Puzzle Learning Centre

How to Play Futoshiki: Inequalities and Beginner Logic

Futoshiki (Japanese for “inequality”) is a small number grid where a handful of less-than and greater-than signs do all the work. It’s quick to learn, and the logic is satisfyingly tidy.

The grid

A Futoshiki puzzle is a square grid, commonly anything from 4×4 to 9×9. A grid of N squares across uses the digits 1 to N: a 4×4 uses 1, 2, 3 and 4; a 5×5 uses 1 to 5. Unlike Sudoku there are no boxes, only rows and columns.

Between some neighbouring squares you’ll see a sign, < or >. Each sign compares the two squares on either side of it.

A finished 4×4 Futoshiki. Each row and each column holds 1, 2, 3 and 4 once, and every sign is true.

Reading the signs

The rule of thumb: the point of the sign faces the smaller number, and the open side faces the larger one, just like < and > in arithmetic.

2 < 4: “2 is less than 4”.
4 > 2: “4 is greater than 2”.
1 above 3: the sign points up, at the smaller 1.
4 above 2: the sign points down, at the smaller 2.

Signs between a square and the one below it are simply turned on their side. Point upwards (∧): the top square is smaller. Point downwards (∨): the top square is larger.

In this guide every sign is also written out in words, for example “R1C1 is less than R1C2”.

The rules

  1. Fill every square with a digit from 1 to N, where N is the number of squares in a row.
  2. Each digit appears exactly once in every row.
  3. Each digit appears exactly once in every column.
  4. Every sign between two squares must be true.

Some puzzles also start with a few digits already in place. A well-made puzzle has exactly one solution.

First moves

Two facts give you a head start on any grid:

  • A square that must be larger than a neighbour can never be 1. Nothing is smaller than 1.
  • A square that must be smaller than a neighbour can never be N, the largest digit.

Here is the puzzle we’ll solve later in this guide. It starts with three digits and six signs:

The practice puzzle: a 4×4 grid with three starting digits and six signs.
  • R1C3 is greater than R1C4.
  • R1C1 is less than R1C2.
  • R2C3 is greater than R3C3.
  • R4C1 is greater than R4C2.
  • R1C2 is less than R2C2.
  • R1C1 is less than R2C1.

Combine the signs with the row and column rules, and some squares lose options straight away:

  • R1C1: the row and column allow 1, 2, 3 or 4, but its sign leaves only 1 or 2.
  • R1C2: the row and column allow 3 or 4, but its sign leaves only 3.
  • R1C3: the row and column allow 1, 2 or 4, but its sign leaves only 2 or 4.
  • R1C4: the row and column allow 1, 2, 3 or 4, but its sign leaves only 1, 2 or 3.
  • R2C1: the row and column allow 1, 2, 3 or 4, but its sign leaves only 2, 3 or 4.
  • R2C2: the row and column allow 3 or 4, but its sign leaves only 4.
  • R2C3: the row and column allow 1, 2 or 4, but its sign leaves only 2 or 4.
  • R3C3: the row and column allow 1 or 4, but its sign leaves only 1.

Inequality chains

When signs line up, they form a chain. If A < B and B < C, then A < B < C: three different digits in increasing order. In a 4×4 grid that leaves very little room:

A chain of three squares in the top row of an empty 4×4 grid, with what each square can still be.
  • Square A can only be 1 or 2.
  • Square B can only be 2 or 3.
  • Square C can only be 3 or 4.

A can’t be 3 or 4, because two larger digits must follow it. C can’t be 1 or 2, because two smaller digits come before it. Longer chains are even more powerful: a chain of four in a 5×5 grid leaves each square just two options (1 or 2; 2 or 3; 3 or 4; 4 or 5).

In a chain of K squares in an N×N grid, each square has only N − K + 1 options. A chain as long as the row is completely solved: 1, 2, 3 … N.

Eliminating candidates

On larger grids, write the possible digits in each square as small pencil marks (candidates). Then cross them off:

  • Remove any digit already in the same row or column.
  • For a sign, the smaller square must be less than the largest option left in the larger square.
  • And the larger square must be more than the smallest option left in the smaller square.

Repeat until nothing more changes. For the practice puzzle, that gives:

The practice puzzle with every remaining candidate pencilled in.
  • R1C112
  • R1C23
  • R1C324
  • R1C4123
  • R2C1234
  • R2C24
  • R2C324
  • R2C41234
  • R3C1134
  • R3C31
  • R3C4134
  • R4C124
  • R4C424

Whenever a square is down to a single candidate, write it in, then go back and cross that digit off its row and column.

A complete worked puzzle

Let’s solve the practice puzzle. Each of the 13 steps names the square, the digit, and the row, column or sign that forces it.

Starting position

The starting grid

Givens: R3C2 = 2, R4C2 = 1 and R4C3 = 3. Press “Next step” to begin.

Read all 13 steps as text
  1. R1C2 is 3. Column 2 already has 2 and 1, so its two empty squares hold 3 and 4 in some order. The sign says R1C2 is smaller than the square below it. The square below can be at most 4, so R1C2 can’t be 4: it’s 3.
  2. R2C2 is 4. Column 2 now has 3, 2 and 1. The only digit left for R2C2 is 4.
  3. R3C3 is 1. Row 3 has a 2 and column 3 has a 3, so R3C3 is 1 or 4. It must be smaller than R2C3 above it. R2C3 can’t be 4 (row 2 has one) or 3 (column 3 has one), so R2C3 is at most 2, which leaves R3C3 only 1.
  4. R2C3 is 2. Column 3 now has 1 and 3, and row 2 has 4. That leaves 2.
  5. R1C3 is 4. Column 3 has 2, 1 and 3, so the last square in it is 4. (It is also bigger than R1C4, as the sign requires.)
  6. R2C1 is 3. Row 2 has 4 and 2, so R2C1 is 1 or 3. But it must be bigger than R1C1 above it, and nothing is smaller than 1, so R2C1 can’t be 1: it’s 3.
  7. R2C4 is 1. Row 2 has 3, 4 and 2. The last square takes the 1.
  8. R1C4 is 2. Row 1 has 3 and 4, so R1C4 is 1 or 2. Column 4 already has a 1, so it’s 2.
  9. R1C1 is 1. Row 1 has 3, 4 and 2, leaving 1. It is smaller than R1C2 and R2C1, as both signs require.
  10. R3C1 is 4. Row 3 has 2 and 1, so R3C1 is 3 or 4. Column 1 already has a 3, so it’s 4.
  11. R3C4 is 3. Row 3 has 4, 2 and 1. The last square is 3.
  12. R4C1 is 2. Column 1 has 1, 3 and 4, leaving 2, and 2 is bigger than the 1 beside it, as the sign requires.
  13. R4C4 is 4. Row 4 has 2, 1 and 3, so the final square is 4. Every row, column and sign now checks out.

Common mistakes

  • Reading a sign backwards

    The point faces the smaller number. If in doubt, read it aloud: 2 < 4, “two is less than four”.

  • Forgetting the signs between rows

    Vertical signs are easy to overlook. Check above and below a square as well as either side.

  • Using digits that are too large

    A 4×4 grid only uses 1 to 4. There’s no 5, however tempting.

  • Looking for boxes

    Futoshiki has no boxes, only rows and columns.

Practice advice

  • Begin with 4×4 and 5×5 puzzles. The ideas are the same at every size.
  • Look for chains first: they narrow several squares at once.
  • Next, look at squares next to a 1 or an N, and at squares touched by more than one sign.
  • Pencil marks repay the effort on anything bigger than 5×5.
  • When you finish, check every sign as well as every row and column.

A puzzle to try

Ready for one on your own? This one is a step up: a 5×5 grid, so each row and column holds the digits 1 to 5. The reasoning from the worked puzzle is all you need: rule out what a square’s row, column and signs forbid, and fill any square left with one possibility.

Futoshiki: a puzzle to try

3 digits given and 7 signs. Look for the chain first.

Answer

From the free guide at northfinchpress.com/futoshiki/how-to-play · Northfinch Press

Show the answer

Quick rules to print

A one-page reminder to keep beside your puzzle book.

Futoshiki: the quick rules

  1. A grid of N × N squares uses the digits 1 to N (1–4 on a 4×4, 1–5 on a 5×5, and so on).
  2. Each row contains each digit exactly once.
  3. Each column contains each digit exactly once.
  4. A sign between two squares must be obeyed. The point of the sign always faces the smaller number; the open side faces the larger.
  5. Signs between squares one above the other work the same way: ∧ means the top square is smaller, ∨ means the top square is larger.
  6. Any starting digits are fixed. A good puzzle has exactly one solution, found by logic alone.

Full illustrated guide: northfinchpress.com/futoshiki/how-to-play · Northfinch Press

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