Futoshiki

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:

Rules of Futoshiki

The rules are pretty easy to understand:

Every row and column must contain the numbers from 1 to the size of the puzzle once and only once. For example, in a 6x6 grid, each row and column must contain the numbers 1 to 6.
Numbers must respect the inequality symbols, which is to say the < (less than) and > (greater than) symbols.

Here’s what the earlier example looks like once solved:

Futoshiki small sample puzzle - the solution

Solving Futoshiki Puzzles

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:

Begin with simple intersection eliminations. For example, if every row but one is missing a 3, you can work out where it must go by seeing which crossing column does not contain a 3.
Try to place the largest number the grid can support (so on a 6x6 grid, the 6), and also the smallest (which is always 1), with the help of the inequality symbols. For example, any cell with a > (‘greater than’) symbol emanating from it, cannot contain a 1 because 1 is not greater than any other number on the grid. So those cells can all be eliminated from the row or column. Similarly any cell with a < (‘less than’) symbol cannot contain the largest number on the grid. These simple eliminations along a row or column are often enough to figure out where the 1 and the largest number must fit, especially when combined with regular intersection eliminations.
Consider whole rows or columns when working out inequality symbols. For example: In column 1 an empty cell has a > sign pointing right at its neighbouring empty cell in column 2. We may not know the contents of the cell in column 2, but if that column already contains a 1, 2, 3 and 4, then we know the empty cell is at least a 5, so the empty cell in column 1 would have to be at least a 6.
Look for chains of inequality. If multiple cells are stacked with the same symbol between them, you may be able to determine the contents of the entire chain, especially when eliminating any other cells in the row or column using the other methods.
In harder puzzles, use candidate notes (small numbers) to help as you eliminate possibilities.

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.

Worked Example

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.

Try Futoshiki Yourself

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

Where To Play

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.