Latin squares with arithmetic to hold them down
Fill each row and column with the digits once apiece, so that every outlined cage hits its target. Each board is proven to have exactly one filling and to be reachable without a single guess, and its difficulty is measured from the deductions it actually needs.
Why subtraction and division only appear on two cells
A cage is a set of cells, not a sequence — there is no first cell and no last one. Addition and multiplication do not care, so they work on a cage of any size. Subtraction and division do care: 9, 3 and 1 give 5 or −7 or 11 depending on the order you take them in.
Some puzzles paper over this with a rule like "largest first", which is arbitrary and, worse, makes the cage impossible to reason about on its own: you cannot list what a cage can contain until you know which of its cells is biggest, and knowing that is most of the deduction. So the two order-dependent operations appear only on pairs here, defined as the absolute difference and the quotient of the larger by the smaller. That is a real restriction and it is stated rather than hidden.
How a board is proven, and why it takes a moment
A random Latin square is generated, carved into connected cages, and each cage given an operation and target derived from the digits it happens to contain. Then almost all of those boards are thrown away.
First a solver that works the way a person does tries to finish it — listing what each cage can contain, crossing off digits, looking for a digit with one home left. A board it cannot finish needs a guess somewhere, which is a legitimate puzzle and an unpleasant one, so it goes.
Then a second, exhaustive solver enumerates fillings and stops at two. The board is published only if it found exactly one. Those two checks are not the same thing and neither replaces the other: the first would happily finish a board whose eliminations were unsound, and it is the enumeration that would catch that.
Difficulty is measured. The band is the hardest technique the first solver genuinely needed — not the cage count, which says very little. Sampling boards from this generator turns up 17-cage boards labelled beginner and 17-cage boards labelled medium.
How to start
Look for the cages with the fewest possibilities rather than the smallest targets. A two-cell cage with a division target is usually the most constrained thing on the board — on a 5 × 5, "3÷" can only be 1 and 3, and nothing else. A large sum spread over four cells is the least constrained, and it is often the last thing you resolve rather than the first.
The second thing worth knowing: two cells of the same cage that sit in one row cannot hold the same digit, and that cuts the possibilities more than people expect. An L-shaped three-cell cage adding to 6 has far fewer arrangements than the same target spread across three separate rows.
After that it is ordinary elimination. A digit already placed in a row is gone from the rest of it; a digit with only one home left in a column belongs there whatever else that cell might have held.
What the difficulty labels mean
Beginner — cage combinations and last-digit-standing, nothing else. Every step is visible once you write out what a cage can hold.
Easy — also needs a digit with only one home left in a row or column, which is harder to see because you are scanning for an absence rather than a presence.
Medium — also needs pairs: two cells in a line that share the same two candidates own them both, so nothing else in that line can take either. The first point at which you have to reason about cells you cannot fill yet.
Hard — also needs a cage confined to a single row or column, which uses a digit up for the whole line even though you cannot say which of its cells holds it.
Nothing above that band is published, because the next techniques up are chains rather than deductions, and a puzzle that needs one is not more satisfying — it is just longer.
Playing without a mouse
Arrow keys move, digits enter, Backspace clears. Holding Shift with a digit writes a pencil note instead. The grid is a proper ARIA grid, so a screen reader announces each cell's position, the digit in it, and the cage it belongs to — including that cage's target and operation, which is the part a purely visual outline does not convey.
Nothing depends on colour. Cage outlines are heavier rules rather than tints, and a wrong digit is underlined as well as recoloured, so the board reads in greyscale and prints correctly.
Related
Sudoku for the same discipline with boxes instead of arithmetic, or Number Forge if it is the arithmetic you came for. Everything else is on the puzzle hub.
Questions people actually ask
Is this KenKen?
It is the same family of puzzle — a Latin square carved into arithmetic cages — built from scratch with its own generator, solver and difficulty measurement. KenKen is a trademark of a company we have nothing to do with, and nothing here is copied from any commercial puzzle set.
Why can't a three-cell cage subtract?
Because a cage is a set of cells with no order, and subtraction has one. Nine, three and one give 5 or −7 or 11 depending on which you take first. Some puzzles fix this with a rule like "largest first", which makes the cage impossible to reason about on its own — you cannot list what it might contain until you already know which cell is biggest. Here subtraction and division appear only on pairs, as the absolute difference and the quotient of the larger by the smaller.
Can a board have two answers?
No, and that is enforced rather than assumed. Every candidate board is handed to a solver that enumerates fillings and stops once it finds two; it is published only when the count is exactly one. Finding one solution proves only that one exists.
Do I ever have to guess?
Never. A board that a technique solver cannot finish — listing cage combinations, crossing off, hunting for a digit with one home left, pairs, and cages confined to a line — is discarded during generation rather than shipped with a harder label. If you find yourself stuck, the hint will usually show a step you have not spotted.
Why does a 7 × 7 hard board take a few seconds?
Because it is rare. Cages are carved at random and the difficulty is whatever emerges, so the generator builds, measures and discards until something lands in the band you asked for. A hard 7 × 7 turns up roughly once in a few hundred boards, and each one has to be proven unique before it can be counted.
Are hints just the answer?
Only if you ask three times. The first press names a cell and the cage doing the work, the second explains which combinations are left and what they force, and only the third fills the digit in. If your grid has drifted from the only valid filling, it says that instead — explaining a deduction about a position that can no longer be reached helps nobody.
Is my progress saved?
In your browser, on this device. There is no account and nothing is sent anywhere. Clearing your browser data clears it.