Rules
The board is a square. Fill each row and each column with the digits once: 1–4 on a 4×4, up to 1–7 on a 7×7. Between some neighbours, across or down, there is a less-than or greater-than sign. The open side faces the larger digit.
- One of each digit. Every row and every column holds each digit once. A 4×4 uses 1 to 4. A 7×7 uses 1 to 7.
- Signs between neighbours. A less-than or greater-than sign sits between two cells, across or down, never diagonally. The open side faces the larger digit.
- Givens stay. A given digit is already filled. You can select it. You cannot change it.
- Finished. The puzzle is done when every row and every column has each digit once and every sign holds. There is no box.
The heavy signs show both directions. Beside the given 3, the open side faces the 3. Under the 2, the open side faces up. Your digits are teal. The given stays bold.
- Less-than or greater-than
- Given
- Your digit
Tap a cell. Its row and column are tinted, and any cell with the same digit is a little darker. Type a digit or use the keypad. Erase, Backspace, Delete, or 0 clears it. Arrow keys move. Notes (the Notes button, orN) are pencil marks. Shift with a digit toggles one note without turning Notes on. There is no auto-pencil.
When you place a digit, that digit leaves the notes in the rest of its row and column. It does not leave a note because of a sign. You strike those. A full row or column that already holds each digit once turns soft teal. Givens can be selected and cannot be edited.
- Undo
- Redo
- Notes
- Hint
- Check
Signs
The small side of a sign cannot be the largest digit, and the large side cannot be 1. A filled neighbour clamps the other cell. Then write the last digit in a cell.
The small end and the large end
Start at a sign. The small side cannot be the biggest digit on the board. The large side cannot be 1. You do not need the other cell to be filled. On a 4×4 the small side cannot be 4, and the large side cannot be 1.
That alone is not always a digit. It becomes one when the row, the column, or another sign has already removed the rest. Hint names only the trim that left one digit. The rest of the reason is in the picture.
The given 3 is in the row, so the cell on its left is 1, 2, or 4. It is the small side of the sign beside the 3, so it cannot be 4. It is also the large side of the sign below, so it cannot be 1. Left: 2. Hint says the cell is greater than its neighbour, so it cannot be 1.
The row holds 2 and 3, so this cell is 1 or 4. It is the small side of the sign under it, so it cannot be 4. Without the sign it would still be 1 or 4. The cell below stays empty. Hint says the cell is less than its neighbour, so it cannot be 4.
A neighbour already filled
Once one side of a sign is filled, the other side has to sit strictly outside that digit. Smaller than a 2 means 1. Larger than a 3 means 4 on a 4×4, or anything above 3 on a bigger grid. This is the same sign as before, read against a digit instead of against 1 or the top digit.
The first row is full, so the column already holds 2. The cell under it is 1, 3, or 4. It is the small side, so it cannot be 4, and it has to be smaller than the 2, so it cannot be 3. Left: 1. Hint says the neighbour is already filled, so this side must sit outside that value.
Only one digit left
After the row, the column, and the signs have spoken, a cell with one digit left takes it. The Sudoku guide already teaches this. On this board the cell is also the last empty cell of the row. There is no second name for that.
Row 1 has 2, 3, and 1, so the last cell is 4. The same sentence fires when the row and the column together have used every other digit while the row still has an empty cell. Hint says only one digit can still go in this cell.
An easy grid finishes with these three looks. A medium grid is where a run of signs, or a digit with one home in the line, still has work to do.
Chains
Walk a run of signs until a cell has one digit left. Then write a digit that fits in only one cell of its row or its column.
A chain of signs
One sign only bans the extremes, or clamps a filled neighbour. A run of signs is stricter. Walk it until a cell has one digit left. Hint writes one cell. The sentence names the chain. The stored edge is only the last sign that trimmed that cell.
From the easy rules alone, the middle cell of the column is larger than the empty cell above it and smaller than the cell below, so it is not 1 and not 4. The cell below is larger than that middle cell and smaller than the cell on its left, so it is not 1 and not 4 either. Both are 2 or 3. The only way the upper one is smaller is 2, then 3. The pencil 3 is not written yet. The next hint is the 1 above, not that 3. Hint says after the chain of inequalities, only one digit remains here.
The only home in the line
Look along a row, or a column, for a digit that fits in only one empty cell. Write it there even when that cell could still hold something else. The Sudoku guide teaches the same idea inside a box as well. This grid has no boxes. A hidden single is a medium step here. On the Killer Sudoku guide it is an easy step, and on the Sudoku guide it is a medium step.
4 is gone from the other three cells, so it goes in the first cell. That cell can still hold 1, 2, or 3. Each neighbour is the small side of a sign, which is the easy ban on 4, gathered into one home. The first cell has no sign. It is not a cell with one digit left. Hint says this digit appears nowhere else in the row, so it goes in this cell.
A 1 is already written in the row, from an earlier one-home digit. In the column, the top cell cannot be 1 because it is the large side of the sign on its right. The cell under it cannot be 1 because it is the large side of the sign below it. The third cell cannot be 1 because the row already holds 1. The 1 has one home, at the bottom. That cell’s own list still has every digit until you write it. Hint says this digit appears nowhere else in the column, so it goes in this cell.
A medium grid adds the chain and this one-home digit. It may show you either one first. It does not ask for a pair or a try.
Pairs
Two cells that share the same two digits claim them. A cell that is smaller than two others in its line has to leave room for both.
Two cells, the same two digits
Two cells in the same row, or the same column, sometimes have the same two digits left and no others. Those two digits cannot appear anywhere else in that line. Hint writes the cell that this clears down to one digit. It points at the pair, not at the cell it writes. The Sudoku guide teaches the pair. The signs are why these cells have those two digits.
On some hard grids, two digits fit only two cells, and the extra digits drop off those cells. That never comes up as the Hint sentence on the grids here.
In the row, two cells are 1 or 2, so 1 and 2 leave the cell that was 1, 2, 4, or 5. It becomes 4 or 5. The cell lower in the column is already 4 or 5: it is larger than the 3 on its left, and the column already holds 3. Those two cells are the pair. Striking 4 and 5 from the rest of the column leaves the ringed cell, which was 1 or 4, as 1. A later hint on the same pair writes another cell. This one does not. The same pass also strikes 1 and 2 down the right-hand column, and it does not write a digit there. Hint says these two cells share the same pair of candidates in this house. Not every hard grid has a pair. This one never tries a digit.
Less than two cells in one line
A cell that is smaller than two other cells in the same column has to leave room for two different digits above it. The same idea works in a row, and it works the other way when the cell is the large one. The grids here only ask for the column, and only on a 7×7.
The row already holds 3, 4, 6, and 7, and the column already holds 1 and 6, so the ringed cell is 2 or 5 before this step. Both neighbours have to be different digits, and both have to be bigger than 5. The cell above can be 4, 5, or 7, so the only digit it has above 5 is 7. The cell below can be 3, 4, 5, or 7, so the only digit it has above 5 is 7. They cannot both be 7. A 2 still fits. Hint says this cell is less than two others in the same column, so its maximum drops further. The cell above had already lost 2 because it has to be larger than a cell that was only 2 or 5. That is the chain from earlier. It is not what places the 2.
The other form is a cell greater than two others in the same column, so its minimum rises further. A row works the same way. Not every hard grid asks for this, and the ones that do are 7×7.
What if
When the signs, the lines, and the pairs have stopped, one cell may still have two digits. Try one. Follow only the rules you already have, and do not start a second try inside the first. If that choice runs a cell out of digits, the cell takes the other digit. If both choices would fill the grid and they agree on some other cell, both ways of filling the split cell put the same digit here. That second case did not come up on the grids here.
Easy and medium never need this. Some hard grids finish on the pair, or on the 7×7 clamp, and never try a digit. Check stays quiet on the trial digit alone: one end of its sign is empty, and the digit is not a repeat yet. The failure shows up only after the forced digits are written in.
The cell on the left has to be smaller than 2, so it is 1. That is the first thing the try forces. It is not the digit you write on the real grid.
Keep going with the same rules. Row 2 now holds 1, 2, and 4. Column 1 holds 5, 2, 4, and 3. Nothing from 1 to 5 can sit in the ringed cell. Check stays quiet: nothing repeats, and every filled sign holds.
Hint says if this cell were 2, that side contradicts, so it is 4. The two digits of the try were 2 and 4. The neighbour is empty, so the pencil list on the cell can still show 1. The solver has already dropped it.
A hard grid adds the pair, and on a 7×7 sometimes the two-neighbour clamp, and on most hard grids this one-cell try. Some hard grids never ask for the try.
Common mistakes
Red marks only the cells a broken rule cites, and a chevron whose two ends are filled the wrong way. A quiet digit stays violet. The given 3 stays where it is.
The two cells do not share a sign. The 1 beside the given 3 is still smaller than 3, so that sign holds. The other sign does not fire, because the cell under it is empty. Check marks the repeated digit. Hint waits: “This board has a mistake, so there is no hint yet.”
Same idea, down the column. The 1 is still smaller than the given 3. The cell between them is empty, so the vertical sign does not fire. The lower 1 has no sign. Use 1 here. A 2 in the top cell is the digit that belongs there, and a 4 would also break the sign beside the 3.
Both ends are filled, and the top cell was supposed to be the larger one. The chevron goes coral with the two cells. The 1 beside the given 3 still holds, so that sign stays violet. The same coral shows up if you put 4 over an empty-looking neighbour and then fill the small side with something that is not larger: 4 beside 2, on the sign that wants the left cell smaller, is only this mistake.
A 4 on the small side, with the other side still empty, does not turn red. The sign does not turn red until both sides are filled. The easy rule already says this cell cannot be 4. The candidate list on the cell still includes 4, because it only clamps a sign when the neighbour is filled. Check says “No mistakes so far. Keep going!” Hint asks for a different cell and does not erase the 4. A 1 on the large side, with the small side still empty, is the same quiet Check.
What to play next
An easy grid finishes with the small side, the large side, a filled neighbour, and the last digit in the cell. A medium grid adds the chain and the digit with one home. A hard grid adds the pair, and on a 7×7 sometimes the two-neighbour clamp, and on most hard grids the one-cell try.
- EasySigns, then the last digitThe daily size. Start with the small side, the large side, and a neighbour that is already filled. Then write the last digit in the cell. A 4×4 easy grid is the same lesson on a smaller board.Play Easy →
- MediumA chain, or one homeWalk a run of signs until a cell has one digit left. Then look for a digit that fits in only one cell of its row or its column. A 4×4 medium grid is the short version.Play Medium →
- HardA pair, or one tryWhen those stop, look for two cells that share the same two digits. On a 7×7, a cell smaller than two others in its column can drop further. On most hard grids, one cell with two digits left is worth one try.Play Hard →
4×4 comes in easy and medium. 5×5, 6×6, and 7×7 come in easy, medium, and hard. The short easy lesson is 4×4 easy.
The daily puzzle is a 5×5 easy grid. Today’s Futoshikiopens the same size and setting as 5×5 easy.
Print a Futoshiki pack. Every shipped size prints four grids to a sheet. Large print is one. The answer key starts on its own page and leaves the signs off.
- GuideSudokuRows, columns and boxes, from singles to triples.
- GuideKiller SudokuCage sums and house totals do the work.
- GuideKakuroOne-way sums first, then short runs and longer lists.
- GuideBinairoSpot pairs, fill the gaps, balance every line.
- GuideHashiCount bridges, never cross, connect every island.
- GuideNonogramsOverlap long clues, cross the gaps, and read the crossing line.
- GuideNurikabeGrow each island to its number and keep one sea.
- GuideLight UpLight every cell. Start beside the 4s and 0s.
- GuideSlitherlinkDraw one loop. Start from the 0s and the 3s.
- GuideStar BattleOne or two stars in every row, column and region.
- GuideMasyuTurn on black pearls, go straight through white ones.