You are given a grid and a list of sets. Place every set into the grid exactly once. Some sets are given.
On a touch screen, a tap does what a click does, and a long press what a right-click does (more ways to right-click).
Every letter has a fixed position in each cell. A position shows its letter once the letter is marked present, a cross once it is cleared, and nothing while it is undecided. A cell whose set is given sits on a lighter surface than the ones you fill.
Click a letter to add it to that cell’s set; again to rule it out, and a third time to leave it undecided. Right-click it to rule it out of that cell’s set, going round the other way. Keep the button down and drag to do the same to every letter you pass over that looked the same as the first.
With the keyboard, the arrow keys move a cursor around the board. Enter does what a click does to the letter under it, and Space what a right-click does. Press Backspace or Delete to leave the letter under the cursor undecided; clicking it round gets there too.
The tally band beside the grid works in both directions. Click a set in the tally and every cell it could still legally go in is spotlighted; if it’s already placed, the cell it lives in is shown in a different color. Going the other way, the inspect badge above a cell — or simply moving the keyboard cursor onto it — highlights every set that cell could still hold.
The answers come from what’s visible on the board: the cell’s own marks and rule-outs, the horseshoes to its decided neighbors, and the rule that each set appears once. The horseshoes rule out two sets on their own: the empty set sits inside every set, so it can only go in a cell that every neighbor’s horseshoe points into, and the full set only in a cell whose horseshoes all point out. Nothing is read off the solution, so the aid can’t do the deducing for you.
Some deductions are about whole sets rather than letters: “{A,C} can’t go in this cell”. To record one, focus the cell (tap its inspect badge), then tap the set in the tally. It is struck through, and drops out of the sets the aid lists for that cell. Tap it again to take it back. While a cell is in focus, tapping the tally rules sets out of it; to spotlight a set instead, tap the badge again to let go of the cell first.
With a keyboard, move the cursor down past the bottom row of the grid into the tally, keeping the cell you were on in focus. The arrow keys move between sets, Enter or Space rules one out, Backspace takes a rule-out back, and moving up from the tally’s top row returns to the grid.
Check & save treats a letter marked or crossed out against the solution as a mistake, and draws it in red. It treats a set ruled out of the cell it belongs in the same way.
This puzzle type was invented by Inaba Naoki under the name サブセットリンク, released as a Java Applet with a puzzle generator.
Hint explains the next step rather than simply making it, one letter or one ruled-out set at a time. It reasons only from the horseshoes, the given sets and your own letters and rule-outs, so it carries on from wherever you are, as long as none of those is wrong. If one is, whether or not it breaks a rule yet, it asks you to fix the highlighted mistakes first.
In its words, to mark a letter present is to add it (left-click its position, or Enter), and to clear a letter is to rule it out (right-click, or Space).
While a hint is on show, the tally strikes through the sets already ruled out of the cell the step is about, just as it does for a cell in focus.
The hint rules sets out too, when a deduction needs it, and says why. The usual reason is a horseshoe: the set at its open end must be strictly bigger than the set at its closed end. So a set can’t go at the open end if none of the sets that can still go at the closed end fits inside it with room to spare, and a set can’t go at the closed end if none of the sets that can still go at the open end holds it and more.
Every puzzle is played on a 4×4 grid over a four-letter universe. That is the one size where the sixteen possible sets exactly fill the sixteen cells — the bijection the puzzle is built on — so the board never changes size. The one thing you can choose is how hard the deductions have to work.
These parameters are available from the ‘Custom type…’ option on the ‘Type’ menu.
Easy puzzles can be solved by reading the horseshoes forwards: a set placed on the closed end of a horseshoe tells you letters the open end must contain, and a letter ruled out of the open end is ruled out of the closed end too.
Normal puzzles also need the argument run backwards. Because a horseshoe forces the closed end to be a strictly smaller set than the open end — two cells can never hold the same set, since each set is placed exactly once — a candidate for the closed end is only viable if some candidate for the open end still contains it. Rule out the last set that could sit on the open end above it, and the closed end loses that option, even though nothing about it changed directly.