數學方陣

Math Square: rules and solving techniques

Use arithmetic cage clues to complete a Latin-square grid. This guide reflects the controls and win conditions in the current browser build. It is reviewed by the PocketJoy game-testing team and maintained by Long-Term New Media Limited.

Puzzle rules

Math Square combines a Latin square with arithmetic. On an N×N grid you place the digits 1 to N so that every row and every column contains each digit exactly once — the same row and column rule as sudoku, but without any box constraint. The extra rule comes from the heavily outlined regions called cages. Each cage carries a target number and an operator in its top-left corner, and the digits inside it must produce that target under that operation. Addition and multiplication cages may hold any number of cells, while subtraction and division cages hold exactly two, with the result taken as larger minus smaller or larger divided by smaller, so order does not matter. A single-cell cage simply states its answer. The grid size is adjustable and larger boards are harder. Press check when finished and the game confirms the square or reports that digits still violate the row, column or cage rules. Your completion time is stored on your device.

Reasoning and solving techniques

The most efficient opening is hunting for forced cages. Single-cell cages are answers outright, so fill them all first. Then examine two-cell subtraction and division cages: on a 6×6 board, a cage marked divide-by-5 can only be 1 and 5, and one marked minus-5 can only be 1 and 6. Prime factorisation is especially useful for multiplication cages: on a 6×6, a two-cell cage of 30 can only be 5×6, because the other factorisations of 30 — 1×30, 2×15, 3×10 — all fall outside the range 1 to 6. Addition cages yield to extreme sums: a three-cell cage totalling 6 with distinct digits must be 1+2+3. Once those certainties are placed, switch to Latin-square reasoning: a row with four digits already placed leaves only two candidates for its remaining cells, and combined with their cage arithmetic that usually resolves immediately. A more advanced technique is cage summing: the digits of a full row always total the same amount, the sum of 1 to N, so if several cages lie entirely within one row, adding their targets and subtracting from the row total gives the sum of the remaining cells, which frequently pins down cells that span rows.

Easy mistakes to make

The most common mistake is importing sudoku's box rule. Math Square constrains only rows and columns, and cages do not require distinct digits inside them — an L-shaped three-cell addition cage may perfectly well contain 2, 2 and 5, provided those two 2s do not share a row or column. Assuming cages must hold distinct digits eliminates a great many legal solutions and leads straight to an apparent dead end. The second mistake is forgetting that subtraction and division cages are order-independent: a cage marked minus-2 may be 5 and 3 or 3 and 5, and which sits where is settled by the row and column constraints rather than by assuming the larger goes on the left. The third is doing only arithmetic without positional reasoning, or only positional reasoning without the arithmetic. Math Square's difficulty lies precisely in alternating between the two, and relying on either alone stalls quickly. The fourth is starting on a large grid; boards of 6×6 and above have many candidates and complex cage shapes, which is hard going before the common factorisations are familiar.

Who will enjoy it

Math Square was devised in 2004 by the Japanese mathematics teacher Tetsuya Miyamoto, expressly as mental arithmetic and logic training for students, which makes it particularly suitable for learners — solving a single 4×4 square works through dozens of mental calculations, each with a logical purpose, which is far more engaging than a page of sums. For adult players it is an excellent sudoku variant: the row and column rule is familiar while the arithmetic layer changes the thinking entirely. The adjustable grid size gives it a wide difficulty range, from a 4×4 solvable in a few minutes and suitable for beginners or children, to 6×6 and beyond which demand a systematic order of reasoning. There is no luck involved: every puzzle has a unique solution and every cell is deducible. Nothing counts down, and the timer is only a reference for comparing with yourself.

Helper features

Grid size is selectable, which adjusts both difficulty and character — small boards lean towards mental arithmetic, large ones towards logical deduction. The records panel holds your fastest time, and the sidebar timer shows your current elapsed time beside that best. The check button validates whether the grid satisfies both the row and column rule and every cage constraint at once, and when it fails it states specifically that digits still violate the row, column or cage rules, telling you which direction to look. A hint button settles one cell when you are stuck. Cage boundaries are drawn visibly differently from ordinary grid lines, with the target and operator in the top-left corner where they cannot be confused with an entered digit. Completion shows your total time and offers to raise the difficulty by moving to a larger board, giving a natural progression.

Controls and saved progress

Control is a two-step action: tap a cell to highlight it, then tap a digit from the pad at the bottom of the screen. The number of keys adapts to the grid size, so a 4×4 shows only 1 to 4 and out-of-range entries are impossible. Phone players never need the system keyboard and can complete a puzzle one-handed, with desktop and touch identical. To change an entry, select the cell again and enter a new digit, which overwrites directly. Your fastest time is kept in the browser's local storage on your own device, never uploaded to any server and requiring no account; clearing browser data removes it. The language button switches between Traditional Chinese and English instantly, updating every prompt and result string. Once the page has loaded, the game itself needs no network connection.

Solving FAQ

Reviewed by the PocketJoy game-testing team · Operated by Long-Term New Media Limited · hklongtermad@gmail.com