Keen (Simon Tatham's Puzzles)

About the puzzle

Keen mixes two old ideas: the Latin square and mental arithmetic. In a grid of size N (6x6 by default), each row and each column holds every digit from 1 to N exactly once. On top of that, the grid is divided into outlined blocks, and each block shows a target number and an operation. The digits inside the block must produce that target with that operation.

There are no starting digits. Everything comes from the clues, and the fun is in how quickly the arithmetic narrows things down: a two-square block marked 30x in a 6x6 grid can only be 5 and 6. Simon Tatham added Keen to his collection at the end of 2009.

Rules

• Each row contains each digit from 1 to N exactly once.
• Each column contains each digit from 1 to N exactly once.
• 15+ means the digits of the block add up to 15.
• 60x means the digits of the block multiply to 60.
• 2- is always a block of two squares: one digit is 2 more than the other, in either order.
• 3÷ is always a block of two squares: one digit divided by the other is 3, in either order.
• A single-square block simply gives its digit.
• A digit may repeat inside a block, as long as the copies are not in the same row or column.

Controls

The game highlights a digit that appears twice in a row or column, and a block whose digits are complete but do not reach the target.

How to solve it

Single squares are free. A one-square block is just a given digit; fill those in first and remove that digit from their row and column.

Factorise the products. A multiplication target often has very few combinations. In a 6x6 grid, 30x on two squares is 5 and 6, 2x on two squares is 1 and 2, and any product containing 5 needs a 5 in the block.

Big differences and quotients are rare. In a 6x6 grid, 5- can only be 1 and 6; 4- is 1-5 or 2-6; 3÷ is 1-3 or 2-6; 2÷ is 1-2, 2-4 or 3-6.

Use whole rows. Each row of a 6x6 grid adds up to 21 and multiplies to 720. If a row is covered by blocks plus a few loose squares, that total tells you what the loose squares hold.

Pencil in the candidates. Press M to fill every square with all digits, then delete the impossible ones block by block. When a digit has only one place left in a row or column, it goes there.

Sizes and difficulty

The Type menu goes from 4x4 Easy to 9x9 Normal, with 6x6 at Easy, Normal, Hard, Extreme and Unreasonable, plus "multiplication only" versions where every block is a product. Each level needs a more advanced kind of reasoning; at Unreasonable you will sometimes have to try a digit and backtrack, though the solution is still unique. Custom sets any size from 3 to 9, the difficulty and the multiplication-only rule.

Playing here

Click Play to open the puzzle. The menu above the board has New game, Restart, Undo, Redo and Solve (which you can undo to keep going on your own), the Type menu with the sizes, and the Game menu, where you can type in a game ID or a random seed, download the current game as a file, load a saved file and change the preferences. Your game in progress is kept in this browser and comes back with its whole undo history the next time you open the page.

On a touch screen, tap a square and use the number keys that appear below the board; turn on the Mark button to enter pencil marks instead.

Credits and licence

Keen is part of Simon Tatham's Portable Puzzle Collection, written by Simon Tatham with code contributed by several other authors, and distributed with its source code under the MIT licence. This page runs a WebAssembly build made by FredyGames from the author's official source snapshot, with no changes to the puzzle; the source and the build script are hosted with the game, and the licence text is linked from this page.