absurdtools

Rolling Block Puzzle

Tip a 1x1x2 block across a grid of holes. One square costs three moves and three squares cost two — the nearest square is not the cheapest.

Tip the block onto the yellow square, standing upright. Arrow keys work too.

One square costs three moves. Three squares cost two.

Nudging the block to the square next door takes three moves — tip it away, walk it round, stand it back up. But tipping it flat and standing it up again lands it three squares along in two moves. Here is the minimum move count to stand on each square of an open 9×9, starting from the top-left corner:

0
3
4
2
5
6
4
7
8
3
6
6
3
6
7
5
8
9
4
6
7
4
6
7
6
8
9
2
3
4
4
5
6
6
7
8
5
6
6
5
6
7
7
8
9
6
7
7
6
7
8
8
9
10
4
5
6
6
7
8
8
9
10
7
8
8
7
8
9
9
10
11
8
9
9
8
9
10
10
11
12

Read along the top row: 0, 3, 4, 2, 5, 6, 4, 7, 8. It falls twice on the way out. The square three along is cheaper than its own neighbour, and the square six along is cheaper than the one at five — the block covers three cells per two moves in a straight line and pays a premium for anything that is not a multiple of three.

The worst square in the neighbourhood is the diagonal one, at 6 moves — more than three squares east, more than three south, and more than the square three-and-three away diagonally, which costs 4. Asking a rolling block to go “one square that way” is the hardest thing you can ask of it.

And a refutation, which is why this page exists

The folklore is that a rolling block can only ever reach a lattice — every third square — and that most of the board is closed to it. On an open board that is simply false. Searching every board from 4×4 to 9×9, the block reaches every single square standing: 16 of 16, 25 of 25, all the way to 81 of 81.

The lattice is real, but it belongs to the puzzles rather than to the block. Put holes and walls in the way, as every rolling-block puzzle does, and the reachable set collapses — block every other square of a 7×7 and the block cannot move at all, because it never has two free squares to lie down on. The restriction comes from the obstacles.

There is no parity shortcut either: the parity of the move count does not track the parity of row plus column, so no quick argument tells you which squares are expensive. You have to search — which is exactly what makes the puzzle a puzzle.

How to use

  1. Tip the block with the arrow keys or the buttons.
  2. Land it standing upright on the yellow square.
  3. Add holes to make it harder — they are what really restrict the block.
  4. Ask for the shortest route if you get stuck.

Frequently asked questions

How many moves does it take to move a rolling block one square?

Three. You have to tip it away, walk it around and stand it back up again. There is no way to nudge a standing block one square across in fewer, which is the whole awkwardness of the puzzle.

Why is three squares cheaper than one?

Because tipping the block flat and standing it up again carries it three squares in two moves. A straight line costs two moves per three squares, so distances that are multiples of three are cheap and everything else pays a premium.

Which square is hardest to reach?

The diagonally adjacent one, at six moves from a standing start. That is more than the square three along, more than three squares down, and more than the square three-and-three away diagonally, which costs four. Going one square diagonally is the most expensive short trip there is.

Can a rolling block reach every square?

On an open board, yes — every square of every board from four by four to nine by nine can be stood on. The common belief that it can only reach every third square is false without obstacles; that lattice belongs to the puzzles rather than to the block.

What do the holes actually do?

They are the real restriction. A block needs two free adjacent squares to lie down on, so blocking squares removes far more than itself. Block every other square of a seven by seven and the block cannot move at all.

Is there a shortcut for working out the distances?

No. The parity of the move count does not track the parity of row plus column, so no quick argument tells you which squares are expensive. The distances have to be searched, which is exactly what makes it a puzzle rather than arithmetic.

How are the puzzles generated?

Holes are scattered, then the target is chosen from squares the block can actually reach, so a generated puzzle can never be impossible. The stated move count is the true shortest, found by breadth-first search.

Does this send anything anywhere?

No. The puzzles and the solver both run entirely in your browser.

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