How to Solve a Futoshiki

One real 5×5 carrying a single number and six signs, from an untouched grid to a finished one. Every move below is a move the solver made, in its own order.

Signs first, sweep second, everything else third

Begin with the signs and only the signs. Follow each one to see whether it joins others into a run, and mark the ends: a cell with two cells forced above it cannot exceed n minus two, and a cell with three below it is at least 4. Nothing on the grid changes while you do this and no clue is involved, which is exactly why it comes first — it is free, and it is the only technique whose entire yield is available before you have written anything.

Then sweep. Take every number that is settled, including the ones printed on the grid to start with, and remove it from the rest of its row and the rest of its column. This is bookkeeping rather than cleverness and it is where most of the answers on an easy grid come from, because it cascades: cross a value out of five cells and one of them is very often left with a single possibility.

Only when both of those have run dry is it worth turning a question round and asking where a value can live, or reading a sign against a neighbour’s remaining candidates, or weighing two cells against each other. Those are the three expensive moves and doing them early means doing them on cells a free sweep was about to settle anyway.

One 5×5, from the first sign to the last cell

The grid below is a 5×5 with one number given and six signs, graded expert — chosen not because a beginner should start there but because its solve happens to use all five rungs of the ladder in its first nineteen moves, in ladder order, which no easier grid does. The five positions shown are the five moments where something new is being used; the thirty-two moves in between are more of the same.

Read each pair of grids left to right. On the left is the position as it stood, with the cells the argument read marked and the cell it decides outlined. On the right is the same position a moment later. The small digits in a cell are its remaining candidates, and most of these moves rub one out rather than filling anything in — which is what a futoshiki actually looks like from the inside, and why a walkthrough that showed only the answers would show you the same picture five times.

1. Before anything is written, follow the signs. The cell at row 2, column 4 is greater than its right-hand neighbour and less than the cell below it, so one cell is forced above it and one below — it can be neither a 1 nor a 5, and that is settled from the printed page alone.

   The position

   	1	2	3	4	5
   1	12345	12345	12345	12345	12345
   2	2	12345	12345	12345>∧	12345
   3	12345	12345	12345	12345	12345
   4	12345	12345	12345∨	12345	12345∨
   5	12345<	12345<	12345	12345	12345

   A moment later

   	1	2	3	4	5
   1	12345	12345	12345	12345	12345
   2	2	12345	12345	234>∧	12345
   3	12345	12345	12345	12345	12345
   4	12345	12345	12345∨	12345	12345∨
   5	12345<	12345<	12345	12345	12345

   Sign chains

2. Nine sign readings later the free pass is spent, and the single given number finally gets its turn. The 2 at row 2, column 1 comes out of the rest of its row and the rest of its column at a stroke — eight cells narrowed, and the bottom-left corner left holding one candidate.

   The position

   	1	2	3	4	5
   1	12345	12345	12345	12345	12345
   2	2	12345	12345	234>∧	123
   3	12345	12345	12345	345	12345
   4	12345	12345	45∨	12345	2345∨
   5	12<	23<	34	12345	1234

   A moment later

   	1	2	3	4	5
   1	1345	12345	12345	12345	12345
   2	2	1345	1345	34>∧	13
   3	1345	12345	12345	345	12345
   4	1345	12345	45∨	12345	2345∨
   5	1<	23<	34	12345	1234

   Singles

3. Now the question turns round. Look along the bottom row and ask where the 5 could still go: three of its cells have been pushed under 5 by the signs and one is already settled, which leaves exactly one home. The cell takes it, even though on its own it still looked like four different numbers.

   The position

   	1	2	3	4	5
   1	345	12345	12345	12345	12345
   2	2	1345	1345	34>∧	13
   3	345	12345	12345	345	12345
   4	345	12345	45∨	12345	2345∨
   5	1<	23<	34	2345	234

   A moment later

   	1	2	3	4	5
   1	345	12345	12345	12345	12345
   2	2	1345	1345	34>∧	13
   3	345	12345	12345	345	12345
   4	345	12345	45∨	12345	2345∨
   5	1<	23<	34	5	234

   Only one home

4. The sign between rows 2 and 3 in the fourth column was read on the opening pass and said very little. It says something now: the cell below has come down to a 3 or a 4, so the cell above must be under 4, and a candidate list reading 3 or 4 loses its 4.

   The position

   	1	2	3	4	5
   1	345	12345	12345	1234	12345
   2	2	1345	1345	34>∧	13
   3	345	12345	12345	34	12345
   4	345	12345	45∨	1234	2345∨
   5	1<	23<	34	5	234

   A moment later

   	1	2	3	4	5
   1	345	12345	12345	1234	12345
   2	2	1345	1345	3>∧	13
   3	345	12345	12345	34	12345
   4	345	12345	45∨	1234	2345∨
   5	1<	23<	34	5	234

   Sign bounds

5. Finally the expert move. In the third column two cells are both down to a 4 or a 5 — between them they will use both values, whichever way round — so no other cell in that column may hold either, and three cells lose candidates at once from one observation.

   The position

   	1	2	3	4	5
   1	345	12345	12345	12	2345
   2	2	45	45	3>∧	1
   3	35	1235	1235	4	235
   4	345	12345	45∨	12	345∨
   5	1<	23<	34	5	234

   A moment later

   	1	2	3	4	5
   1	345	12345	123	12	2345
   2	2	45	45	3>∧	1
   3	35	1235	123	4	235
   4	345	12345	45∨	12	345∨
   5	1<	23<	3	5	234

   Pairs and sets

what the reasoning read what it decides what it wrote

Thirty-two more moves of the same five ideas, and every cell is down to one number. Here is the finished grid.

When the grid will not move

Sweep again before concluding anything, and start with whatever you settled last. By a wide margin the commonest cause of a stuck futoshiki at this level is a number written in five minutes ago whose row and column were never cleared of it. That is free to fix and it usually unlocks two or three cells in a chain.

If that comes up empty, go along the rows and then the columns asking about each value in turn: is there exactly one cell left in this line where the 4 could go? That question is slower than looking at a cell, and it is worth asking exhaustively rather than where you suspect, because a line about to give way and a line with plenty of room look identical until you ask.

And if both of those fail, go back to the signs with fresh eyes. A sign you read on the first pass said very little when both its cells could be anything; the same sign now sits between a cell that can only be a 2 or a 3 and one that can only be a 1 or a 2, and it settles them both. What the grid never wants is a guess — every grid published here was checked to be finishable without one.

Common questions

More Futoshiki pages

• Futoshiki puzzleUnlimited grids
• Daily futoshikiA new grid every day
• Futoshiki rulesThree rules, in full
• Futoshiki techniquesThe whole ladder
• Sign chainsTechnique — the opening
• The line sweepTechnique — easy
• Only one homeTechnique — medium
• Sign boundsTechnique — hard
• Pairs and setsTechnique — expert
• Guessing at futoshikiTechnique — refused
• Printable futoshikiFor paper
• Futoshiki archiveEvery past grid