§5 annex: More about sufficiency. This page elaborates on §5 of the home page.

§5X. The board-filling procedure starts with placing on the board any tile not all of whose hexagons lie in a straight line. Suitable are all the factory tiles, and tile 50 invented here as a minimal example:

figure 5a — tile 50

Now tiles are added:

figure 5b1 install tile 51 via 50a-50b	figure 5b2 install tile 52 via 50b-50c	figure 5b3 install tile 53 via 50c-51e
figure 5b4 install tile 54 via 52d-53d	figure 5b5 install tile 55 via 53e-54e	figure 5b6 install tile 56 via 54d-55d

At this point, the method should be clear. Here is the second time around the board:

figure 5c1 — add tile 57	figure 5c2 — add tile 58	figure 5c3 — add tile 59
figure 5c4 — add tile 5Z	figure 5c5 — add tile 5Y	figure 5c6 — add tile 5X

Third time around the board:

figure 5d1 — add tile 5W	figure 5d2 — add tile 5V	figure 5d3 — add tile 5U
figure 5d4 — add tile 5T	figure 5d5 — add tile 5S	figure 5d6 — add tile 5R

Very little of tile 5R, the final tile, is visible in figure 5d6.

Now the board is full, and figure 5e merely unifies the colors:

figure 5e

Tile 5Q in figure 5f below is another minimal tile that could start the board-filling procedure. If either of these tiles is placed elsewhere on the board, the procedure will still work, although the possible sequences of adding tiles might be greatly different.

figure 5f	figure 5a — as above