Arithmetic Cages

How to play

Tap a cell and then a number, or use the keyboard: arrow keys move, the number keys write, Backspace clears and N switches to pencil marks. On a 4×4 you are placing 1 to 4, on a 5×5 1 to 5, and on a 6×6 1 to 6.

Every cell belongs to exactly one outlined cage. The number in a cage’s corner is what its digits have to produce, and the sign says how — except in the no-operator mode, where working out the operation is part of the puzzle. A cage of a single cell simply is that number.

Nothing is judged until the grid is full. A half-filled cage that cannot come right is not marked as a mistake, because you may be partway through working it out.

Rules and controls

• No digit repeats in any row or any column.
• Every cage’s digits must produce its target.
• Minus and divide cages read either way round, and cover exactly two cells. Plus and times cover any number.
• Divide cages always divide exactly — there are no remainders.
• A digit may repeat inside a cage if the row and column rules still hold.
• Pencil marks never count as answers.

Modes

• 4×4 Beginner — The smallest board and the place to start.
• 5×5 Standard — The usual board: more room, longer chains of reasoning.
• 6×6 Expert — Thirty-six cells and cages up to four.
• No operators — Targets without the signs — work out the operation as well.
• Daily — One 5×5 board, the same for everybody, seeded by the UTC date.
• Choose your own — Pick the board and how hard it should be.

Tips

• Start with the single-cell cages. They are free digits, and each one removes a possibility from a whole row and a whole column.
• Small targets on big cages, and big targets on small ones, are the most informative. A 3+ over two cells can only be 1 and 2.
• A divide cage is usually the tightest clue on the board — very few pairs divide exactly.
• Every row and every column holds each digit exactly once, so the sum of a full row is fixed. That often pins the last cell of a cage that straddles two rows.
• Use pencil marks on the cells with two possibilities and come back to them; the cage that resolves them is usually somewhere else.

Privacy and local play

Everything runs in your browser. Your current puzzle, your settings and your best times are kept on this device only, and the button under the board clears them without touching any other Zerdly tool. No account, no sign-in, and nothing about your play leaves the machine you are on.

Questions

Which way round do a minus or divide cage go?

Either way, and that is a rule rather than a leniency. A cage is a group of squares, not a sequence — nothing about it says which square comes first, and you cannot tell by looking. So a 3− cage is satisfied by 5 and 2 in either arrangement, and a 2÷ cage by 6 and 3. It is also why minus and divide cages only ever cover two cells: there is no order-free reading of “a − b − c”, so the same three numbers could make several different answers depending on an order the grid never states. Plus and times cover any number of cells, because they do not care about order.

Can a digit repeat inside one cage?

Yes, as long as the row and column rules still hold. A cage that bends around a corner covers cells in different rows and columns, so the same digit can legitimately appear twice in it. The only rules are: no repeat in a row, no repeat in a column, and every cage hits its target.

Do divide cages ever leave a remainder?

No. A divide cage always divides exactly. If two digits do not divide evenly they are not a divide cage at all, and the generator will not make one.

What do the difficulty names mean?

A band is the weakest kind of reasoning that finishes the board, measured by actually solving it rather than a label attached by the generator. Gentle means you can finish by taking each cage together with the digits already in its row and column. Steady also needs you to spot that a digit has only one place left in some row or column. Tricky cannot be reasoned out one step at a time — at some point you have to try something and follow it through.

Does the hint ever guess?

No. It only offers a cell that follows from what is on the board by the same deductions a player could make, and it says which reasoning got there. When nothing follows in a single step it says so rather than showing a digit it could not justify. If the board already contradicts itself it tells you that instead of advising from a wrong digit.

Why does Check not tell me which digits are wrong?

Because that would be a hint wearing a different label, and the route already has one of those. Check tells you how many digits cannot be right, which is enough to know whether to look again without doing the looking for you.

Can I read the answer out of my browser storage?

No. The board is not stored at all — only its seed, so it is rebuilt each time — and what is stored beside it is an eight-character hash of the solution rather than the solution. The hash is there so a restored board can be confirmed to be the board that was saved. It is not a security measure and is not offered as one; anybody who wants to solve the puzzle themselves can have the answer.

What is the Daily?

One 5×5 board, chosen by today’s UTC date, the same for everybody on the same version. Your time and hint count are kept on this device and nowhere else, along with a streak. The key includes the generator version, so a result stays attached to the puzzle it was earned on.

Is anything sent to a server?

No. The generator, the solver, the hints, the sounds and your saved puzzle all run in your browser. There is no account, and clearing this game’s data leaves every other Zerdly tool untouched.

Related interactives

• Codebreaker — deduction against a hidden code, with a hint that cannot cheat.
• Kakuro — cross sums, the other arithmetic grid on the site.
• Sudoku — the other Latin-square puzzle, without the arithmetic.
• Nonogram — picture logic, also with exactly one solution per board.