Most broken Futoshiki grids come from one of six mistakes, and the biggest is putting a number in before it’s certain. The others are misreadings: a sign the wrong way round, a sign read on one side only, signs down the columns overlooked, a sign taken to mean “one bigger”, and notes that have fallen out of date. Each one is below, with how to spot it.
If you’re new, the rules and how to solve pages explain the ideas these mistakes go wrong on.
1. Putting a number in before it’s certain
When a square could be one of two numbers, it’s tempting to pick one and see. Usually nothing tells you straight away. We took every one of the game’s levels, put a wrong number in one empty square (one that broke no rule yet), and kept making honest moves until the game’s red mark appeared:
| Track | Red mark appeared | Moves before it (median) | Longest |
|---|---|---|---|
| Easy 5×5 | 37 of 50 | 15 | 24 |
| Medium 6×6 | 33 of 50 | 16 | 32 |
| Hard 7×7 | 12 of 50 | 21 | 41 |
On the rest, the red mark never appeared at all: the solve jammed first, with no move left to make or a square with no number it could take. Here’s one on Medium level 3, with a wrong 1 in the top-left corner:
Sixteen honest moves later, a sign at the bottom of column 3 breaks. Neither of its squares is the wrong one.
How to avoid it: only put a number in when a technique makes it certain. If a square could still be two things, write both as notes and work somewhere else.
2. Reading a sign the wrong way round
The point of a sign faces the smaller number; the open side faces the larger, just as 2 < 5 in maths. Read one backwards and every move that follows from it is wrong.
How to avoid it: say it to yourself as you place: “this one is smaller than that one”. On a column, turn your head: the point still faces the smaller number.
3. Reading a sign on one side only
A sign limits both squares. It’s easy to see that the smaller side can’t be the top number and forget that the larger side can’t be 1, or the other way round.
How to avoid it: every time you use a sign, ask what it says about each side in turn. See reading the signs.
4. Overlooking the signs down the columns
Signs down a column are easy to miss when you’re reading across. On the game’s levels, half the signs are in columns, and 76% of the moves that come from the signs use at least one of them.
How to avoid it: sweep the columns for signs as carefully as the rows, and follow chains that turn corners: a chain can run across a row and then down a column.
5. Thinking a sign means “one bigger”
A sign only says which number is the smaller. 1 < 5 holds just as well as 4 < 5. On the game’s levels, 42% of signs are between numbers more than one apart.
How to avoid it: use a sign to rule numbers out, never to fill one in from its neighbour plus one.
6. Notes that have fallen out of date
Notes are only as good as the last time you checked them. Cross a number out by mistake, and the square can look settled when it isn’t. When you place a number, the game takes it out of the notes in its row and column, but it doesn’t redo the signs for you.
How to avoid it: after you place a number next to a sign, look at the sign again and trim the notes on its other side.
How to recover from a mistake
- A red number or sign means a rule is broken, but the mistake may be elsewhere, sometimes a long way back. Undo to the last move you were sure of.
- Press Stuck? It rings a wrong number, or a square whose notes leave out its number, before anything else, so you know exactly where.
- Stuck with no red mark? Go over every sign again, rows and columns, then look at when you get stuck.