How to play
Futoshiki
A little room
for each number.
Start with a small sign. Let it narrow the possibilities. Then let every number help the next.
Small = possibleStroke = ruled out
Neither square has its answer yet.
The point picks the smaller.
An inequality sign compares the two squares beside it. Its pointed end faces the smaller number. Its open side faces the larger.
Using 1–4, the left square cannot be 4: it needs a bigger neighbour. The right cannot be 1: it needs a smaller one. Equal is never allowed.
Rule study: A < B, with numbers 1–4. Neither number is fixed.
Which end of the sign faces the smaller number?
Choose an answer to check your reason.
Read the answer
Yes. The point faces the smaller number. The open side faces the larger. Equal numbers never satisfy a sign.
A rule study using 1–4, before the full puzzle.
A little order. Still a little room.
Add a third square. Each step must increase, but it does not have to increase by exactly one.
Two larger numbers must fit after the first square, so it can only be 1 or 2. The middle can be 2 or 3. The last can be 3 or 4. Keep the possibilities small; none is certain yet.
Rule study: A < B < C, with numbers 1–4. The possible starts are 1 and 2.
In a three-cell increasing chain using 1–4, is the first cell definitely 1?
Choose an answer to check your reason.
Read the answer
Exactly. Both 1 < 2 < 3 and 2 < 3 < 4 work. Even 1 < 2 < 4 works: an inequality does not require a gap of one.
A short chain alone does not tell you exact numbers.
Four places. No room to spare.
Now four squares increase using only 1–4. Three larger numbers must follow the first square. It can only be 1.
Each position needs room for every smaller and larger neighbour. The second is 2, the third 3, the fourth 4. Watch each alternative go before its survivor becomes an entry.
On a 5 × 5 board the range is 1–5, so a four-cell chain alone leaves room.
Keep the row. Open the grid.
That same row becomes the top of our 4 × 4 puzzle. Use 1, 2, 3 and 4 exactly once in every row and exactly once in every column. There are no smaller Sudoku boxes.
The underlined 2, 4 and 3 are fixed clues. Keep them. All other blank squares are still unknown. Every inequality must also hold.
Our ordinary game has 4 × 4 and 5 × 5 boards. Size describes the range, not a difficulty rating.
Let two directions meet.
Look at row 2, column 4. Its row already contains 2. Its column already contains 4 and 3. Cross those candidates out.
Only 1 remains. This proved number now helps the next square. Look at row 2, column 2 and check both directions again.
A highlighted source rules out a candidate in its row or column.
Row 2 has 2 and 1. Column 2 has 2 and 4. Row 2, column 2 is still blank.
Now what belongs in row 2, column 2?
Choose an answer to check your reason.
Read the answer
3. Row 2 already has 2 and 1, so neither can repeat. Column 2 has 4, so that goes too. Only 3 survives.
Rows and columns share their facts. You never need a box rule.
One certainty helps the next.
In row 2, column 2, the row removes 1 and 2; the column removes 4. Only 3 remains.
Now row 2 contains 2, 3 and 1. Its last square, column 3, must be 4. Keep each new number as evidence for later moves.
A highlighted source rules out a candidate in its row or column.
A settled entry is a proved fact, not a guess.
Carry those facts down.
Row 3, column 1 cannot repeat the 1 and 2 above, or the given 4 beside it. That leaves 3.
Column 4 already has 4, 1 and 3, so row 3 needs 2 there. Row 3 then has 3, 4 and 2: its last square must be 1. Three small reasons, in that order.
A highlighted source rules out a candidate in its row or column.
Watch the highlighted source number and the candidate it rules out.
Every last space has a reason.
The first three rows are complete. Column 1 lacks 4; column 2 lacks 1; column 3 lacks 2. Settle each last space.
Now every row and column contains 1–4 once, the fixed clues are intact, and the three signs read 1 < 2 < 3 < 4. This is a finished Futoshiki, earned one reason at a time.
A highlighted source rules out a candidate in its row or column.
Take the reasoning into a different little puzzle below.
Try a fresh puzzleA new little grid.
Your own reasons.
A different 4 × 4. Follow the bent chain from row 1, column 2. How many numbers have to fit in order?
Practice stays here while this page is open. Your regular games and saves are separate.
3 of 16 numbers entered4 × 4
The pointed end of each sign faces the smaller number. Start with the bent four-cell chain.
Underlined = fixed. Select a blank square, then enter a number. Big numbers are entries. Small numbers are your own candidate notes.
Notes affect hints: a square with notes permits only those listed numbers during hint checking. No notes permits all numbers. Your notes are never filled or corrected automatically. Erase clears the selected value and notes.
Clear your entries and notes for a fresh attempt? The fixed clues stay. You can undo the reset.
Controls and hints
Keyboard: Tab into the board. Arrows move, Home/End move along the row. Type 1–4 to enter. N toggles Notes; Delete or Backspace erases. Ctrl/Cmd+Z undoes; Ctrl/Cmd+Shift+Z or Ctrl/Cmd+Y redoes. Tab leaves the board. Number buttons also work with Enter or Space.
Explain a move checks your current entries and notes. It may use constraint elimination or check alternatives by bounded search. A contradiction asks you to revise your numbers or notes; a search limit means no move was proved. Apply is always separate. Asking for an explanation marks this attempt as assisted, even if you undo it.
Notes can be added only to blank, editable squares. To replace an entry with notes, erase it first. Guide progress is page memory only; leaving or reloading starts fresh.
A little less unknown.
Read the point. Leave room along a chain. Bring row and column facts together. There is always another little grid to think through.
Play Futoshiki Return to your practiceFree to play. No account needed.
The quick rules.
- Fill all squares with 1–4 on a 4 × 4, or 1–5 on a 5 × 5.
- Use every number once in each row and column. There are no smaller Sudoku boxes.
- Obey every inequality. The pointed end faces the smaller number; equality is not allowed. A vertical sign follows the same rule.
- Keep the underlined fixed clues. You can change or erase your own entries and notes.
One more thing?
Must neighbouring numbers differ by one?
No. 1 < 4 is just as valid as 1 < 2. The sign asks for strict order, not a particular gap.
Does any increasing chain start at 1?
No. A chain of three in 1–4 can be 2 < 3 < 4. A full four-cell chain in 1–4 has no spare number, so it is exactly 1, 2, 3, 4. On a 5 × 5 board, a four-cell chain alone does not fix exact values.
Does a vertical sign work differently?
No. Look at its tip. If the tip points up, the upper square must be smaller. If it points down, the lower square must be smaller. Our starter has both a horizontal and a vertical comparison.
Why can a note cause a contradiction?
When you list notes in a square, hints allow only those numbers there. If the needed number is missing, your notes may block every solution. Erase that square’s notes, revise them or Undo. The guide never silently corrects them.