Futoshiki is a number-based puzzle that shares many similarities with Sudoku, but adds additional clues in the form of inequality symbols.
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Here’s what a small Futoshiki puzzle looks like:
The rules are pretty easy to understand:
Here’s what the earlier example looks like once solved:
Simple Futoshiki puzzles can mostly be solved using basic Sudoku methods, in other words, elimination based one row and column intersections.
Anything beyond the most easy puzzles requires us to look at the inequality symbols, though. Here are some tips to help you get started:
As puzzles increase in difficulty, you’ll need to combine techniques to narrow down possible cell contents, and that’s when notes come in handy.
Theory is all well and good, but there’s nothing like putting pen to paper and solving a real puzzle, so let’s do that now. We’re going to work on an easy level 1 puzzle so that this example doesn’t become too unwieldy, but it will show the basic techniques we can use to solve Futoshiki. Please bear in mind the objective here is to demonstrate techniques, not to solve the puzzle in the most efficient way possible.
This is the puzzle we are going to be solving. It’s a simple level 1 puzzle that will adequately demonstrate the approach we can take to solving.
The easiest of easy wins on this board is this row at the bottom. It’s only missing one digit, so it’s a real no-brainer! We can put a 1 in here.
The 1 we’ve just placed is going to help us solve the middle cell in this column. That cell requires a digit that is smaller than 4 and smaller than 3. The only numbers that fit the criteria are 1 and 2, and seeing as we just placed the 1, we know that can’t be it! So we have to put a 2 in here. And seeing as there’s only one other empty cell in the column, we know that will have to contain the 5.
In this row we are missing a 1 and a 4, and we have two cells ready and waiting. Cell A says it contains a number bigger than 2, so we can’t put the 1 in there, it has to contain the 4, and the 1 goes into B.
In the last column we need a 2 and a 4. If we look at the row that intersects with cell A, it already has a 4, so that cell can only contain the 2. The 4 will have to go in cell B.
There’s only one empty cell left in this row, so that’s our 5.
How can we solve this column? Well, we’re missing a 1 and a 5. According to our inequality symbols, cell A contains a number smaller than 3, so it must contain the 1. That leaves the 5 in cell B.
This is a very easy puzzle, so there are loads of ways of figuring out the last few cells. Let’s look at the top row. We’re missing a 2 and a 4. Cell A must be bigger than 1, so that doesn’t help us. But cell B must be smaller than 3, so the 1 must go in there, leaving the 4 to go into cell A.
There are only three empty cells remaining, and each one is the final empty cell in its column, so they require no effort to fill…
That’s it, all done. I did say it was an easy one! It might be simple, but these kinds of eliminations are at the heart of how we solve Futoshiki puzzles. By combining them in their different forms, and cascading implications between rows and columns, we can solve all but the most complex of puzzles without resorting to candidate notes. If you want to have a go yourself, keep reading for a free taster pack to download.
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 Futoshiki? We’ve got you covered!
Puzzle Weekly Presents: Futoshiki is a collection of 120 puzzles set over seven levels of difficulty. Grab a copy here.
Our favourite four-block teaser is back this week, with Tetromino. And there's more spatial fun to be had with Thermometers. If numbers are your thing, there are plenty in Suguru. And of course, there are all the usual features including:
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:
