Indexing the archive…
Your Universe of Digital Possibilities
Cover the board with dominoes — each hides two squares, and because neighbouring squares always differ in colour, every domino drinks one black and one white. Cut the two opposite corners and they share a colour, so 62 squares remain split 32 and 30; a full tiling of 31 dominoes would need 31 of each, and the difference of two can never be arranged away (Max Black, 1946). The machine strands two same-colour strangers every run. The one door is Gomory’s necklace — cut one of each colour and the rook circuit tiles it, wherever you cut. This instrument ends at that wall, on purpose.
On a chequered board every domino, however laid, covers exactly one black and one white square — so any region that tiles must hold equal counts. Opposite corners share a colour; cutting both leaves 32 to 30, and no arrangement closes a gap of two.
Thread all 64 squares onto one closed rook’s tour, colours alternating like beads. Remove one black and one white and the necklace falls into two arcs of even length — each tiles trivially along its own strand. Balance the colours and the wall opens, wherever the knife falls.
The purest two-liner in the cycle — a proof a child can hold: every domino covers one square of each colour, opposite corners share a colour, so 32 against 30 can never be tiled. McCarthy filed it in 1964 as a tough nut for resolution provers that refuse to count colours — a wink across the rack at The Machine. The door is Gomory’s exact necklace, a Hamiltonian rook circuit whose beads alternate; tiling as generation — growing tilings rather than forbidding one — belongs to The Tile (the edge between them: the Tile grows tilings, the Tiling forbids one), and the counting style is The Fifteen’s parity worn in two colours. As with every wall in this edition, the impossibility is the finding.