This puzzle belongs to the family of cryptarithms (also known as alphametics), the brain teasers in which the digits of an arithmetic operation are replaced by letters and you have to reconstruct the mapping. The ancestor of them all is the famous SEND + MORE = MONEY, first published in 1924; since then the genre has become a classic of recreational mathematics and of puzzle magazines.
Here the idea is taken onto a 3×3 matrix: not a single operation, but six interlocking operations that must all balance at the same time. No luck is involved and you never have to guess: every puzzle has a guaranteed unique solution, reachable by logic alone.
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The grid holds nine numbers arranged in a 3×3 matrix, but their digits are hidden behind letters. There is a single golden rule:
The three horizontal operations and the three vertical ones must all balance at the same time. The operators (+, −, ×, :) are always visible: your job is to work out which digit hides behind each letter, and the correct mapping is always the only one.
Here is a real puzzle, with letters in place of the digits:
B + D = E × + : AC − AB = A BD − AE = E
Each row is an operation read left to right; each column is an operation read top to bottom (the third row holds the results of the three vertical operations). The letter A appears in four different numbers: in all of them it stands for the same digit.
How do you start? The letter A opens the numbers AC, AB and AE, so it cannot be 0. Furthermore AC − AB = A: two two-digit numbers that both start with A, whose difference is a single-digit number — so the subtraction only involves the second digits, which means C − B = A. The first row too, B + D = E, ties three letters together as single digits. Cross-referencing these constraints leads to the one and only solution: A=1, B=2, C=3, D=6, E=8, that is
2 + 6 = 8 × + : 13 − 12 = 1 26 − 18 = 8
Check it for yourself: all six operations balance at the same time, horizontally (2+6=8, 13−12=1, 26−18=8) and vertically (2×13=26, 6+12=18, 8:1=8).
Not all operations are equally informative. A multiplication or a division narrows things down far more than an addition: only a few pairs of digits give a product compatible with the length of the result. Look for the operation with the fewest letters in play, or the one whose result has one digit less than its operands: those are the best footholds to begin with.
The first letter of a multi-digit number cannot be zero. That is a free exclusion, to be marked in the matrix straight away: crossed with the others, it is often the one that closes the first line of reasoning.
As in Sudoku, real progress does not come from "trying a number" but from eliminating candidates. If you prove that a letter cannot be 7, mark it; when only one free cell is left in a row of the matrix, that letter must take that digit.
The symmetrical argument — "this digit is possible for only one letter, so it is hers" — looks just as obvious, but on its own it does not hold: there are 8, 9 or 10 letters in play, so that digit might not appear in the puzzle at all. It only becomes valid once the missing digits have been ruled out, which is exactly the job of the option described further down.
Look at the last digit of the results. In an addition, the final digit of the result depends only on the final digits of the addends (plus any carry). In a multiplication, if the result ends in 0 then one of the factors must contain a 5 or a 0. Reasoning of this kind rules out many combinations without working anything out in full.
Every assignment enables more: a fixed digit removes itself from all the other letters and blocks all the other digits for that letter. In the game this propagation is automatic, but it is worth understanding: it is the engine that turns a single deduction into a chain of them.
Before each game you choose how large the numbers in the puzzle will be. There are always 8, 9 or 10 letters in play in all three cases: what changes is only the size of the numbers, and therefore how many digits are on show in the grid.
| Choice | Numbers | What to expect |
|---|---|---|
| Small | 1 to 3 digits | Short sums, nearly all doable in your head. This is how the game has always behaved. |
| Medium | 1 to 4 digits | The middle ground, and the default choice: three-digit numbers show up, and now and then a four-digit one. |
| Large | mostly 3 and 4 digits | The setting closest to the puzzles in the magazines: heavier arithmetic, but also more digits in sight. |
The labels say what changes; they do not promise an order of difficulty. Larger numbers mean longer sums to work out, but also more visible digits and therefore more footholds for deduction: which of the three is really the toughest depends on how you reason. In all cases the solution stays unique: guessing is never necessary. Best times are kept separately for each choice, since they would not be comparable.
The matrix below the grid is the tool that actually solves the puzzle: one row per letter, one column per digit from 0 to 9. The game is won by elimination, exactly as you would do it with a pencil on a newspaper.
Next to the matrix there are two modes, which decide what a click on a crossing does:
The Assign mode is momentary: once the assignment is made the game switches back to Rule out on its own, which is what you want 90% of the time. While it is armed you cannot miss it: the button lights up blue, the matrix takes a pulsing blue frame, and a blue dot ● follows the pointer. It is always the same blue the square will take once assigned: the colour previews the result, while red in this game stays the colour of a mistake. So you never have to look at the buttons to know what the next click will do.
To undo a mark, just click it again, whichever mode you are in: this works both for the × marks and for assignments, right or wrong ones alike. That is why the old "Clear" mode is gone. A letter that is already assigned, on the other hand, cannot be moved straight onto another digit: you have to cancel it first, so a stray click cannot throw away a piece of reasoning.
Assigning a digit propagates the exclusions by itself: that digit is crossed out for all the other letters, and all the other digits are crossed out for that letter. These automatic exclusions appear as a fainter dot ·, to tell them apart from the × marks you reasoned out yourself; if you remove the assignment, they disappear on their own. When only one free cell is left in a row, the game highlights it in yellow: the deduction is closed, that is the digit.
Besides the matrix, you can select a letter by clicking it at the start of its row, by clicking a cell of the grid, or by pressing the corresponding key on the keyboard; then press a digit 0-9 to assign it. The digit appears in every position where that letter occurs. If a digit is already assigned to another letter, the game will not let you reuse it.
The = signs in the grid work as a visual check: they turn green when the operation balances, and red when all the numbers are revealed but the arithmetic does not add up. An operation that still contains letters stays neutral.
Below the buttons there are two options, both remembered by the browser and changeable at any time, even mid-game.
Flag wrong exclusions and assignments right away. With the option on, a wrong assignment is shown in red and counts as a mistake, with a 30-second penalty. The same goes for an × exclusion placed on the correct digit: the square turns red, and that red is help in all but name (it tells you the answer is right there), so it costs a mistake and 30 seconds too. It is charged once per square, and only when the exclusion is placed: removing it and putting it back adds no further penalty. Turn it off and you play "the old-fashioned way": you will only notice the mistake from the arithmetic that fails to add up.
Cross out the columns of the digits missing from the puzzle. There are 8, 9 or 10 letters in play, so one or two digits may not appear at all. This makes reasoning "by column" lame: a column with a single free cell proves nothing as long as that digit might be out of play. With the option on, the dead columns are crossed out from the start and the matrix becomes a full square again, where reasoning by column is worth as much as reasoning by row. It is a help in every sense, so it is off by default; switching it on and off crosses out and frees the columns, and never touches the × marks you made yourself.
The game is saved automatically in your browser after every move: if you close the page or the computer, next time you will resume exactly where you left off, with time, mistakes, assignments and exclusions preserved.
© 2026 Luciano Manenti