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Inequality grid

A Latin square with < and > signs between cells: place each number once per row and column so every sign holds. 4×4 up to 7×7.

  • Logic
  • Untimed
  • 11 levels

How it plays

Each row and each column holds every number from 1 up to the size once. A sign between two cells says which is smaller: the narrow point faces the smaller one, for < and > across and ∧ and ∨ between rows.

Signs give bounds before anything is placed: a cell that is bigger than a neighbour cannot be 1, and a chain a < b < c leaves a at most n − 2. Combine those with the numbers already in each line.

Levels take digits and signs away until only what uniqueness needs is left, then grow the board. Every puzzle has exactly one solution, checked by a solver.

Keys

Space starts and skips ahead · 1–9 pick · Enter submits · Esc stops. On a phone, tap.

What it trains

This practises chaining order relations and elimination together. It is practice at these puzzles, not a measured gain in reasoning elsewhere.

Learn

How to get better at inequality grid: the method, the techniques, what each new level asks of you, and the traps the wrong answers are made of.

Your approach to each question

  1. Read the signs before the digits. The narrow point always faces the smaller cell: 2 < 3 across, and between rows ∧ means the upper cell is smaller, ∨ that it is larger.
  2. Mark the bounds the signs give. A cell larger than a neighbour cannot be 1; a cell smaller than one cannot be the top number. Longer chains push the bounds further.
  3. Now bring in the given digits. Strike out every value already in a cell’s row and column, and every value its signs rule out against a given neighbour. Fill any cell left with one value.
  4. Then turn it round and ask, for each row and column, where each number can go. The top number cannot sit in any cell that is smaller than a neighbour, and 1 cannot sit in any cell that is larger than one. Often one place is left.

Techniques and tricks

  • Count the chain

    A cell with k cells chained below it by signs is at least k + 1; with k chained above it, it is at most the size minus k. Follow the signs as far as they go in any direction.

    e.g. On a 5×5, a > b > c: a is at least 3, c is at most 3, and b is 2, 3 or 4

  • The full chain

    A chain that runs the whole length of a line, all one way, fixes the line outright.

    e.g. A 4×4 row reading _ < _ < _ < _ is 1 2 3 4

  • Next to a given

    A sign beside a given digit cuts the open cell to one side of it. Beside a 2, “smaller” means 1; beside the top number minus one, “larger” means the top number.

    e.g. On a 6×6, a cell > a given 5 must be 6

  • Where can the top number go?

    In each row and column, cross off every cell that is smaller than some neighbour — the top number cannot sit there. Do the same for 1 with every cell that is larger than a neighbour. One cell left means placed.

    e.g. A 4×4 column whose top three cells are each smaller than a neighbour: the 4 goes at the bottom

  • The locked pair

    If two cells in a line can only be the same two numbers, those numbers are used up there: strike them from every other cell in that line.

    e.g. Two cells in a row each 1 or 2: no other cell in the row is 1 or 2

  • Squeeze between two signs

    A cell with a larger neighbour on one side and a smaller one on the other sits strictly between them. When both neighbours are known, the gap may leave a single value.

    e.g. 2 < _ < 4 is 3

As the levels climb

  • The ladder starts on a 4×4 with about a quarter of the digits and half the signs given. It then cuts to fewer of each, and then to the fewest uniqueness allows.
  • The board then grows to 5×5, 6×6 and 7×7, each starting with more clues and cutting back to the fewest. The 7×7 has only two rungs: about an eighth of the digits and a third of the signs, then the fewest.
  • On the fewest-clue boards, digits go before signs, so what is left leans on the signs. Chains and the “where can the top number go?” question carry those boards.
  • With “Given digits” on “Signs only, where possible”, no digit is given unless one is needed for a single answer. Every bound comes from the signs, so start from the longest chain.
  • The clock is off unless you turn on Time pressure; then a board gets 60 seconds plus 12 seconds for each open cell.

Traps to avoid

  • Reading ∧ and ∨ the wrong way. The point faces the smaller cell: under ∧ the lower cell is larger; under ∨ the upper cell is larger. Say “point to small” each time.
  • Using a chain one step too short. In a > b > c > d, a is at least 4, not 3 — count every cell after it, not every sign.
  • Forgetting that signs do not wrap or jump. A sign compares only the two cells it sits between; a > b and c > d in one row says nothing about b and c.
  • Filling a line by its signs and forgetting the column. A value that suits the row’s signs can still repeat a digit in its column; check both before placing.

How to practise

  • Set “Largest board” to 5 while you learn to read chains, then raise it to 7 once the 5×5 fewest-clue boards fall without a guess.
  • Switch “Given digits” to “Signs only, where possible” when the digit-led boards feel routine. It forces the chain and top-number moves you might otherwise skip.
  • Before placing any digit, pencil the bounds of every cell in a chain in your head. It is slower on the first board and faster on every one after.
  • Leave Time pressure off while learning, and try Relaxed once a 6×6 comes out right first time.

Levels

It is untimed unless you turn the clock on. There are 11 levels with the default settings, and the counter stops at the real top. Your level is judged on your last ten answers, every ten: 90% moves you up, below 80% moves you down, which holds most people near four in five right.

All of it is in the settings beside Start — how many answers a judgement looks at and how often it recalibrates, the thresholds, the time, which questions, how many options, and whether you see each verdict at once or review them together at the judgement. You can also jump to any level.

More logic

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