Inequalities strategy: following the signs

An inequalities puzzle looks like a sudoku with the boxes taken away and a few signs scattered between the squares. There are usually very few printed numbers, so the signs carry most of the logic. This guide covers every technique our hints use, grouped by the difficulty level that first needs it, with small examples you can check by eye. The names match the wording of the hints on Inequalities, so when a hint mentions a pair or a square that can be “at most 3”, you will know where it comes from.

Reading the board

A grid N squares wide holds the numbers 1 to N once in every row and every column. A sign between two neighbouring squares says which of the two is larger: the open side faces the larger number, and the point faces the smaller one. Signs between squares in the same column work the same way, pointing up or down. A sign says nothing about how far apart the two numbers are: 1 and 5 satisfy it just as well as 3 and 4.

The hints count rows from the top and columns from the left, starting at 1, and describe a neighbour by where it sits: “the square to its right”, “the square below it”. This guide does the same. The numbers a square could still hold are its candidates. You can pencil them in as notes (use the Notes button, press N, or hold Shift while typing a digit), and placing a number removes it from the notes in its row and column. The full controls are in How to play.

What Easy puzzles need

Easy puzzles are 4×4 or 5×5 and can be finished with three ideas: sign bounds, naked singles and hidden singles. Each one looks only at a square, its line and the signs that touch it.

Sign bounds

Every sign gives you two facts for free. The square on the smaller side can’t be N, because nothing is larger than N. The square on the larger side can’t be 1, because nothing is smaller than 1. When the neighbour is already placed, the sign becomes much stronger: a square that must be smaller than a placed 3 can only be 1 or 2, and one that must be larger than a placed 3 in a 4×4 grid can only be 4.

A square with signs on two sides, larger than one neighbour and smaller than another, can be neither 1 nor N. On a 4×4 grid that already leaves only 2 or 3.

Tip: the quickest wins are signs next to a placed 2 or a placed N−1. Smaller than a 2 means 1; larger than N−1 means N. Scan for those before anything else.

Naked single

A naked single is a square with only one candidate left once you have crossed out the numbers already in its row and column and applied the signs that touch it. The example above is the simplest case, where one sign does all the work. More often the reasons combine. In a 5×5 grid, say the row already has 1 and 4, the column has 3, and the square is smaller than its neighbour, so it can’t be 5 either. Only 2 is left.

Tip: count the numbers a square can’t hold rather than the ones it can. Squares in busy rows and columns that also touch a sign are the likeliest to run out of options.

Hidden single

A hidden single turns the question round. Instead of asking what a square can hold, ask where a number can go. Every row and column needs each number once, so if all but one open square in a line is ruled out for, say, 4, then the 4 goes in the remaining square, even if that square has other candidates of its own.

The largest and smallest numbers are the ones to hunt for, because signs rule them out so easily. Any square on the smaller side of a sign can’t hold N, and any square on the larger side can’t hold 1.

Tip: for each row and column, glance along it asking “where can N go?” and then “where can 1 go?”. On a sparse board those two questions find more placements than any other.

What Medium puzzles need

Medium puzzles are 5×5 or 6×6. Everything from Easy still applies, and one new idea opens them up: following a sign bound through a square that is still empty.

Chain bounds

Suppose a square must be smaller than its neighbour, and that neighbour, for whatever reason, can be at most 4. Then the first square can be at most 3. The bound has travelled across a sign without either square being filled. Repeat this along a line of signs pointing the same way and you get a chain: if A is smaller than B, B smaller than C and C smaller than D, then A is at most N−3 and D is at least 4. In general, the smallest end of a chain of k signs is at most N−k and the largest end is at least k+1.

Two details make chains more useful than they first appear. They don’t have to be straight: a chain can turn a corner, as long as each step keeps going from smaller to larger. And the bound doesn’t only come from the edge of the number range. If a neighbour’s own row already rules out its top candidates, the limit it passes on is lower. Our hints phrase this as “it must be smaller than the square below it, which can be at most 4”.

Notice that the chain alone would not have been enough. Without the 2 already in row 1, the square at row 1, column 2 could have been 2 or 3. The step came from mixing the chain with the line rule, and that combination is typical of Medium puzzles.

Tip: look for runs of three or more signs, and for chains that end in a placed number. Trace each one from the large end to the small end, writing the maximum next to each square as you go, then trace back the other way for the minimums. A square whose maximum and minimum meet is solved.

What Hard puzzles need

Hard puzzles are 6×6 or 7×7. They still use every earlier technique, but at some point no single square or number is forced by one line and its signs. The step that breaks the deadlock is a pair.

Naked pair

If two squares in the same row or column can only hold the same two numbers, those two numbers are used up between them. You may not know which way round they go, but no other square in that line can hold either number. Once those candidates are gone, the signs often finish the job.

The example is small for clarity, but it has the shape you will meet in bigger grids: nothing else on that board can be placed with the Easy or Medium techniques, and the pair unlocks it. In a 7×7 grid the pair usually comes from a mixture of reasons, such as one square limited by its column and the other by a chain of signs.

Hidden pair

The mirror image of a naked pair is a hidden pair: two numbers a row or column still needs that fit in only the same two squares. Those squares must hold those two numbers, so any other candidates in them can be removed. Signs make hidden pairs common at the top and bottom of the range. If a row of a 6×6 grid still needs 5 and 6, and chains of signs keep every open square but two at 4 or below, those two squares must hold the 5 and the 6, whatever else their notes say.

Pairs feed the chains

Removing candidates with a pair changes the bounds that travel along the signs, and new bounds can reveal a new pair. Our solver repeats the two until nothing more changes, and a Hard hint always names the pair it relied on. When you find a pair, re-trace any chain that runs through its line.

A good solving routine

1. Start with signs next to placed numbers, especially a 2 or N−1 on the far side of the sign.
2. For each row and column, ask where N can go and where 1 can go.
3. After every placement, look at its row, its column and any sign it touches before moving on.
4. When the singles dry up, trace the longest chains and note the bounds they give.
5. Pencil in candidates for squares with two or three options, then look along each line for pairs.
6. After a pair, go back to step 1: the easy techniques usually take over again.

Common mistakes

• Reading a sign backwards. The open side faces the larger number. With vertical signs it is easy to slip, so say it to yourself: “the point is the small end”.
• Treating a sign as “one more”. A sign only orders two numbers. A chain of two signs on a 6×6 grid could be 1, 2, 3 or 2, 4, 6.
• Following a chain through a reversed sign. A chain only continues while each sign points the same way along it. Where the direction flips, the bound stops.
• Forgetting the line rule inside a chain. A chain gives a range; the numbers already in the square’s row and column narrow it further, as in the bent chain above.
• Trusting old notes. Notes are only as good as the reasoning behind them. The hints and the Check button look at the numbers you have placed, never at your notes, so a wrong note goes unnoticed until it misleads you.

Using hints while you learn

A hint always shows the easiest step available from your current board and names the reasons: the numbers already in a line, the sign bounds, the chain with its “at most” or “at least”, and any pair. Try to finish the step from the explanation before pressing Reveal. If a hint says something doesn’t fit, one of your entries is wrong; Check will outline it. The general advice in Getting started applies here too.

The best way to learn the chains is to start with a few Medium puzzles on Inequalities and trace every sign bound by hand, even the ones you could guess. After a dozen grids you will start seeing the forced squares without writing anything down.