How to play Shikaku
One rule, two deductions, and no luck. Here is the whole game, worked through on a real 5×5 board.
By Gonzalo Fuentes, who built Kaku Updated
The rule, in one sentence
Divide the whole grid into rectangles so that each rectangle contains exactly one number, and that number equals the number of cells the rectangle covers.
That is the entire game. There is no trivia, no arithmetic beyond counting cells, and no luck. Rectangles may not overlap, and when you finish they cover every cell. A number that says 6 lives in a rectangle of six cells: 1×6, 2×3, 3×2 or 6×1, whichever fits.
Notice that the numbers add up to 25, the number of cells. They always do, in every Shikaku ever made. That is not a hint, it is a consequence: the rectangles tile the grid, so their areas sum to it.
Deduction one: the forced single
Take a number. Write down every rectangle of that exact area which covers it and no other number. If there is only one, you have found a forced single, and you can place it with no risk at all.
Look at the 3 in the top row. A three-cell rectangle is either 3×1 or 1×3. Standing it upright would swallow the 3 directly below it, which is illegal: one number per rectangle. Sliding it left would swallow the 2. Sliding it right runs off the board. Exactly one rectangle survives.
Forced singles cascade. Every rectangle you place steals cells from the board, so a number that had four options a moment ago may now have one. On an easy board, this single technique carries you to the end.
Deduction two: cell ownership
Eventually every number has two or three options and nothing is forced. Now flip the question round. Instead of asking where can this number go, pick an empty cell and ask which numbers could possibly reach it.
If only one number can reach a cell, that number must end up covering it. Nobody else can: the cell has to belong to somebody, and every rectangle holds exactly one number. Often that single fact collapses the number's options to one.
These two techniques are not folklore. They are exactly what Kaku's own solver uses to grade a puzzle before it ever reaches you, and a board is only called Beginner if forced singles alone are enough to finish it.
Finishing the board
Alternate between the two. Place what is forced, then hunt for a cell with a single owner, which forces something else. Our 5×5 falls in seven placements without a single guess.
Three habits that make it faster
- Start at the edges and corners. A number near a wall has fewer legal rectangles, because half its options run off the board.
- Start with the awkward numbers. Primes like 5 and 7 can only be a single row or column, so they are long, thin, and easy to pin.
- Never guess. A well-made Shikaku has exactly one solution and never requires a guess to find it. If you are stuck, you have missed a deduction, not run out of information.
Where to go next
If you want the deductions in more depth, including what happens on a board where neither technique fires, read Shikaku techniques. If you are wondering how it compares to the puzzle everyone knows, we wrote Shikaku vs Sudoku. And if you want to know why a 12×12 is harder than a 7×7 for reasons beyond its size, see the difficulty ladder.