Prime Numbers in Shikaku

The rule

A rectangle with area n has width × height = n. A prime number has no factors except 1 and itself, so the only shapes are 1×n (a row) and n×1 (a column).

Why it works — and how much it saves

Around a number in open space, every shape can sit in as many positions as its area. A 5 has two shapes, so it has 5 + 5 = 10 possible rectangles. A 6 has four shapes (1×6, 6×1, 2×3, 3×2), so it has up to 24. Primes start with far fewer options than their neighbours.

Those counts were computed by our solver on an empty 11×11 grid: the 6 has 24 options there, the 5 only 10. (Why exactly these numbers? See Shikaku and maths.)

Using it in a real puzzle

1. Measure the room. A 5 needs five free cells in a straight line. If the grid edge or another number leaves fewer than five in one direction, that direction is out.
2. Check for swallowed numbers. A strip may not pass over another number. A single number sitting in the way can kill a whole direction.
3. Count the remaining positions. Often just one position survives — draw it. If two survive, look for overlap cells they share.

A real example

This 6×6 puzzle comes straight from our generator. Its first logical move is a prime:

Large primes are gifts

On big grids, 11, 13 or 17 look scary, but they are among the easiest numbers: a long strip rarely has room to fit in more than one or two places. Look for them early on 15×15 and larger boards.

Questions