Futoshiki

Read the inequality chains first

Most people fill in Futoshiki like a small Sudoku and only glance at the greater-than signs when they get stuck. Flip that habit. The signs are the strongest clues on the board, especially where several link into a chain. Here is the key idea, with a worked case. In a five-wide grid, suppose four cells in a row read a < b < c < d. The cell a sits below three others, so it cannot be 4 or 5, or even 3, because b, c and d must all squeeze in above it. So a is 1 or 2. The same squeeze from the other end says d is 4 or 5. A chain of all five cells pins every single value: it must be exactly 1, 2, 3, 4, 5 in order. Solvers call this the chain squeeze, and the longer the chain, the harder it squeezes. Hunt for the longest chain on the board before you write anything else.

A lone sign earns its keep too. A cell on the large side of a > can never be 1, and its smaller neighbour can never hold the grid's top value. Every arrow you honour trims candidates from two cells at once, which is why ignoring them is such a waste.

Then bring in the Latin square

The rest is the familiar rule: each number from 1 up to the grid's size appears exactly once in every row and every column. Fold that together with the signs and cells resolve in a chain reaction. Borrow two habits from Sudoku. A naked single is a cell with only one candidate left, so fill it. A hidden single is a number with only one home left in its row or column, so place it, even if that cell still shows other candidates. The signs feed both. If a row has two spots left for its 5 and one of them sits on the small side of a greater-than sign, that spot is dead and the 5 goes in the other one without further thought. Write candidates lightly in the corners of empty cells and cross them off as the signs and lines kill them. On the bigger grids, the pencil marks are not optional; they are where the solve actually happens.

• Find the longest inequality chain and pin its two ends before anything else.
• Whenever you place a number, recheck both neighbours across any sign. One fill often triggers the next.
• Look for the grid's extremes: the biggest number can never sit on the small side of a sign, and 1 can never sit on the large side.

Where solves go wrong

Two mistakes cause nearly every broken grid. The first is satisfying a sign locally while breaking a line, like placing a 3 > 2 pair that reads fine on its own but doubles the 3 in that column. Check the line every time, not just the arrow. The second is reading a sign as if it applied to a whole region. Each sign binds only the two cells it sits between, nothing further. When you do hit a contradiction, resist erasing everything. Walk back to your most recent placement, since the error almost always lives there, and the candidates you pencilled earlier are usually still sound. A quieter third slip is tunnel vision on one corner of the board. If a region stalls, leave it. Solve somewhere else and come back, because a number placed three rows away removes candidates everywhere along its row and column, and stuck corners have a habit of opening themselves while you are not looking.

The not-equal puzzle

Futoshiki takes its name from a Japanese word meaning inequality, and it is sometimes published in English as Unequal or Hutoshiki. It grew up in Japanese puzzle magazines and reached a big Western audience in the 2000s, when British newspapers took it up as a daily puzzle alongside Sudoku, The Guardian among them. It is younger and less famous than its cousin but scratches the same itch, with one twist that keeps it fresh: the answer is not hidden only in what is missing from each line, but in the relationships the signs pin down between neighbours. A well-made grid has exactly one solution and never needs a guess. Even a small board can put up a proper fight once the chains start pointing in awkward directions, which is exactly when the chain squeeze pays for itself.