VectorPaint invented domino chains: match adjoining counts and flipped orientations

English translation of the Chinese original, translated on 2026-11-06. Original publication dates, attribution, code and version notes are retained.

A pip chain needs both half identities preserved after flips, not merely dots on tiles. Invent four two-cell tiles with only equal-count adjoining ends, without importing a real game’s full rules. This static layout/counting exercise implements no play, winning logic or automatic completion.

Give four tiles independent IDs and ordered values

Create a 1200 by 600 pixel canvas with A=(1,3), B=(3,2), C=(2,0), D=(0,1). Record left value first externally; (1,3) and (3,1) are not identical orientations. IDs track sources, not counts or matches. Give each horizontal tile two cells and a divider. A blank zero cell still belongs to its tile. State the sole equal-adjoining-end rule without implying complete domino legality. Leave gaps/label space before any circular arrangement.

Build sources for zero through three pips

Use blank for 0, one centre pip for 1, two diagonal pips for 2 and that diagonal plus centre for 3. Save this convention with equal pip sizes clear of divider/edges. Inspect cell ownership with temporary bounds, then hide them for direct reading. Other arrangements are variants; this is no manufacturing standard. Name pip groups and preserve size/cell spacing when copying rather than shrinking three pips independently. Do not decorate zero with a faint pip that makes its value ambiguous.

Assemble each ordered pair inside two cells

Place A 1-left/3-right, B 3/2, C 2/0, D 0/1. Count each half before whole tiles; equal totals do not establish identity. Manage frame, pip groups and external IDs separately as needed. Flips must include complete tile faces while external text may remain upright. Save all four sources instead of changing values repeatedly in one master. Test backgrounds so zero cells remain distinguishable from outside space through frames/fills; white patches cannot make a cell cease to exist.

Check three joins in A–B–C–D order

Arrange A–B–C–D: A-right 3 matches B-left 3, B-right 2 matches C-left 2, C-right 0 matches D-left 0. Three small gaps may separate tiles without elaborate lines crossing them. Start/end both 1, but they are not drawn joined, so do not call this a closed loop. Record each touching pair rather than approving appearance. A vertical arrangement needs a new top/bottom explanation instead of blindly reusing horizontal left/right rules.

Reassess the chain after flipping one tile

Rotate B 180 degrees in a copy, producing (2,3). Both joins fail: A-right 3 versus B-left 2, and B-right 3 versus C-left 2. Changing the label back cannot repair graphics. Rotating pip positions preserves counts within halves but exchanges left/right identity. Restore or separately save the broken-chain example with failed rules marked. Objects do not check legality automatically; a drawn rotation is not a game engine or simulated play.

Use twelve total pips as a supplementary audit

Tile totals are 4,5,2,1, summing to 12. Flipping B preserves 12 but breaks the chain, so totals cannot replace end matching. Count eight half-cells including zeros before pips. Read ordered values with IDs hidden, then restore identity checks. Overlapping duplicated pips may require selecting source groups instead of visual counting. Remove test copies and retain four tiles, eight halves and three joins without unnamed filler tiles.

Preserve direction and readable zero cells after export

Save tile masters, chain and broken test. Check eight half-values and three joins at actual export size, preserving dividers/pips. Test zero-cell frames on changed backgrounds; white faces do not become holes. Name invented four-tile chain, direction and revision and confirm B is restored in the actual file. Individual-tile delivery still needs IDs/ordered values for rearrangement. Retain the limited task scope instead of implying competition rules or an automatic solver.

Check orientation when counts are right but joins fail

For a 3-to-2 join, inspect ordered values/rotation instead of silently changing a tile’s pip groups. Extra totals suggest duplicate copies; a 0 resembling 1 suggests decoration/background problems. Separate external text from face orientation so correcting inverted IDs does not flip counts again. Recheck all three joins after rearrangement, not merely the obvious failure. Explain both ends, their swapped orientation and matching pairs rather than presenting arbitrary dotted rectangles.