Shikaku strategy: eight techniques, from 1s to look-ahead

Technique 1 · basic

1s and numbers with one fit

Start every board the same way: draw the 1s, then hunt for numbers that fit exactly one rectangle. A 1 is always a single tile, so it costs nothing and turns that tile into a wall for its neighbors. Next, look at each small number near an edge or a corner and list its rectangles: a 3 in a corner has only two (along the row or up the column), and if another number sits in one of them, the other is the answer. A 2 squeezed between two numbers or against a finished parcel often has a single way out too.

Each parcel you draw here is a free fact, and each one removes options from its neighbors. That is why the empty board is worth a full first pass: one sweep of 1s and boxed-in small numbers usually gives you four or five parcels before any real thinking starts.

Where to look: the four corners, then every 2 and 3 that touches an edge or another number.

Technique 2 · basic

Big numbers first

A big number feels hard, but it is often the easiest place to start. A 10 needs ten tiles in a rectangle that holds no other number, and on a small board that is a lot to ask. List its factor pairs (10 is 2 by 5 or 5 by 2; 1 by 10 is too long for a 6 by 6 board) and slide each shape over every position that still covers the number. Most of those positions will swallow another number or run off the board, and often only one survives.

In open space a number can sit in up to as many positions per shape as its own value, so big numbers look free. They aren't: every extra tile is another chance to hit a neighbor. Check the biggest number on the board early, and come back to it after each new parcel nearby.

Where to look: the largest number on the board, then any number of 6 or more near an edge.

Technique 3 · basic

Factor pairs and primes

Every number's shapes come from its factor pairs. 4 is 1 by 4 or 2 by 2. 6 is 1 by 6 or 2 by 3. 8 is 2 by 4 (or 1 by 8 on a big board), 9 is 1 by 9 or 3 by 3, and 12 is 2 by 6 or 3 by 4. Primes have no pair at all: a 5, 7 or 11 can only be a straight line, one tile wide. On a 5 by 5 board a 5 is a whole row or a whole column, and if another number already sits in its row, it is the column.

Knowing the pairs by heart speeds up every other technique, because you stop drawing shapes that can't exist. It also catches long thin numbers: a 7 on a 7 by 7 board, or a 6 that can't be 2 by 3 because the space around it is only one tile wide.

Where to look: every 5 and 7, and any 4 or 6 in a gap only one tile wide.

Technique 4 · intermediate

Corners and edges

Every tile ends up in some parcel, and a corner tile is the hardest one to reach. Only rectangles that stretch all the way to both edges can cover it. So pick a corner that is still empty and ask which numbers could reach it at their size. Often one number is close enough and the others are too small, too far or blocked. Then the corner belongs to that number, and every shape of it that misses the corner is out.

Edges work the same way, one step weaker: a tile on the edge can only be reached from three sides. Sweep the border after each parcel you draw, because a new parcel next to an edge often leaves the tiles beside it with only one possible owner.

Where to look: empty corners first, then edge tiles beside a parcel you just drew.

Technique 5 · intermediate

The only number in reach

This is the corner check applied anywhere on the board. Pick an empty tile and list the numbers whose rectangles could still cover it. If only one number can, the tile is that number's, even though you may not know its shape yet. Drop every shape of that number that misses the tile, and see what is left. Good candidates are tiles in a narrow gap between parcels, or tiles far from every number but one, because distance and size rule out the others fast.

It pairs well with factor pairs: a number with three shapes left that must reach a tile two rows down is usually down to one. It also works in reverse. When a tile has two possible owners and one of them is settled elsewhere, the other one takes it.

Where to look: empty tiles hemmed in by parcels, and tiles far from every number but one.

Technique 6 · intermediate

Overlap: tiles every shape shares

Sometimes a number has two or three shapes left and you can't pick one yet. Lay them on top of each other in your head. Any tile that every shape covers belongs to that number for sure, so no other number may use it. That rarely settles the number itself, but it often settles a neighbor: a shape of another number that needed that tile is gone, and that number may now have just one shape left.

Overlap is strongest for a number with two shapes that share most of their tiles, like a 4 that can be a 2 by 2 square or a strip. The tiles next to the number itself are the usual shared ones. Look at what they take away from the numbers around them.

Where to look: numbers with two shapes left, and the neighbors that want the same tiles.

Technique 7 · advanced

Blocking: don't wall in a tile

Before you draw a shape, look at the tiles it leaves behind. If the shape would wall in a tile that no remaining number can reach, or take the last room a neighboring number needs, the shape is wrong. Try each shape a number has left and run this one-move check. When all but one fail, you have the parcel. The trap is usually a single tile pinned between your shape, an edge and a finished parcel.

A quick count helps here. Any closed pocket of empty tiles must hold numbers that add up exactly to its area. A pocket of three tiles with no number, or with a 2 inside, is a dead end, so the shape that made it can't be right.

Where to look: a number with two shapes, where one of them hugs an edge or a finished parcel.

Technique 8 · advanced

Look-ahead: space pressure

On hard boards, the basic checks run dry. Then pick a number with only two shapes, imagine one of them drawn, and play the simple techniques forward from there. If that leads to a number with no room at all, or a tile nobody can reach, the shape you imagined is wrong and the other one is right. Keep the chain short: two or three forced parcels, not ten. If it runs longer than that, try a different number.

Choose the number with the fewest shapes and the most crowded surroundings, because that is where space runs out fastest. Corners and the gaps between big numbers are the best places to try. You never need to guess: a chain that ends in a contradiction is a proof.

Where to look: a number with two shapes in a crowded corner, or between two big numbers.

Common mistakes

• Drawing a shape because it fits. Fitting is not enough. Draw a parcel only when every other shape of that number is ruled out.
• Forgetting a factor pair. A 4 can be a 2 by 2 square, a 9 a 3 by 3 square, and a 12 two different blocks. Missing one gives you a wrong only-fit.
• Leaving a pocket that can't be filled. A closed group of empty tiles must hold numbers that add up to its exact area. If it doesn't, an earlier parcel is wrong.
• Not looking again after each parcel. A new parcel is a new wall. Check the numbers and tiles right around it before you go hunting elsewhere.