How to Solve Shikaku Puzzles

Shikaku, invented by Yoshi Anpuku, and originally published by Nikoli, a game and puzzle magazine in Japan, is an area-division type of puzzle.

Your goal is to divide the grid into rectangles, in which every rectangle contains one and only one of the number clues, and has an area equal to that number. For example, the number 8 will be in a rectangle containing 8 squares. It might be a 2×4 shape, or a 1×8 shape, in either a horizontal or vertical orientation. All grid squares must be used.

Rectangles may only border each other – they cannot cross. Some knowledge of multiplication and a few prime numbers can help, but is not required to solve the puzzle.

Let’s start in the upper left corner, where we have a 3 and a 5 given as clues. Neither of these can be more than 1 square wide, because they are both prime numbers – nothing divides into them evenly, so a rectangle can only be 1×3 and 1×5 for them.

Using blue for the 3 and red for the 5, you can see where the different placement options would cross each other. Because that is not allowed, we only have a few possible placements for these rectangles which do not cross.

This placement doesn’t work, because it leaves the top left grid square unused

This one fails because it forces the clue for the 12-square rectangle to be only 1 square tall, and the grid isn’t 12 squares wide.

This means this orientation must be the correct one.

Next, let’s look at these two squares. Remember that every square in the grid must be used in a solution rectangle.

The only nearby numbers are 6 and 16. While the 16 could be part of a 4×4 square to reach them, it would have to cover the 6 to do so. Therefore the 6 must be the clue we will use.

Because the rectangle must include the number, and the total number of squares from the top row to the 6 is three, we must conclude that it is a 3-square tall by 2-square wide rectangle.

The same concept applies to this cluster of squares. They have to be covered by a rectangle, and the only nearby number which won’t cross another rectangle is the 20.

Because this gap is only 2 squares wide, it means the entire rectangle must be 2 squares across. Fortunately, the grid is 10 squares tall, so we know we have a narrow rectangle that is 2×10 containing 20 squares.

After drawing in the 20-square rectangle, it’s easy to see that the entire bottom row follows the same pattern. The only numbers close enough to reach those squares without crossing other rectangles can only do so by creating a rectangle that is 2-squares wide.

The sixes are each 3 squares away, while the eights are each 4 squares away. 6/3 = 2, and 8/4 = 2, so we can fill this area with four different rectangles, each two squares wide.