# Futoshiki Rules

### A brief guide

This is an example of a completed Futoshiki grid. The rules for Futoshiki are similar to the rules for Sudoku. This is a 7x7 puzzle, and for this puzzle, each row and column must contain the numbers 1-7 once and only once. This is very similar to the rules of Sudoku. Futoshiki puzzles come in different sizes, so a 5x5 puzzle would need to have the numbers 1-5 in every row and column, for example.
Additionally, we also have arrows that give us additional clues. The arrows between the cells tell you which cells have the bigger number. A < symbol means the number on the left is smaller than the number on the right. A > symbol means the number on the right is smaller than the number on the left, and this symbol works the same way when it is vertically between cells.
This is an example of a Medium Futoshiki puzzle - in a harder Futoshiki you are given fewer starting cells, and you are also given fewer > and < relations.
This is one of our easy puzzles. The best place to start with a Futoshiki puzzle is to look at inequalities where you are given one side already, or where you have a few inequality signs chained together.
I have highlighted three cells that we can fill in straight away. Look at the highlighted cell to the left of the '2', the < tells us it's smaller than 2, so it has to be 1! That was easy!
The highlighted cell to the right of the '2' is a little harder. By chaining the inequality signs, we know that the highlighted cell must be bigger than 5 other cells. All 5 of those cells are in the same row, so we also know that none of them can be duplicated, so this highlighted cell must be a '6'.
Once you have filled in all the obvious cell, you will have to start using pencil marks. The highlighted cell at the bottom of the puzzle is a good candidate. We know it has to be bigger than 4 other cells, but you need to be careful, they're not all in the same row/column, some of those might be duplicated, i.e. with just this information, the smallest this cell could be is 4, and not 5 as you might expect. However, the existing '2's in the puzzle mean that this cell can't be 4, so this cell must be 5 or 6.
This is an example of a medium starting grid. You will need to make extensive use of pencil marks on hard Futoshiki puzzles.
Look at the highlighted cell, by chaining the inequality signs together we can see that this cell must be bigger than 3 other cells, giving us a minimum value of 4. The 9 already in the column, and the 8 already in the row, gives us a maximum of 7. This gives us pencil marks of 4, 5, 6, 7 for this cell. We can also write in pencil marks for the other cells in this chaing, the cell to the left would be 3, 4, 5, 6, for example.
By filling in the pencil marks using this information, we can then start to apply more advanced pencil-mark techniques on this puzzle.