Hoshi is a variant of Sudoku, based around triangles instead of squares.
In a hurry? Jump to: Rules / Tips / Download Free Hoshi Puzzles / Books
Here’s what an easy Hoshi puzzle looks like:
Normal Sudoku rules apply to Hoshi, but instead of being constrained by rows, columns, and blocks, we have to fill in the puzzle without repeating digits in triangles and lines.
Here’s what the earlier example looks like once solved:
Below, we've highlighted the six triangles, to make it easier to see those constraints. They are the equivalent of blocks in regular Sudoku:
Here we’ve highlighted examples of different lines, which are the equivalent of rows and columns in the normal version of the puzzle:
You will notice that although the six triangles that comprise the Hoshi grid each contain nine numbers, not all the lines in the puzzle do; some only contain eight. However, the rules still dictate that a line cannot contain duplicate digits. For example, the red line at the bottom only contains eight digits, but it does not contain the digits from 1-8. Instead, there is a 9 but no 7. As long as no digit is repeated, then it’s within the rules.
You will also see that four of the lines in Hoshi cross the hexagonal hole in the middle. In this example, the yellow line is split across the middle, yet still forms a single valid line containing the digits 1-9 once and only once.
Hoshi is a variation of classic Sudoku, so if you’ve played Sudoku before, you’re all set to go. If not, be sure to read through our comprehensive three part tutorial – it teaches you everything you need to know to solve regular puzzles, and the same techniques apply in Hoshi.
Here are some additional tips you can use to help with Hoshi:
Are you ready to have a go at Hoshi yourself? We’ve put together free taster pack with a couple of easy starter puzzles, and a couple of harder ones, too. Download the PDF below. Solutions are included so you can check your results.
Click Here To Download The Taster PDF
Want to play Hoshi? We’ve got you covered, we have books of puzzles ready and waiting to be solved.
Puzzle Weekly Presents: Hoshi is a collection of 120 puzzles set over seven levels of difficulty. Grab a copy here.
Got a Kobo? We’ve got Hoshi puzzles for that too! Hoshi for Stylus Devices presents 100 puzzles over five levels, and is designed especially for Kobos with stylus. Find out more here.
It's a new month, and we have a brand new puzzle for the magazine: Trilogy! It's all about shapes, and not a digit in sight. If numbers are your thing though, don't worry, as Sudoku makes a guest reappearance. And for those who prefer to combine numbers with spatial reasoning, Snake should keep you busy. Plus, all the usual features including:
Noodle is a number puzzle played on a square grid. There’s a little bit of calculation involved, but not as much as something like Number Cross or Calcudoku!
In a hurry? Jump to: Rules / Tips / Worked Example / Download Free Noodle Puzzles / Books
Here’s what a small Noodle puzzle looks like:
The objective of Noodle is to place numbers onto the grid according to the following rules:
Here’s what the earlier example looks like once solved:
Solving these puzzles involves elimination and some simple calculations. Here are some tips to help you get started. Below we’ll work through a sample puzzle from start to finish to see how to apply them.
Let’s put all of that into practice and solve a puzzle. This is a very simple level 1 puzzle, so it’s easy to follow along with and does not involve logic chaining or thinking ahead. This example isn’t intended to show the most efficient way of solving the puzzle, but is to demonstrate the strategies we can use to solve Noodle. This is just one way of getting to the solution.
This is the puzzle we are going to solve. As mentioned, it’s an easy level 1 puzzle, just enough to get the idea, not to stretch our brainpower to the limit!
As we have a couple of zeros here, we can immediately eliminate the first column. It cannot contain any digits, so we can strike through all the cells in that column.
Let’s move onto the second column. The top number tells us we must place a single digit in this column, and the bottom number tells us that the contents of the column must add up to 7. In other words, we know we have to place a single 7 somewhere in the column. Therefore, we can eliminate any cells where putting in the 7 would bust a row. As we can see, none of the bottom three cells can contain the seven, as they belong to rows that add up to 2, 6, and 5. We can put strike marks in to eliminate them.
The top three cells could all potentially hold the 7, so we’ll have to come back to this column later.
We’ll finish looking at the ‘1’ columns, as they give us more easy eliminations. Here we know we have to place a 4, so we can eliminate the cell in the row that sums to 2…
…and in this one we must place a 6, so we can eliminate the cells in the intersecting 2 and 5 rows.
Now we can turn our attention to the rows. We must place a single 8 in the first row. We’ve already eliminated the first cell, and in fact we can eliminate the next four as to put an 8 in any of them would bust the totals for the intersecting columns. That just leaves the last cell, so we must put the 8 into that.
Let’s take a look at the end column now. The top number tells us we need to place two digits, and the bottom number says they must add up to 15. We’ve placed the 8, so we know the second digit must be a 7 (because 8+7=15). As there’s only one cell that can take a 7, that’s where we must put it. We can eliminate the remaining cells in that column, which will help complete the other digits.
The 7 we just placed means that we have completed the row in the green box, so we can eliminate all the remaining cells in that.
As we only have one free cell in the row in the blue box, we can place the 2 that is required in that row.
The knock-on effect of placing the 2 is that we now know we must place a 5 in this column. That’s because the column requires two digits that add up to a total of 7. There’s only one valid cell to put the 5, so we place it and eliminate the remaining cells.
For completeness, we can eliminate the last cell in this row as we’ve complete it.
Let’s take a look at this row. We can solve it using simple arithmetic. We need two digits, they must add up to 11. From the three available, we must use the 7 and the 4. Placing these digits completes a couple of columns, too.
Only one digit left to place, and it’s easy because it’s the only one left! That’s it, puzzle solved.
Obviously that was a very simple example. It’s a starter puzzle, useful for getting to grips with the rules. If you want to try something harder, check out our free taster pack below – it includes a couple of easy puzzles like this, and some harder ones, too. And if you want even more, find our selection of Noodle books at the bottom of this page.
Ready to have a go yourself? We’ve put together free taster pack with a couple of Level 1 puzzles, and a couple of harder ones, too. Download the PDF below. Solutions are included so you can check your results.
Click Here To Download The Taster PDF
Would you like even more Noodle? We’ve got you covered!
Puzzle Weekly Presents: Noodle is a collection of 120 puzzles set over seven levels of difficulty. Grab a copy here.
Got a Kobo? We’ve got Noodle puzzles for that too! Noodle for Stylus Devices presents 100 puzzles over five levels, and is designed especially for Kobos with stylus. Find out more here.
It's Halloween week! What could be more terrifying than a level seven puzzle? What about three level seven puzzles? Don't worry, you get the whole week to work up to them. This week it's Pipelink, Tents, and Calcudoku providing the chills, along with all the usual features including:
We've got another new puzzle for you this week: Arrows. It's quite different to anything we've published before, and rather addictive – you have been warned! Also this week, Killer Sudoku, Hidato, and all the usual features including:
Trilogy is a pure logic puzzle that uses symbols on a square grid. The aim is to fill the empty cells on the grid with the correct missing symbols.
In a hurry? Jump to: Rules / Solving / Tips / Worked Example / Download Free Puzzles / Trilogy Books
Here’s what a small Trilogy puzzle looks like:
There are three symbols used in the puzzle: circles, squares, and triangles. There are only two rules to remember:
Here’s what the earlier example looks like once solved:
Solving these puzzles requires the application of logic. There are lots of patterns you can use to help you, and the more puzzles you do, the more easily you’ll spot the patterns.
These patterns are all “forced moves” – they allow you to immediately place a new symbol on the grid. Easier puzzles have more of these patterns, but they’ll help with harder puzzles too; as you add more symbols, more patterns will emerge.
In this tutorial we will use the notation A, B, C, and x, where A, B and C stand in for the three symbols and x is the cell we are trying to solve.
There are two kinds of pattern. The easiest is a simple row of four cells with a gap, allowing you to place a symbol:
AxCC pattern. In this case, the x must be the same as the A symbol. It can’t be B because otherwise we’d have (ABC)C, which isn’t allowed, and it can’t be C because otherwise we’d have A(CCC), which is also not allowed.
In this example, A is a square and C is a circle, so the empty (x) cell must be a square.
ABxB pattern. In this case the empty cell must contain the A symbol, because (ABC)B isn't allowed, and A(BBB) is not allowed.
In this example, A is a square and B is a triangle, so the empty (x) cell must be a square.
These patterns are easy to spot as you can quickly look for doubles on the grid (two symbols the same next to each other), and doubles with a gap between and another symbol next to them.
Next, you can look for intersection patterns. These are patterns where the empty cell we are trying to solve is at the intersection of two patterns. The simplest intersections are a confluence of doubles. For example, if you have AAx in one direction, and BBx in another, x must obviously be C, otherwise we’d be making a triple, which isn’t allowed.
In this example, we have two circles together vertically, forming an AAx, intersecting with two triangles together diagonally, forming a BBx. The intersecting cell (x) must be a C to avoid three in a row. Therefore, in this case it would be a square.
There are lots of these patterns to find in Trilogy. The following table lists some common intersections and what they resolve to. It should be fairly easy to work out why the answer is what it is in each case.
| Solves To: | ||
| AAx | BBx | C |
| AAx | ABx | B |
| ACx | BAx | A |
| AxA | ABx | B |
| AxC | ABx | A |
These are not necessarily all the possible patterns – they are the most common, and a good starting place. As you get deeper into this fascinating puzzle, you will find more that you can add to your arsenal.
Here are some further tips to help you solve Trilogy:
Let’s put all of that into practice and solve a simple puzzle. This is not intended to show the most efficient way of solving this puzzle. The purpose of the example is to demonstrate the strategies we can use to solve Trilogy, and is just one way of getting to the solution.
This is the puzzle we are going to solve. We’ve deliberately chosen a simple one to keep this tutorial manageable. The same concepts apply whatever the size of the grid, though.
Let’s begin by looking for the easiest patterns. The first of those is the AxCC. There are three of those in this puzzle, all in the vertical plane. We know that the empty cell in this pattern must be the A symbol. If we put a C in there, we’d have three Cs in a row, which isn’t allowed. And if we put a B in there, we would have A, B and C in a row, which also isn’t allowed. So from the left, the first empty cell must be a square, the second also a square, and the third one, on the right edge, must be a circle.
Filling in those three symbols has created a a new AxCC pattern (in yellow), so we know the empty cell must be a square. It’s also given us an ABxB pattern (green). This must solve to A, for exactly the same reason as before. So the empty box in the green area must be a circle.
We’ve exhausted the simple gap patterns for now. Let’s look for intersections. The easiest to spot are the doubles (it’s quick to find two symbols the same on the grid). Here’s one example, right at the top. We have an AAx intersecting with a BAx. We know this must solve to B, because if we tried to put an A in the intersecting cell we’d have AAA which isn’t allowed, and if we tried to put a C in there, we’d have BAC which is also not allowed. So the empty cell must contain a circle.
Here’s another, almost the same. This time it’s ABx / AAx, but it solves to the same solution for the same reason. We have to put a circle in the intersecting cell.
Here’s a different kind of intersection. This is an AxC / ABx. From our table above, we know it resolves to A. That’s because if we tried to put a B in the intersecting cell, we would be creating an ABC run in the green area, and if we put a C in there, we’d be creating an ABC run in the yellow one. A is the only option. The empty cell contains a square.
This is an easy puzzle, and there are lots of intersections and patterns appearing all over the place. Let’s look at this AxCC for no other reason than because it’s a diagonal, and we haven’t done any of those yet! It’s important to remember to check the diagonals, particularly when puzzles get harder, as they add two extra dimensions and can often be the key to unlocking the grid.
Being an AxCC pattern, we know the empty cell solves to A, so it’s a circle.
We can do intersections on diagonals as well. Here’s another AAx / BAx, with the AAx on the diagonal. We know it solves to B, so the empty cell at the intersection must be a square.
Here’s another diagonal, an AxCC. It must be a square.
And here’s an ABxB. As we fill out the cells, the simple patterns keep popping up again, so it’s worth keeping an eye out for them. This must be a circle.
Let’s speed up a bit. We’ve got two patterns here. The blue is another ABxB, so that’s a circle. The yellow and green intersection is an AAx / AxC, so that has to solves to C - a circle in this case.
Nearly there. Here are two more. The yellow and green intersection is an AAx / ABx, so must solve to B. It’s a square.
The blue intersection is an AxA / ABx. It must solve to B, so it’s a circle.
A quick AxCC here. It must be a circle.
Just two more to go. This is a new intersection: ABx / ACx. This must solve to A (square). If we put a B in here, we’d be creating an ACB run on the diagonal. if we a C in there, it would be an ABC on the vertical. So, it’s a square.
That just leaves us with a simple ABxB pattern along the bottom. It must be another circle.
That’s it, we’ve completed the puzzle. And we didn’t even have to draw any triangles! I said it was an easy one. If you want to try some puzzles like this yourself, read on – there are some freebies to download below…
This was a very simple puzzle, but it shows the basic techniques used in solving Trilogy. Harder puzzles have far fewer starting symbols on the board, and fewer (if any) initial patterns. They require using more advanced logic, and notes in the cells as you eliminate possibilities.
Are you hooked? Ready to have a go at some Trilogy puzzles yourself? We’ve put together a sample pack of four grids for you – a couple of easy ones like the above example, and a couple that are more challenging. Download the PDF below. Solutions are included so you can check your results.
Click Here To Download The Taster PDF
Would you like even more Trilogy? We’ve got you covered!
Puzzle Weekly Presents: Trilogy is a collection of 120 puzzles set over seven levels of difficulty. Grab a copy here.
Got a Kobo? We’ve got Trilogy puzzles for that too! Trilogy for Stylus Devices presents 100 puzzles over five levels, and is designed especially for Kobos with stylus. Find out more here.
Brand new to Puzzle Weekly, Hitori makes its debut today. It's quick to learn and great fun to solve – we hope you like it. As if that wasn't enough, we've also got Star Battle and Yagit, plus all the usual features including:
On World Architecture Day we are celebrating the mathematical underpinnings of building. And doing lots of lovely new puzzles, of course! This week we've got Number Cross, Binairo, Sudoku, and all the usual features, including:
Arrows is a logic puzzle in which you must place arrows outside the grid according to certain rules.
In a hurry? Jump to: Rules / Tips / Worked Example / Download Puzzles / Books
Here’s what a small Arrows puzzle looks like:
Your task is to place arrows in the empty boxes around the grid in such a way that each box contains a single arrow that points to at least one number in the grid. The numbers tell you how many arrows point to that cell.
Here’s what the earlier example looks like once solved:
Solving these puzzles is all about elimination. They can seem daunting at first (especially larger ones), but by taking a methodical approach and applying logic, we can always reach the correct solution.
Before we get into specific tips, here are some useful things to bear in mind about Arrows puzzles.
We can attack a puzzle on two fronts: by looking at the numbers and trying to work out from which directions the arrows must point at them, or by looking at the arrow cells and working out where they must point. Considering that most numbers have eight potential arrows pointing at them, but any given arrow cell only has three possible directions in which it can point, we’re better off working from the arrows inwards. If we can eliminate two out of three directions that an arrow could point, we can complete the arrow cell.
Here are some tips to help you get started with solving Arrows puzzles. In a moment, we’ll work through a puzzle from start to finish and put these into practice.
Let’s put some of the tips above into practice and work through a puzzle from start to finish.
This is the puzzle we are going to work through. We’re using a very simple puzzle here as anything larger would make this walk-through way too long. Being an easy puzzle, we are provided with clue numbers in all the cells. Harder levels have fewer clues.
Before we get started, a quick word on notation. As we’ll be eliminating possible arrow directions, we need a consistent way of doing so. Any given cell has three possible arrow directions: up / down / horizontal for the sides, and left / right / vertical for the top and bottom. Here we have the three possible side arrows shown. As we eliminate possible placements, we can place an X in the relevant position to rule it out.
The very first thing we can do on any puzzle is eliminate ‘illegal’ arrow positions in the corners. The rules state that arrows must point towards at least one number. That means that the arrows shown here are not acceptable as they don’t point at any numbers….
…so we can put Xs in all those positions to rule them out. It’s not strictly necessary, but it makes things easier later on.
Now let’s look at the 0. No arrow can point to it, so we can add in more Xs as we eliminate all the positions that could point to it. That means no vertical arrows in the column the 0 is in. I’ve eliminated those with the purple Xs.
It also means no horizontal arrows for the row the 0 is in, so we eliminate those with Xs too.
And of course we mustn’t forget the diagonals. That gives us four more arrow positions to eliminate.
With all those eliminations, we’ve got some places to put some arrows on the board. We’ll start at the top. With the vertical arrow and diagonal right arrows not an option, we know we must place an arrow going diagonal left, pointing at the cells highlighted here.
At this point we are going to add another kind of note to the board. Each of the three highlighted cells now has an arrow pointing at them, so we’ll mark each one with a dot. This will help us keep track of which cells are ‘complete’, which will be essential later on.
Working around clockwise, we can place a horizontal arrow pointing to this row. Again, we add a dot to each of the cells it’s pointing to.
Still going clockwise, we have another placement here. And again, we add dots to all the numbers the arrow is pointing to.
And lastly for now, we can put in this vertical arrow.
So where next? We’ve used up all of the known placements, so how can we eliminate any more? We’re going to have to find some constrained numbers – those with limited placement opportunities that can help us out.
We know that numbers on the main diagonals are already more constrained than the other numbers on the board, as they can only have a maximum of six arrows pointing at them, not eight. The largest number on a main diagonal is this 5. It’s already got two arrows pointing to it, so it needs three more. There are four more potentials, but we don’t know which three are the ones we need to use.
Note: Although we don’t know which of those four arrows will be used, we do know that at least one of the ones on the diagonal will have to be used. Had this 5 been deeper in the board and not in a corner, that information could have allowed us to add a dot to all the other numbers on the same diagonal, which may have helped us move forward. In this puzzle, with the 5 being in the corner, it’s no help, but this is an essential technique when working on harder level puzzles.
The next largest number on a main diagonal is this 4. It’s already got one arrow pointing to it. Out of the remaining five possibles, two have been eliminated (highlighted here in yellow). We need three arrows, and lo and behold there are only three places to put them!
We’ll draw them in one at a time, taking care to add dots to our number tallies as we go. First the top one…
…then this bottom one. That’s ‘completed’ two of the numbers in this column. The dot tallies let us see at a glance that the 2 and the 3 both have the requisite number of arrows pointing at them. We’ll draw circles around those numbers to remind us we can’t point any more arrows at them. That will allow us to eliminate more potential arrow positions in a moment, but first…
…we’ll just draw in that last arrow for the 4. Adding the dots confirms the 4 is now complete, so we can circle it.
Now we can use these three newly circled cells to eliminate more arrow positions. The 2 gives us three new eliminations (purple Xs). I haven’t drawn in the one to the left of the row as that arrow is already placed, so we don’t need it.
The completed 3 also eliminates three arrow positions (purple Xs).
We know the 4 doesn’t eliminate anything because we already used up all the remaining positions.
We’re really motoring now. We’ve got lots more arrows we can place. Going around the board clockwise again, we’ll start with this diagonal. That completes the 1 in the corner, which in turn allows us to eliminate an arrow position on the diagonal (purple X).
This next new arrow puts three more dots on the board…
…and this one completes two numbers, which in turn give us two new eliminations.
We can place all three arrows up in this corner. They complete several new numbers, which in turn eliminate further positions (again, the purple Xs).
We know where to place the final three arrows. I’ve added the dot tallies to the numbers, but there’s no need to circle them, we’re all done!
Phew! How did you get on? Did you race ahead and finish before the end of the example? If you want to have a go and try some more, including some harder puzzles, there’s a taster pack to download below.
This was an easy puzzle, but it shows the basic techniques used in solving. Harder puzzles have fewer clue numbers, and require using more advanced logic like partial-elimination when you know the direction of an arrow (horizontal, vertical or diagonal), but not from which side it originates.
Ready to try some puzzles yourself? We’ve put together a sample pack of four grids for you – a couple of easy ones like this example, and a couple that are more challenging. Download the PDF below. Solutions are included so you can check your results.
Click Here To Download The Taster PDF
Ready for even more Arrows? We’ve got you covered!
Puzzle Weekly Presents: Arrows is a collection of 120 puzzles set over seven levels of difficulty. Grab a copy here.
Got a Kobo? We’ve got Arrows puzzles for that too! Arrows for Stylus Devices presents 100 puzzles over five levels, and is designed especially for Kobos with stylus. Find out more here.
Sort out your horizontals from your verticals and diagonals, with Line Segment, in this week's free Puzzle Weekly. We're also path-finding with Hidato. Plus: all the usual features, including:
