Shikaku

Let the numbers pick their rectangles

Shikaku asks you to carve the whole grid into rectangles, one number per rectangle, where the number equals the rectangle's area in cells. Drag across the grid to draw a rectangle; drawing over an existing one replaces it, and right-clicking or tapping a rectangle erases it. The way in is to start with numbers that have only one possible shape. A 2 can only be a 1x2 domino, so it has just a few placements. A prime number like 5 or 7 must be a thin strip, one cell wide, so it either lies flat or stands up, and the walls of the grid often leave it a single legal position. Think in factor pairs. A 12 could be 1x12, 2x6, 3x4, 4x3, 6x2 or 12x1, but the free cells around it usually kill most of those before you start.

A worked corner

Here is the kind of reasoning the whole puzzle runs on. Suppose a 9 sits one cell away from the top-left corner. A 9 is either a 1x9 strip or a 3x3 square. If the row it sits in has another number three cells to its right, the long strip is dead, because a rectangle may never contain a second number. That leaves the 3x3 square, and there are only a few ways to hang a 3x3 around that cell. If two of them would swallow a neighbouring number, the last one is forced, so draw it. Every rectangle you fix this way becomes a wall. The cells it covers are gone, and the numbers around it now have fewer directions to grow, which forces more rectangles in turn. Corners and edges are the best early targets for exactly this reason: a rectangle there has fewer ways to sit.

Two habits that finish hard grids

First, remember the number does not have to sit in a corner of its block. It can be anywhere inside, which is why a rectangle sometimes extends in a direction you overlooked. When a number seems stuck, slide its rectangle around the number in your head, holding the shape fixed, and check every offset. Second, interrogate empty cells directly. Every cell must end up inside exactly one rectangle, so pick a bare cell far from the action and ask which numbers could legally reach it. Reaching it means a rectangle of that number's size can cover both the number and the cell without crossing another number. Often only one candidate survives, and that tells you where its rectangle must stretch, even before you know its exact final shape. If you find an orphaned cell late in the solve that nothing can reach, an earlier rectangle is the wrong size, and the orphan tells you which neighbourhood to rethink. One more sorting habit helps: treat perfect squares with suspicion and respect. A 4 can be square or strip, a 9 can be square or strip, and the square option is easy to forget when your eye has been drawing skinny rectangles all game. Big numbers cut the other way. A 16 or an 18 is so large that the grid's edges usually allow only one or two placements, so the biggest number on the board is often the best opening move after the primes.

• Open with primes and small numbers; strips with one legal spot cost nothing.
• List factor pairs for big numbers, then cross out shapes the walls and other numbers forbid.
• Check lonely empty cells early. A cell only one number can reach is a free deduction.
• Two numbers can never share a rectangle, and no cell may be left out. Both rules cut both ways.

A Nikoli puzzle

Shikaku comes from Nikoli, the Japanese publisher whose magazine Puzzle Communication Nikoli has been the wellspring of grid logic puzzles since 1980, Sudoku's worldwide fame included. The full Japanese name, Shikaku ni kire, means roughly divide by squares, which is the whole rule in four words. Nikoli's founder Maki Kaji, often called the godfather of Sudoku, championed exactly this kind of design: a rule you learn in ten seconds, hand-tuned grids, and no arithmetic beyond a little multiplication. A well-made Shikaku never needs a guess. The pleasure is watching a blank grid resolve into a clean mosaic of rectangles where every piece is forced, and the moment the last block clicks into the last gap is as tidy an ending as puzzles offer.