the way to the end.

The Bigger Picture

Before we do that though, looking at the puzzle as a whole shows us there are some easy cells to fill in - single free cells within rows, columns, blocks, or a combination:

9x9 rack example 16

Filling all of those in makes the grid look like this:

9x9 rack example 17

Now we’ve got even more single free cells to fill in. I said this was an easy puzzle! If we filling those, we get to this:

9x9 rack example 18

With no more single free cells, we can go back and work the ranks and stacks. At this late stage in the puzzle, we’ll find that often filling in a number as we work through the racks and stacks provides us with some easy single free cells we can fill in as well along the way. 

Following everything we’ve covered so far, by re-working the racks, stacks, and single free cells, we can quickly complete the puzzle.

9x9 rack example 19

The End?

Congratulations! You now know enough sudoku to complete easy-level puzzles. Of course, there’s more to sudoku than just checking racks and stacks. Anything harder than easy-level grids will require some extra techniques.

To reach intermediate level, we are going to have to take things a little further. Part three of this series dives into a new technique that will take your sudoku-solving skills to a whole new level.

In the meantime, if you want to practice what you’ve learned so far, we’ve put together some easy puzzles you can download and print out, along with the solutions in a separate PDF.

And of course, we have loads and loads more puzzles in our books. Our Pocket Sudoku - Classic series is perfect for beginners. With five different levels of difficulty, you’ll find plenty of easy puzzles like the example we’ve just looked at. And as you learn more techniques, the higher levels will keep you busy too. All in a beautifully decorated, pocket-sized book you can slip into a pocket or bag and take anywhere.

Practice Grids To Print

9x9 Sudoku Easy Practice — Grids

9x9 Sudoku Easy Practice — Solutions

Right click or long-tap and Download Linked File or click or tap to open in a new window then choose Print from your browser.

Part One — Super-Simple Sudoku

If you’ve never played sudoku but want to get started, or if you’ve ever looked at a sudoku grid and found it intimidating or impenetrable, then you’ve come to the right place! This tutorial series assumes zero knowledge of the popular puzzle game. We’ll take you from scratch to being able to solve most puzzles up to intermediate level.

Sudoku is great fun, and is actually really easy to learn. It’s a good way to relax, to keep your mind sharp, and can be the perfect distraction from the difficult times we find ourselves in.

Contrary to popular belief, sudoku is not a mathematical puzzle. That it uses numbers is almost incidental. The numbers are just symbols, and in fact any kind of symbols can be used. Some kids sudoku grids use shapes or icons. Some sudoku puzzles use letters (these are usually called wordoku). Numbers are by far the most common symbol used in sudoku though, so that’s what we will stick with here.

The aim of the game is very straightforward: to find all the missing numbers in a partially filled grid of numbers.

To get started, we’re going to begin with a smaller than usual grid of numbers. It makes it easy to grasp the rules. What’s that, you say? What are the rules? Great question!

The Rules

The rules of sudoku are encapsulated in its name, which comes from the Japanese Sūji wa dokushin ni kagiru, which means the numbers must be single. In sudoku, each number in a grid can only appear once in each row, column, and block. 

Before we go any further, let’s look at what a grid looks like, and explain some of that terminology.

The Sudoku Grid

Here's an empty sudoku grid. We call this a 4x4 grid because it’s four cells across and four down. Small grids like this are great for kids.

Note: If you have kids who you think would enjoy sudoku, we recommend Amelia Baker’s excellent Sudoku For Kids series. It starts with small grids like this and works up to full size ones, with fun tutorials along the way. It’s all wrapped up in a beautifully decorated book that kids will love.

The grid has four rows (each comprising four cells), four columns (of four cells), and four blocks - also made up of four cells. Some sudoku designers refers to blocks as boxes regions, or if they want to get really fancy, nonets. At Puzzle Genius we prefer to stick with calling them blocks because, well, they’re kind of blocky.

Here’s a 4x4 grid with some numbers filled in:

The aim of the game of sudoku is to fill in the empty cells with the missing numbers. We can always do this with logic - there’s no need to guess. In fact, guessing is very bad and should never be part of solving a sudoku puzzle. That’s because if we guess at the contents and get it wrong, we won’t necessarily know until we are much further into the puzzle. By the time we realise something is wrong, we won’t know where our mistake occurred and the only way forward would be to start over from the beginning.

Remember that sudoku comes from the Japanese for the numbers must be single? That means that:

  • Each row in the grid must include all of the numbers 1,2,3 and 4.
  • Each number is allowed only once in each row.
  • Each column in the grid also has to contain all the numbers from 1 to 4.
  • Each number can only show up once in each column.
  • Every block in the grid also has to have every number from 1 to 4.
  • And of course, each number can only show up once in each block.

Armed with the numbers already present, and these simple rules, we can solve the puzzle.

How To Solve Sudoku Puzzles

With a tiny grid like this, the logic to solve the puzzle is almost self-explanatory. But let’s lay it out there anyway, because making sure we understand the basics means we’ll have a good foundation for solving bigger, more complex grids.

The easiest place to start is with a part of the grid that’s already almost complete. Looking back at our example, which is super-simple, there are a few places we could begin. Let’s take the highlighted column which only has one number missing:

Given that column already has a 1, a 2, and a 4, the missing number is a 3. It’s the only number that can go in the blue square. We can write the 3 like this:

Now we have a block that’s only missing one number - the 2. Writing it in makes something interesting happen:

Look at that - there’s a block and a column that are both missing the same number. Fortunately they are missing the same number - the 3. If they were missing different numbers we would know we have a problem somewhere, because we’ve only got space to put one. Here’s what the grid looks like now:

 

The next number is a 1:

That gives us a block and row with the same empty cell, which has to be a 4.

It’s easy to keep filling in the single missing number each time until the whole grid is filled in. This is what the final solution looks like:

Many Roads

This example took one route to solving the puzzle, but it wasn’t the only route. There are no rules that say we had to start with the right-most column. It would have been just as easy to start with the top row, or the top-right block. In a puzzle this simple it’s possible to start in almost any cell. There’s no right or wrong way to proceed.

Checking The Result

The sudoku puzzle has been completed, but are we sure we got the result right? Checking it is easy - we just have to apply the rules of sudoku to each row, column, and block. Working top to bottom, we can read off each row to make sure that they all contain the numbers 1-4 once and only once. Then we can work left to right doing the same for the columns. Finally we can triple check everything is hunky-dory by verifying each of the blocks also contains each number once and only once.

Sales pitch alert! Checking 4x4 puzzles takes seconds. Checking regular sudokus, which are 9 cells by 9 cells, takes longer. That’s why every Puzzle Genius puzzle book contains complete solutions at the back, to make it much faster to check your results.

More Logic

The example we’ve just looked at was unusually easy. If every sudoku was that simple, the game would be very boring indeed. Let’s look at a more realistic puzzle, though we’ll stick with a 4x4 grid for this introductory tutorial.

This time there are no rows, columns, or blocks that are missing only one number. The best we can do is the block on the top left, which is missing two.

Logic is our friend. We have a 4 and a 2 in the block, which means we are missing a 1 and a 3. Which one goes in which empty cell? It’s easy enough to work out by considering other parts of the grid. For example, if we were to consider the top row, we would notice this:

There is already a 1 in the top row, and we know we can’t put the same number into a row twice. That means we can’t put a 1 into the green square - we’d be breaking the rules. Therefore the only place the 1 can go in our highlighted block is the blue square.

Now that block is only missing a 3, so that’s what goes in the green square.

That little bit of login has opened up the puzzle. By filling in single missing cells we get to this stage.

At first glance, an entirely empty block might be difficult to solve. But it’s only a problem if we consider the block in isolation. Taking into account the rows and columns makes it a doddle. 

By looking at the intersection of the row and column highlighted, it’s obvious there’s only one number that can go in there. We can’t put a 4 or a 1, because they are already in the column, and we can’t put a 1 or a 2, because they are already in the row, so we’re only left with a 3.

Once again, that simple logical leap has opened up the puzzle, and the remaining three cells are easy to fill in.

A Logical Foundation

This introduction might be over-simplified, but it has been designed to give you the basis of logic required to solve sudoku puzzles. 

Bigger puzzles, like those you’ll find in our books, can be solved with the same principles, although harder puzzles need some extra techniques. You’ll find lots of more advanced methods in the pages of this site.

Practice Puzzles

In part two of this series we’ll look at regular sized sudoku, and introduces some helpful tricks you can use to solve puzzles up to intermediate level. 

If you’d like to practice the very basics first though, we’ve put together a page of 4x4 puzzles below that you can print out.

Practice Grids To Print

4x4 Sudoku Practice — Grids

4x4 Sudoku Practice — Solutions

Right click or long-tap and Download Linked File or click or tap to open in a new window then choose Print from your browser.

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