The checkerboard trick
3-minute read
There’s an old puzzle that goes like this. Cut two opposite corners off a chessboard, leaving 62 squares. Can you cover what’s left with 31 dominoes? People try for a long time before they see the answer, and the answer is a single observation: opposite corners are the same colour. Every domino covers one dark square and one light one, so 31 dominoes cover 31 of each. The board has 32 of one colour and 30 of the other. It can’t be done.
The same trick works in Fill the Space, and because pieces don’t turn, it works better here than almost anywhere.
Shade the room
Imagine the room shaded like a checkerboard. Here’s the warm-up room from Fill the Space #2 (September 24, 2026), sixteen cells, eight shaded and eight light.
Now look at each piece in the tray the same way. Wherever it goes, it covers some shaded cells and some light ones:
- The S covers two of each, wherever it sits. It’s balanced.
- The T covers three of one and one of the other: it leans by two.
- The small L covers two of one and one of the other: it leans by one.
- The P-shape covers three of one and two of the other: it leans by one.
Which way a piece leans depends only on where it lands. Slide it one cell sideways and it flips. Because pieces don’t turn, that’s the only thing that changes it, so every spot a piece can take is either a shaded-leaning spot or a light-leaning one.
The leans must cancel
The room has eight shaded cells and eight light ones, so the four pieces together must cover eight of each. The leans have to cancel out. The S leans nowhere. The T leans by two, so the small L and the P, which lean by one each, must both lean the other way.
That’s a real deduction, and you can make it before your first drop. Once the T is in, you know which kind of spot the L and the P need, and roughly half of their spots are gone. In the finished room, the T covers three light cells and one shaded, and the L and the P each cover one more shaded cell than light.
When the room itself is uneven
The trick gets sharper when the room isn’t balanced. The pillar room from the same day is a six-by-five rectangle with one cell blocked by a pillar, and the pillar sits on a shaded cell. That leaves 14 shaded cells and 15 light ones.
Five of its six pieces are balanced wherever they go. Only one leans: the five-cell piece shaped like a square with a tail. So that piece must cover the extra light cell. It has to cover three light cells and two shaded, and that rules out 4 of its 10 possible spots before you’ve placed anything.
How to use it in play
You don’t need to shade every room. The trick earns its keep in a few situations:
- When a room has a pillar or a notch, check whether it removes a shaded or a light cell. An uneven room forces at least one piece to lean.
- When one piece leans by two, like a T, look for the pieces that must lean against it.
- Late in a room, when two or three pieces are left, shade the empty cells in your head. If the leftover cells don’t balance the way the pieces can, a piece has already landed on the wrong kind of spot.
It’s a tool for ruling things out, not for finding the answer on its own. Pair it with the corners and a quick count, and the rooms get much smaller.
Next guide: Every drop counts