C.40:5.15 - Keep different tiles, then change what distinguishes them

Consider a stipulated design exercise with four adjacent cells, each black or white. Represent a tile by four bits, with 1 meaning black. The available variation flips one cell. For this exercise only, quality is the number of neighboring pairs with different colors. It ranges from 0 to 3; it makes no claim about people’s aesthetic preferences. The designer initially wants a good sparse tile, with at most two black cells, and a good dense tile, with three or four.

The retained sparse tile is 0101. It has two black cells and three transitions, so its quality is 3. Flip its first cell to obtain 1101: three black cells and two transitions, quality 2. Global winner-only selection would discard it. Local comparison puts it in the previously empty dense group. The returned material now supports both intended kinds, although the second tile has a lower score. The actual bits and flip operation make both candidates available for continued editing.

From 1101, flipping its last cell gives 1100: two black cells and one transition. It belongs to the sparse group and fails to replace 0101 there. Its parentage does not keep it in the dense group. This failed local replacement supplies no reason to discard the retained dense parent. The designer can choose a tile now or continue from either retained tile if another useful difference warrants the work.

All sixteen tiles can be enumerated cheaply in this exercise. Doing so finds maximum quality 3 in the sparse group and 2 in the dense group. It also gives several tied candidates. Direct enumeration is therefore a sufficient alternative to an ongoing evolutionary search here. The example explains the local retention connection without claiming a need for expensive search.

Now change the receiving request: the two kinds are tiles whose first cell is white and tiles whose first cell is black. Reclassifying the retained 0101 and 1101 puts one in each new group; the observed tiles themselves have not changed. Their prior black-count groups no longer answer the new request. The two retained qualities are still 3 and 2, but the dense-group winner’s previous qualification does not show that it is best among first-cell-black tiles. The available enumeration finds 1010, quality 3, in that group, so 1101 is replaced for this receiving use. Retaining 1101 for some future density-sensitive use requires that separate reason. A note saying only “two groups filled” would hide this changed comparison.