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Puzzle Quarry

Futoshiki Solving Techniques

Five arguments, in the order to reach for them. The hardest one a grid forces on you is that grid’s difficulty — on this site, and in the code that measures it.

Ordered by what it costs your attention

The ladder is an order to try things in rather than a list of tricks to memorise, and the order is deliberately not the one a machine would pick. Reading the printed signs is free for a person because the signs never move and are right there on the paper; asking where a single value could live in a column costs a deliberate scan of five or nine cells. A computer finds the second of those easier. The ladder models attention, not processor time.

Each rung is also made cheaper by the rungs beneath it having been exhausted. Weighing two cells against each other while every cell in the row still has five candidates means considering everything; the same two cells after one more sweep may be down to two candidates apiece and answer themselves. Players who find this puzzle tedious are nearly always doing an expensive move while a free one is still on the table.

The order pays off in the grading too. Because the solver always takes the cheapest available move, the hardest move it was ever cornered into is a genuine statement about the grid: it means nothing simpler existed at that moment, rather than that the solver failed to notice something.

Where the label on a grid comes from

Every published grid is run through the same solver the hint button uses. It applies the cheapest argument that fits, over and over, until every cell is down to one candidate, and it records the highest rung it ever stood on. Easy stops at the sweep. Medium reaches the where-can-this-value-live question. Hard reaches a sign read against live candidates. Expert reaches a set argument.

Nothing is graded by how many numbers were given away and nothing is relabelled afterwards. The carve is capped at the level’s hardest allowed technique, so a grid can never come out harder than its label; it can come out easier, and when it does it ships under the level it measured rather than the level that was asked for. Ask for expert at 4×4 often enough and you will be handed a hard grid, because sixteen cells sometimes never produce a set worth arguing about.

There is a sixth rung and nothing here is allowed to stand on it. Writing a number in, following it until something breaks and rubbing it out will finish any grid whatsoever, which is exactly why it cannot be part of a difficulty scale. Its page explains how to do it properly if you meet one elsewhere, and why the generator throws away everything that needs it.

Common questions

How many techniques does a futoshiki need?

Five, and most grids need three of them. Reading the signs and sweeping the lines will finish an easy grid on its own; the other three are what each further level of difficulty means here.

Which futoshiki technique should I learn next?

Whichever one your current level stops at. Stalling on medium means asking where a value can live is not yet a habit; stalling on expert means set arguments are not. The hint names the argument it used, which is the fastest way to find your own gap.

More Futoshiki pages