When a block puzzle ends, the board is usually not full. Empty cells can be plentiful and still useless, because no piece in hand fits the space they sit in. What matters is not how many cells are empty but how they connect. This post counts the two shapes of trouble: holes and islands.
Two words, defined
A hole is an empty cell whose neighbours above, below, left and right are all filled, or are the edge of the board. An island is a group of empty cells joined edge to edge and walled in on every side. A hole is simply an island of one cell.
The rule that follows is plain and checkable: an island of n cells can only ever take a piece of n cells or fewer, and only one whose shape fits. A hole can take a piece of exactly one cell, and nothing else. A typical piece such as a straight bar, a square or an L is out of the question, so the cell stays empty until something around it clears.
How many neighbours it takes to seal a cell
The board edge does some of the sealing for free. A corner cell has 2 neighbours, a cell on an edge but not a corner has 3, and an interior cell has 4. Count them across the grid and they add up: 4 corners, 28 edge cells (7 on each of four sides) and 49 interior cells make 81.
So the cheapest hole to make is in a corner, where two blocks seal it. An edge hole takes three and an interior hole four.
Why one hole costs three groups
A hole is not only a lost cell. It sits in a row, a column and a 3x3 box, as every cell does (the earlier post on the 9x9 grid has the counting), and none of those three groups can clear while it is empty. One hole blocks 3 of the 27 groups.
The double-clear arithmetic has the same limit: gaps can be filled together only if they connect. A walled-in gap is always a job for a single cell.
How a block shrinks the places a piece can go
Before any holes appear, a board full of room is generous. On an empty 9x9 grid a piece with a bounding box of h by w cells can be placed in (9 − h + 1) × (9 − w + 1) positions. A piece that can be turned adds the positions of each orientation:
| Piece | Orientations | Places per orientation | Total places |
|---|---|---|---|
| Single block (1x1) | 1 | 81 | 81 |
| Domino (1x2) | 2 | 72 | 144 |
| Bar of three (1x3) | 2 | 63 | 126 |
| Bar of five (1x5) | 2 | 45 | 90 |
| Square (2x2) | 1 | 64 | 64 |
| Square (3x3) | 1 | 49 | 49 |
| L of three cells (2x2 box) | 4 | 64 | 256 |
Bigger pieces have fewer places; the 3x3 square has 49, seven starting columns by seven starting rows. Now put one block on the board. The square is blocked wherever it would overlap that block, and how often that happens depends on where the block sits:
The reason is the count of 3x3 squares that include a given cell. A corner cell is inside 1 of them, an edge cell such as the middle of the top row is inside 3 (one row of squares, three columns), and the centre cell is inside 3 × 3 = 9. The centre block costs nine times what the corner block does.
The same effect hits long pieces. A bar of five has 90 places; a centre block removes 10 of them, five horizontal and five vertical, while a corner block removes 2. Blocks near the middle are the ones that fragment a board.
The checkerboard: space with no room
Fill every cell whose row and column numbers add up to an odd number, a checkerboard pattern. That is 40 filled cells and 41 empty ones, more than half the board. Yet no two empty cells touch along an edge, so no domino fits anywhere. All 41 are holes, and only single blocks can go in.
Three habits that follow
- Keep the empty space in one piece. Before placing, ask whether the move splits the empty region in two, or leaves a cell with fewer than two empty neighbours. A new block that creates a 1-cell or 2-cell pocket is the expensive kind.
- Prefer the wall. A block in a corner removes the fewest places for a big piece, as the chart shows.
- Treat a hole as three blocked groups. Its row, column and box cannot clear until something around them does. If a hole is going to stay, put it where it blocks groups you are not building.
An island of n cells takes only pieces of n cells or fewer. A hole takes one cell. Count the island before you decide the board is full.
Practising it in Big Ben Blocks
Big Ben Blocks: Block Puzzle uses a 9x9 grid where completing rows, columns or 3x3 boxes clears space. In Classic and Adventure you can use undo, hints, rotate and bomb, and the Daily Challenge and Time Attack keep those aids switched off, so a hole there is a hole you have to live with. Untimed Classic is the easiest place to practise the checks above.
Big Ben Blocks: Block Puzzle offers untimed Classic play or a three-minute Time Attack, and the Daily Challenge changes at midnight UK time. The ten Adventure levels have line-clearing and combo goals and a fixed starting board and piece sequence. Blocks carry shape symbols as well as colours. It plays offline with no ads and no in-app purchases, and the coins you earn have no cash value.