Snake (sometimes also called *Tunnel*) is a spatial awareness path-finding logic puzzle. The aim is to connect the snake’s head with its tail by filling in grid cells following certain rules. Here’s what a very easy Snake puzzle looks like:

The rules of Snake are easy to remember:

- The snake’s body can be filled in horizontally and vertically
- The body must never touch itself, not even diagonally
- The numbers outside the playing grid tell you how many cells must be filled in for a row or column

In easier level puzzles, every row and column is labelled. As the difficulty ramps up, some rows and columns are blank, making it harder to determine the snake’s path.

Here’s what our earlier example puzzle looks like once solved:

Solving these puzzles requires a combination of elimination and forced placement based on logical deduction. There is no single ‘correct’ way to solve a Snake puzzle, rather, you need to have a variety of techniques available to you.

Although the goal is to connect the snake’s head to its tail, there is no particular need to start at either the head or the tail; sometimes it makes more sense to start somewhere in the middle, solving different sections at a time until eventually they all link up.

Here are some tips to help you tackle Snake puzzles, then we’ll look at a worked example below.

- As you find cells that you know cannot contain part of the snake, it is useful to mark them off. As you narrow your options in this way, you will find places where the snake must go.
- Check for extremes. If a row or column is marked as 0, you can eliminate all the cells within it. Similarly, a row or column labelled with a number that matches the dimension of the grid (eg: labelled 6 in a 6x6 grid) can be fully filled in. However, both these scenarios are very rare and unlikely to appear in anything but the simplest puzzles.
- Look for rows and columns labelled as 1. It is useful to highlight these because they effectively split the grid into unique sections. You can think of these rows and columns as one-way crossings – once the snake crosses one, it cannot return. Thus it becomes necessary to complete each side of a ‘1’ column or row before crossing it.
- Bear in mind that crossing a ‘1’ column or row requires at least three consecutive cells to be filled in. This further narrows your possible crossing places.
- Rows or columns labelled 2 are also constrained. You cannot perform a ‘U-turn’ within them, because that would require at least 3 cells due to the fact the snake cannot touch its own body. This can reduce potential crossing places.
- If you locate three or more filled cells in a row, you can block out the middle cells to either side. Again, the rule about the snake not touching its own body means the snake cannot return to those cells.
- Remember that as soon as the snake body touches the head or tail, it cannot go any further. So be careful not to run into either too soon.
- Cross out clue numbers outside the grid as you complete rows and columns; it helps to remind you where you can no longer go.

We will work through a sample puzzle from beginning to end. As with all examples, remember that there’s no single path to the solution. This is just one way to get to the end.

This is the Snake puzzle we are going to solve. At 8x8, and with every row and column labelled, it’s quite easy.

There are no extreme values on this 8x8 grid (no rows or columns labelled with a 0 or an 8).

We do however, have a ‘1’ row and a ‘1’ column. Because these are one-way crossings, it’s useful to highlight them. Once the snake’s body crosses each of these, there’s no turning back.

This allows us to immediately cross off three cells. The two to the right of the tail cannot be used, because it takes three cells to cross a ‘1’ column, and that would violate the 2 for that row. We can also eliminate the cell to the left of the head, for the same reason. That means the body extends from the head either up or down. Either way will allow us to fulfil the 2 clue for the column containing the head. Which way to go?

We have to go down. Why? Because the row above the head requires three cells to be filled in. Crossing the ‘1’ column would use our full complement. We would have to get back down to the bottom row to fill in the six cells that must be filled there, and that would violate the ‘2’ row, and the ‘3’ row on our way back. The only way to fill in the ‘6’ row is to go there directly from the head.

Once on the bottom row, we know we fill three cells to cross the one-way ‘1’ column. In fact, we can also deduce that we must place all six required cells in that row because there’s no valid way to leave the row and return. In other words, we couldn’t go across three cells, up a couple, across again, and back down, because that would violate the rows above.

This one move has completed a row and two columns for us. Where do we move next?

We know we have to move up the ‘7’ column, because we’ve filled the ‘6’ row. Going up one cell completes the ‘2’ row (with the head), so we *must* go up another cell – there’s no choice.

Now we do have a choice though. Because we have a one-way barrier ahead of us (the highlighted ‘1’ row), we know we must complete the ‘3’ row before moving upwards. So do we go left or right?

The one-way barrier means that once we turn upwards, we will have to continue for at least another two cells. Going right would violate the rules in two ways. Firstly, we would be filling three cells in a ‘2’ column. Secondly, we would end up touching the snake’s tail (diagonally). As the snake cannot touch itself, this would be an invalid move.

So we have to go left…

…then we have to cross our one-way barrier row by going up three cells, completing another column.

We can’t go up any more, so we have to go right…

…and then upwards, because we’ve completed the three cells allowed in this row, and also because we have to complete our ‘7’ column. We’ve knocked off quite a few rows and columns now.

It’s a clear run to the end. We have to fill four cells in the top row, and there’s only one way to do that…

…which means we can drop down to the tail, completing the puzzle. Easy! Ready to have a go yourself? Read on!

We’ve put together a taster of four puzzles for you to try out, including the example above. You can download and print the PDF below. Solutions are included, but no cheating!

Finished the taster and want more Snake in your life? No problem! Get 120 carefully crafted puzzles set over seven levels in *Puzzle Weekly Presents: Snake* – it's great value!

We publish Snake puzzles in *Puzzle Weekly* from time to time. *Puzzle Weekly* is our **free weekly puzzle magazine** – find out more, and get your copy, here

**This Week:**

Pipes - lots of pipes! Will Pipelink drive you round the (u)bend? Also, Calcudoku (KenKen), and the usual features, including:

- Crossword
- Sudoku
- Kid's puzzles
- Colouring pages

Pipelink is a spatial awareness puzzle from Japan. There are no numbers, there’s no complex mathematics, and no fancy formulas. The objective is very simple: to link all the cells in the grid with a single, continuous pipeline. Here’s what a very easy Pipelink puzzle looks like:

The rules are straightforward:

- The pipe must pass through every cell in the grid.
- It must use all the sections that are provided.
- You cannot add to given cells.
- The pipe may cross itself, i.e. create an intersection of perpendicular segments, but it cannot
*turn*at an intersection.

Here’s what our earlier example puzzle looks like once solved:

These puzzles are solved through logical deduction and spatial reasoning. We begin with what we know — the given pipe segments – and using that knowledge we work out where we can and cannot extend them. As we grow the pipeline a segment at a time, we slowly reduce the options for where subsequent segments can be placed.

Here are some tips to help you get started.

- Begin by ruling out routes the pipeline cannot take. Because we must not add to given segments, we can use them to block certain paths.
- Next, extend all the given segments. Every given segment must extend to the middle of the next cell. For example, a horizontal segment extends at least to the middle of the cell to its left, and to the middle of the cell to its right.
- Then look for forced placements – places where your extensions must connect because there’s no other valid option.
- Look for incomplete crossings. If you have a cell with three entries/exits, you know you must add a fourth for it to be a valid crossing.
- Keep an eye out for dead ends. If extending a pipe in a certain direction is going to create a dead end with no possible way to connect back to the main pipe, you’re making a mistake.
- If you get stuck, consider lightly pencilling in a candidate segment (or use a different colour if playing on screen) to make it easy to go back and erase it if it turns out to be wrong.

We will work through a sample puzzle from beginning to end. As with most puzzles, remember that there’s no single way to solve it. The larger the puzzle, the more possible paths to reach the final solution.

This is the puzzle we will be working on. It’s an easy grid, but sufficient to demonstrate the techniques used to solve Pipelink puzzles.

We can begin by marking out routes we know the pipe *cannot* take. This will be helpful later when we come to start extending it. Because we cannot add to the given segments, we know our pipe can’t go anywhere we’ve marked with a red cross.

Next we can extend all the given segments into the middle of the adjoining cells. Doing so has connected some sections of the pipeline together – we’re making progress already!

We can continue extending the pipe. The segments in the highlighted cells can only be extended in a single direction…

…so we can extend those.

Where next? Let’s examine the grid further…

The two highlighted cells have only one possible direction to extend. We *have* to extend them both downwards.

But this leaves us with a problem. We’ve got two ‘islands’ that are isolated from the rest of the pipeline below. We’ll deal with the smaller one first. If we are to connect it to the rest of the pipeline, we can only do so from the highlighted cell. There’s simply no other option. Do we go down, or to the left? The answer, of course, is both! We cannot have a cell with three segments entering/exiting, it’s either two or four (because either the pipeline traverses the cell, or it crosses itself within the cell). So we can draw those extensions in now.

That’s connected the top right-hand island, but it’s given us a two new problems – the highlighted cells are invalid because they have three entries/exits. Of course, these aren’t really problems, they’re opportunities, because we know we have to add a fourth segment to both of them. They must be crossings.

We’re almost done. We just have one gap, and it’s an easy one to connect up…

…which completes the puzzle. It’s fairly easy to see that this solution is valid just by looking. With larger puzzles, it can be harder to be sure your solution is valid. To check a solution, it can be worth tracing the entire pipeline with a different colour to make sure you can cover off the whole thing, and every cell, without lifting the pencil or stylus.

Ready to have a go yourself? We’ve put together a taster of four puzzles for you, including the example above. You can download and print the PDF below. Solutions are included, but no cheating!

We publish Pipelink puzzles in *Puzzle Weekly* from time to time. *Puzzle Weekly* is our **free weekly puzzle magazine** – find out more, and get your copy, here

**This Week:**

Binary – it's not just for computers!

- Binairo
- Thermometers

Plus the usual features, including:

- Crossword
- Sudoku
- Kid's puzzles
- Colouring pages

Binairo, which is sometimes also called Takuzu, is a pure logic puzzle played on a variable sized grid. Using just 1s and 0s, it might be quick to learn but it can be a real challenge to solve! Here’s a small example puzzle:

The goal is to fill the grid with 1s and 0s, following certain rules.

There are only three rules to remember:

- No number can appear more than twice consecutively in a row or column.
- Each row and column contains the same number of 1s as 0s.
- No row or column can be duplicated within the puzzle.

Here is what the example puzzle looks like once completed:

Every puzzle has a single valid solution, though of course there may be lots of ways to reach that solution.

Unlike many kinds of logic puzzle, Binairo is not necessarily easier to solve from the outside in; even the corners aren’t helpful. On the contrary, the easiest wins come from clusters of numbers.

Here are some tips on how to approach solving a puzzle. Below, we will work through an example from beginning to end.

- As with any puzzle, look for ‘easy wins’ first. There are three kinds of easy placement in Binairo:
- Doubles – i.e. two 1s or two 0s adjacent within a row or column. You can immediately place the opposite number either side of a double (otherwise you’d be putting three in a row, which violates rule 1).
- Empty cells with the same number either side of them. As with doubles, you can immediately fill the cell with the opposite number.
- Single remaining cells within a row or column. You’re unlikely to find these off the bat, except in very easiest puzzles. As you progress through a puzzle though, they crop up a lot. Simply count up the 1s and 0s already present in the row or column to work out what the final missing number must be.

- When you’re out of easy wins, look for forced placements based on the number of 0s and 1s you know you must put in a row or column. For example, if you know you have to place three 0s and you have four cells remaining, if three of them are in a row, the three 0s cannot all go in there together. Therefore one of them must go in the other free cell.
- Remember after filling a cell to check for any consequences. Often placing even just one number will have a ripple effect, giving you an easy win which leads on to more easy wins.

As with all puzzles, there’s not a single correct way to solve a Binairo. The example presented here is just one way, and is a demonstration of the general techniques used.

Here is the grid we are going to work through. This is a level 2 puzzle, so though it’s easy to complete, it still presents enough of a challenge to give an overview of the techniques used to solve these puzzles.

We’ll begin by looking for all the easy wins. Remember there are three kinds of easy win. First up, doubles (two adjacent numbers the same).

There are six doubles. We know that the numbers either side of them must be the opposite of the numbers forming the doubles, otherwise we’d be trying to put three numbers in a row, which isn’t allowed. So between them, these doubles allow us to solve seven cells. We’ll put those numbers in now.

Filling those in gave us three new doubles. We’ve also got an almost completed column (another easy win). We can fill in the yellow cell straight away; we know we must have an even number of 0s and 1s in each row and column. That column already has four 0s, but only three 1s. Therefore the yellow cell must be a 1.

Now the top row is almost complete, so we can figure out what the one remaining (yellow) cell must be. We’ve got all the 1s, so it must be a 0.

We can also work out the two remaining cells in the second column must both be 0s, for the same reason.

We’ve used up all the easy doubles for now, so let’s check on another kind of easy win - empty cells with the same number either side. There are two such opportunities available here, so we can fill out those cells.

That’s opened up two new opportunities: there’s a single remaining cell in column three (yellow), and a new double in column five (green). Even without the double, we could complete that column just based on the count of 0s and 1s.

The penultimate row has a new double (which is also a single cell surrounded by two numbers the same).

Row five has a full complement of 1s, so we know the last two cells must both be 0.

We’ve got one more easy win for now (the green highlight). There are a couple more cells we can solve here too.

First, the blue column. It’s already got three 0s, and as this is an 8x8 grid we know we need one more. Although there are three empty cells, there’s only one into which we can place the missing 0 – the bottom cell marked with the blue X. If we put it in either of the others, we’d be making a row of three.

Next, the yellow row. We know that one needs three more zeros. We can’t put *all* of them in the last three cells, so one of them *must* go in the cell with the purple X.

Now we’ve got a double and a single cell, although we don’t need them anyway because as there are four 0s already, both remaining cells in that column must be 1s.

Here’s another single we can fill in.

That gives us a row with one empty cell.

Here’s an easy double.

That gives us an almost complete row.

We’re out of doubles, and empty singles. In fact, we’re left with several pairs of cells and no obvious answers as to which way around to fill them.

But we do have a way to move forward. We can use several pieces of logic to solve the blue and green cells. We can combine our knowledge of what numbers we still need to place, along with the fact that no two columns can be the same. The column with the blue cells is shaping up to look a lot like column two. And the column with the green cells is looking like its neighbour to the left so far. These facts limit our placement options.

Let’s consider, for example, the end column. We must place three 0s somewhere. If we put them both in the top two free cells, that would make the two blue cells at the top of column six both 1s. But that would make it impossible to fill in the remaining free cells in rows two and three correctly:

This is wrong! Clearly that cannot be the solution...

In fact, if we work through the options, there’s only one way we can fill in the blue and green cells in such a way that the columns don’t duplicate others, and that allows us to complete the last two cells correctly. And that’s it – we’ve completed the puzzle.

Ready to have a go yourself? We’ve put together a taster of four puzzles for you, including the example above. You can download and print the PDF below. Solutions are included, but no cheating!

Finished the taster and want more Binairo in your life? No problem! Get 120 carefully crafted puzzles set over seven levels in *Puzzle Weekly Presents: Binairo* – it's great value!

We publish Binairo puzzles in *Puzzle Weekly* from time to time. *Puzzle Weekly* is our **free weekly puzzle magazine** – find out more, and get your copy, here

**This Week:**

Another week, and another new puzzle! And an old favourite returns, too.

- Line Segment
- Number Cross

Plus the usual features, including:

- Crossword
- Sudoku
- Kid's puzzles
- Colouring pages

**This Week:**

We've got not one but **two** brand new types of puzzle for you:

- Tents
- Hidato

Plus the usual features, including:

- Crossword
- Sudoku
- Kid's puzzles
- Colouring pages

It's October, which means another quarter of the year is behind us. And that means there's a brand new volume of Puzzle Quarterly!

With all the puzzles from the July, August, and September 2024 issues of *Puzzle Weekly*, this bumper compendium includes:

- 392 puzzles
- 15 different kinds of puzzle
- Seven different levels
- 98 kids puzzles
- 14 colouring pages

The large 8.5x11" format means all the puzzles have plenty of space for notes, candidates, doodles, and and of course, solutions!

All the details are here!

Thermometers is a logic puzzle played on a square or rectangular grid filled with thermometer shapes. Each thermometer has a base (the bulbous end) and a top. In easy to intermediate level puzzles, thermometers may be placed horizontally and / or vertically. In more difficult levels, the thermometers may be ‘broken’ such that they span more than one column or row.

The objective of the game is to fill the thermometers sufficiently that the number of cells filled in a row and column of the grid corresponds to the numbers on the outside of the grid.

Here’s a small example Thermometers puzzle:

The thermometers are filled (or not) according to the following rules:

- Thermometers can be entirely unfilled, partially filled, or completely filled.
- Thermometers always fill from the bulb toward the top. This is irrespective of the thermometer's actual orientation on the grid.
- Each filled segment of a thermometer counts as one filled cell.

Here is what the example puzzle looks like when it has been solved:

These puzzles are solved through a combination of forced positioning and elimination. Here are some tips to get you started.

- Look for rows or columns with a 0 written outside. This means no cell in that row or column is filled, so all thermometers crossing it remain unfilled beyond that cell. Zeros are rare in all but the easiest puzzles though.
- Look for total fills. For example, a clue of 5 for a row of five cells would mean every cell in that row or column is filled. Again, such clues are rare outside of easy puzzles.
- Look for forced fills. If a thermometer segment in a row or column is filled, then all segments below it (toward the bulb) must also be filled.
- Check for forced ‘empties’. If you know for sure a segment is unfilled, then all segments above it (away from the bulb) must be unfilled too.
- Partially fill where you can. Often you can work out that
*at least*a certain number of cells must be filled, so fill them. Even if it doesn’t complete the row or column, having filled part of the thermometer can help solve rows or columns crossed by it. - Cross out cells you have determined
*cannot*be filled. Eliminating places you cannot fill is just as important as determining those you can. It reduces options for filling other rows and columns. - It can also be helpful to cross off the numbers outside the grid as you complete columns and rows.
- Highlighter pens work well for filling thermometers quickly if you are solving on paper.

We are going to work through a sample puzzle from beginning to end. This is a level 2 puzzle, so easy to complete but a good demonstration of the techniques required. Remember that there’s no single path to a solution; this is just an example of one way to solve this puzzle.

This is the grid we are starting with. As a level 2 it’s fairly small, and the thermometers only run horizontally and vertically. We’ll start off by looking for the easiest fills. Being a 7x7 grid, those 6s look promising.

The final column is an easy win. Out of seven cells, six need to be filled. It’s easy enough to work out the one that cannot be filled – it’s in a row labelled 2, at the end of a thermometer four cells long. Obviously the end two cells of that thermometer can never be filled. To make life easier, we can cross them out. We know all the other cells in the column are filled. We can strike through the 6 at the top of the grid too, so we know we’ve done that column.

Let’s look at the first row, which is also a 6.

Because we’ve already got one cell filled in that row, we have to fill the whole of the other thermometer. We cannot possibly fill the cell in 1-column, because, well, that column can only contain one filled segment and that thermometer is four cells high! In fact, while we are at it, we will strike out the top three cells of that thermometer because we know they can never be filled. That row is done, so we can strike out its label, too.

Now we know how to complete the first column.

We can fill up the first five cells of the thermometer, reaching our target of 6. We strike out the bottom cell as it can’t be used, and strike out the label because that column is complete.

Now we’ve done the easy ones, let’s do some elimination. We’ll see if there’s anything we know we definitely cannot fill.

We can take out four more cells. Now, consider the row highlighted in yellow. We need to fill five cells, and have already filled two. There are four cells remaining. We know we’ll have to fill the bulbs of the two thermometers that start in that row, even if we don’t yet know where the fifth cell will be.

Putting those in completes the 1-column. Striking out the cell at the bottom of that column leaves us only one place to put our second cell in the 2-row (bottom row), so we can fill that in as well. These last placements have more knock-on effects: we can strike out cells in both the 3-columns because they are impossible to reach now.

Where next? Well, that second row (5) is looking promising.

We needed to fill three cells, and there were only three left. We’ve completed that row. We’ve also completed the 3-column, so we can cross out the last two cells in that one.

The cell we filled in the 4-column is useful, because it’s at the top of a thermometer. We *must* fill the cells below it.

That’s completed the 4-column, so we cross out the remaining cell. That leaves us only one place to fill in the last cell in the 5-row highlighted...

…and because that’s the top of a thermometer (even if it’s upside down), we have to fill the cells below it. That completes a whole bunch of columns and rows! In fact, we only have one cell left to fill in to complete the puzzle…

That last cell completes the remaining 2-row and 3-column. All done!

That was easy, wasn’t it? Of course it was, it was a level 2 puzzle! We publish seven levels of difficulty, with bigger grids, longer thermometers, and even thermometers that go around corners. Harder grids also have rows or columns without labels. Plenty to give your brain a proper workout. Read on to find out more.

Ready to have a go yourself? We’ve put together a taster of four puzzles for you, including the example above. You can download and print the PDF below. Solutions are included, but no cheating!

Download Our Thermometers Taster

Fancy filling some thermometers? We publish this puzzle occasionally in our free Puzzle Weekly magazine. You should totally sign up for that if you haven’t already, as it puts at least 28 brand new puzzles in your inbox every week.

You can also find lots of Thermometers puzzles in our *Jumbo Adult Puzzle Book* – which happens to include more than *500 puzzles* of 20 different varieties.

**This Week:**

- No Four in a Row
- Suguru
- Crossword
- Sudoku
- Kid's puzzles
- Colouring pages