How to Solve Futoshiki: Beginner Strategies Step by Step

To solve Futoshiki, you do not need to try numbers and see. Every LogikLand grid can be solved by three kinds of logic, used over and over: lines, signs and chains. This page explains each one, then applies them to a real 5 x 5 grid, step by step, so you can check every deduction. The rules themselves are in Futoshiki rules if you need a reminder.

Three kinds of reasoning

1. Lines

This is the same logic as in Sudoku. A number that is already placed cannot appear again in its row or its column, so you cross it out of the other squares of that row and column. And if a number can only go in one square of a row or a column, it goes there. In the generator, these two moves are the first thing that the solver does at every pass.

2. Signs

A sign gives each of its two squares a limit. A square that is smaller than a neighbour cannot be the largest number of the grid. A square that is bigger than a neighbour cannot be 1. More exactly, the smaller square must stay below the largest number the bigger square can still take, and the bigger square must stay above the smallest number the smaller one can still take.

3. Chains

Several signs in a row, such as a < b < c, form a chain. A chain is much stronger than its signs taken one by one, as the next section shows.

The chain formula

In an n x n grid, suppose k squares form a chain of increasing numbers, a1 < a2 < ... < ak. The numbers must fit between 1 and n, so:

• The first square is at most n - k + 1.
• The last square is at least k.
• The i-th square is at least i and at most n - k + i.

In a 5 x 5 grid, a chain of three squares limits the smallest one to 3 at most and the largest one to 3 at least. A chain of four squares is even tighter: the middle squares have only two possible numbers each. These limits are then combined with the line logic, which often leaves just one number.

Solving an easy grid

Here is an easy grid with four given numbers and five signs, and its solution. Rows are counted from the top and columns from the left, starting at 1.

Step 1: the chain of four

Four squares form a chain: the square at row 2, column 5 is smaller than the one at row 3, column 5, which is smaller than the one at row 3, column 4, which is smaller than the one at row 4, column 4. By the formula, the last one is at least 4. Row 4 already has a 4 and a 2, and column 4 has a 1 at the bottom, so that square can only be 3 or 5. At least 4 leaves only 5.

Step 2: the squares in the middle

The square at row 3, column 4 must be smaller than 5. It must also be bigger than the square at row 3, column 5, which must be bigger than the one at row 2, column 5. Column 5 already contains a 2, so the square at row 3, column 5 cannot be 1 or 2: it is at least 3. Then the square at row 3, column 4 is at least 4, and below 5. It is 4, and the square at row 3, column 5 is 3. The smaller square at row 2, column 5 is below 3 and cannot be 2 (the column has it already): it is 1.

Step 3: a pair at the bottom

In the last row, the square at column 2 is smaller than the one at column 3. Column 3 already holds a 5 and a 4, and row 5 holds a 1, so the bigger square can only be 2 or 3. The smaller one must then be below it and cannot be 1 (taken in the row): it is 2, and the bigger one is 3.

Step 4: a pair in row 4

Row 4 now holds 4, 2 and the new 5. The two squares at columns 1 and 2 share the leftover numbers 1 and 3, and the sign says that column 1 is smaller. So row 4 starts with 1, then 3.

From here the grid finishes by line logic alone: each new number removes candidates from its row and column, and the remaining squares fall one after the other. The generator's solver places the last number in its fifth round on this grid.

Pencil marks

Candidates are the numbers that a square could still hold. Write them in small figures in the corner of the square. Start with the line logic: in the grid above, every empty square of row 4 loses the candidates 4 and 2 at once, because the row already holds them. Then apply the sign limits. When a square has one candidate left, write it in full and update its row and column.

On bigger grids

The same three kinds of logic solve every LogikLand grid, because a grid is only kept if the generator's solver, which uses exactly these moves, can finish it. What changes with the level is how much you must combine.

Bigger grids ask you to keep several candidate lists in mind. The tricks for that are in advanced Futoshiki techniques.

Where to practise

Print the printable easy Futoshiki puzzles and apply the three steps above. To make a whole book with 5 x 5 to 8 x 8 grids, answer pages and a "How to play" guide, create a free account to make a full book and open the Futoshiki assistant.