Each Shikaku number is an area, and each possible rectangle is a factor pair of that number. A 12 can be 1×12, 2×6, 3×4, 4×3, 6×2 or 12×1. Numbers with more factors have more shapes, which is why 12, 18 and 24 are harder than primes.
Area and factor pairs
A rectangle’s area is width × height. To list a number’s possible shapes, list its factor pairs — each pair in both orientations. Squares (4, 9, 16, 25, 36) have one pair that reads the same both ways.
How many options does a number have?
A shape of area n can sit in n different positions while still covering its number’s cell (the number can be in any of its n cells). So in open space a number has n × d(n) possible rectangles, where d(n) is its number of divisors. We checked this formula with our solver on an empty 41×41 grid for 6, 7, 12 and 16.
Example: 6 has divisors 1, 2, 3, 6, so d(6) = 4 and a 6 in open space has 6 × 4 = 24 options. A 7 has d(7) = 2 and only 14.
Rectangle shapes for 1–36
| Number | Shapes (width×height) | Shapes | Options in open space | Note |
|---|---|---|---|---|
| 1 | 1×1 | 1 | 1 | Single cell |
| 2 | 1×2, 2×1 | 2 | 4 | Prime |
| 3 | 1×3, 3×1 | 2 | 6 | Prime |
| 4 | 1×4, 2×2, 4×1 | 3 | 12 | Can be a square |
| 5 | 1×5, 5×1 | 2 | 10 | Prime |
| 6 | 1×6, 2×3, 3×2, 6×1 | 4 | 24 | |
| 7 | 1×7, 7×1 | 2 | 14 | Prime |
| 8 | 1×8, 2×4, 4×2, 8×1 | 4 | 32 | |
| 9 | 1×9, 3×3, 9×1 | 3 | 27 | Can be a square |
| 10 | 1×10, 2×5, 5×2, 10×1 | 4 | 40 | |
| 11 | 1×11, 11×1 | 2 | 22 | Prime |
| 12 | 1×12, 2×6, 3×4, 4×3, 6×2, 12×1 | 6 | 72 | |
| 13 | 1×13, 13×1 | 2 | 26 | Prime |
| 14 | 1×14, 2×7, 7×2, 14×1 | 4 | 56 | |
| 15 | 1×15, 3×5, 5×3, 15×1 | 4 | 60 | |
| 16 | 1×16, 2×8, 4×4, 8×2, 16×1 | 5 | 80 | Can be a square |
| 17 | 1×17, 17×1 | 2 | 34 | Prime |
| 18 | 1×18, 2×9, 3×6, 6×3, 9×2, 18×1 | 6 | 108 | |
| 19 | 1×19, 19×1 | 2 | 38 | Prime |
| 20 | 1×20, 2×10, 4×5, 5×4, 10×2, 20×1 | 6 | 120 | |
| 21 | 1×21, 3×7, 7×3, 21×1 | 4 | 84 | |
| 22 | 1×22, 2×11, 11×2, 22×1 | 4 | 88 | |
| 23 | 1×23, 23×1 | 2 | 46 | Prime |
| 24 | 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1 | 8 | 192 | |
| 25 | 1×25, 5×5, 25×1 | 3 | 75 | Can be a square |
| 26 | 1×26, 2×13, 13×2, 26×1 | 4 | 104 | |
| 27 | 1×27, 3×9, 9×3, 27×1 | 4 | 108 | |
| 28 | 1×28, 2×14, 4×7, 7×4, 14×2, 28×1 | 6 | 168 | |
| 29 | 1×29, 29×1 | 2 | 58 | Prime |
| 30 | 1×30, 2×15, 3×10, 5×6, 6×5, 10×3, 15×2, 30×1 | 8 | 240 | |
| 31 | 1×31, 31×1 | 2 | 62 | Prime |
| 32 | 1×32, 2×16, 4×8, 8×4, 16×2, 32×1 | 6 | 192 | |
| 33 | 1×33, 3×11, 11×3, 33×1 | 4 | 132 | |
| 34 | 1×34, 2×17, 17×2, 34×1 | 4 | 136 | |
| 35 | 1×35, 5×7, 7×5, 35×1 | 4 | 140 | |
| 36 | 1×36, 2×18, 3×12, 4×9, 6×6, 9×4, 12×3, 18×2, 36×1 | 9 | 324 | Can be a square |
Why some numbers are harder
Highly composite numbers — 12, 24, 36 — have many divisors and therefore many shapes. They sit in the middle of the grid on hard puzzles precisely because they are slow to pin down. Primes and squares of primes (4, 9, 25) have few shapes and fall quickly.
The sum rule
Because the rectangles tile the grid exactly, the numbers always add up to the grid’s area: 25, 49, 100, 225… The same holds for any sealed-off region — the basis of region counting.
Questions
What maths is used in Shikaku?
Multiplication, factor pairs and area. Solving also uses logical reasoning and a little counting.
Which Shikaku numbers have the most shapes?
Numbers with many divisors, such as 12, 24 and 36. 36 has nine shapes, from 1×36 to 36×1.
Is Shikaku good for learning times tables?
It gives a reason to recall factor pairs quickly, which makes it a popular, playful practice tool.