Shikaku 四角に切れ

Shikaku (四角に切れ, shikaku ni kire, roughly “cut into squares”) is a Nikoli puzzle that Nikoli also publishes in English as Divide by Box. A grid is scattered with numbers, and the task is to cut the whole grid into rectangles so that every rectangle holds exactly one number and that number is the rectangle’s area. A 6 might be a 1×6 strip, a 2×3 block or a 3×2 block; a 9 is a 3×3 square or a 1×9 strip; a 1 is a single cell. Nothing may overlap and no cell may be left out.

LogicPuzzles.io publishes a new 8×8 Shikaku every day plus an unlimited supply of 6×6, 8×8 and 10×10 grids. Every puzzle is run through an exact solver before it is stored: each one has a single solution, and a deduction grader confirms it can be finished by reasoning alone, so no guessing is ever required. Draw a rectangle by dragging from one corner to the opposite corner, or on a phone tap one corner and then the other. Your progress, timer and best times are saved in your browser; there is no account and nothing to install.

Start with the numbers that have the fewest possible shapes. A 1 is its own cell. A prime such as 2, 3, 5 or 7 can only be a strip one cell wide, so look along its row and its column and see how far a strip can run before it meets another number or the edge of the grid. Numbers in corners and along edges have far fewer placements, and a number boxed in by its neighbours may have only one shape left.

Next, look for cells that only one number can reach. A rectangle may never contain a second number, so each number’s reach is limited by the numbers around it. If an empty cell can be reached by exactly one number, that cell belongs to it, and every shape of that number that misses the cell can be discarded. The same idea works in reverse: if every remaining shape for a number covers a certain cell, that cell is settled before the exact shape is, and no other number may use it.

Large numbers are often the most constrained. A 12 on a 10×10 grid must be a 3×4 or 2×6 block in one of its orientations, and there may be only one place such a block fits without swallowing a neighbour. Once a big rectangle is fixed, the strips squeezed between it and the edge usually fall into place.

The usual mistakes are drawing a rectangle that covers two numbers, forgetting that a square counts as a rectangle, and stretching a strip one cell too far. The board turns a rectangle red as soon as its area disagrees with its number or it holds two numbers. Press Check when you are unsure: it tells you whether every rectangle drawn so far is part of the solution and how many are still to go.

Shikaku is an original puzzle of Nikoli, the Japanese publisher of the puzzle magazine Puzzle Communication Nikoli, and Nikoli’s English site presents it as “Shikaku (Divide by Box)”. The Japanese name 四角に切れ (shikaku ni kire) means “cut into squares”. Nikoli describes it as a puzzle with a very simple rule where the fun lies in finding rectangles by logic rather than by guessing, and points out that it doubles as multiplication practice for children, since a 12 can be 1×12, 2×6 or 3×4. Outside Japan it is also known as Rectangles, the name it carries in Simon Tatham’s Portable Puzzle Collection.