Solving Futoshiki
A Range of Techniques to Crack Even the Hardest Puzzles
Futoshiki forms a Latin Square, ie each row and column must contain one of each of the numbers 1 to max, where max is the size of the grid. So for a 9x9 grid these would be the numbers 1 to 9.
Since this is basically a Sudoku grid (without the additional Region constraint), the row/column solving techniques for Sudoku can be used with Futoshiki. To refresh yourself on these techniques visit the Sudoku Solving Techniques page.
The addition of the Greater Than/Less Than constraints in Futoshiki add some additional techniques that will help, particularly at the start of a puzzle. Here are the main techniques.
Inequality Chain
Finding a chain of inequalities equal to the size of the grid will allow placement of all numbers in that chain. The chain does not need to be in a straight line, but all the inequality signs do need to be in the same sense.
The chain of inequalities shown in green can be filled working back from the 1 to 7.
It is rare to find this in even the easiest of puzzles, but it does occur and if you find it, its an easy way to complete a lot of placements.
Min/Max
If any square is less than 2, it must be 1! In the example the green square is shown as less than 2 and since the only possible value less than 2 is 1, we can confidently place a 1 in the green square.
If any square is greater than max-1, where max is the size of the grid, then that square must be the maximum value possible. In this 7x7 grid we have the pink square shown as greater than the 6 (7-1) in the red square. Since the only possible value in the grid over 6 is 7, we can confidently place a 7 in the pink square.
Exclude Min/Max
If all squares but one in a row or column are shown as greater than neighboring squares, then the remaining square must contain a 1. Any square that is greater than another cannot be 1 since it is the lowest value in the grid.
Similarly, if all squares but one in a row or column are shown as less than a neighboring square, then the remaining square must contain a 7 (or whatever the maximum for the grid size). Any square less than another cannot contain a maximum value since by definition its the highest in the grid.
Not all squares in a row or column need an inequality sign to find such situations. Filled squares with values between 2 and max-1 already exclude those squares from being 1 or max.
Combining Techniques
Its often necessary to combine both Sudoku solving and Futoshiki solving techniques to find a confident placement. On a row where its possible to eliminate all but two potential placements for a 1 using the Exclude Min/Max method, there may be a 1 elsewhere in the same row or column of one of these two positions (basic Sudoku technique). This eliminates another possibility and leaves us able to place the 1 with confidence in the remaining location.
Forcing Chains
Similar to the way these are used in Sudoku, where you have two possible numbers for a square you can try a placement and follow all the implications. Then try the other placement and follow those implications as well. If one fails but the other succeeds then the successful placement can be confidently included.
We hope this helps you solving our Futoshiki puzzles.
