VectorPaint modulo-eight clocks: compare step-three coverage with two step-two cycles

English translation of the Chinese original, translated on 2026-11-12. Original publication dates, attribution, code and version notes are retained.

Eight positions on a circle do not imply every fixed jump visits all eight. Manually draw addition by three and by two modulo eight, retaining visit order, return time and omitted positions. This is no real timer or automatic graph/number-theory tool: the circle is a layout for residues.

Fix residues zero through seven clockwise

Draw two identical eight-point circles with zero at top and one through seven clockwise, 45 degrees apart. Keep labels upright and do not mirror the second view. Cambridge NRICH’s modular introduction supplies wraparound context; this task uses residues 0–7 rather than twelve-hour labels. There is no separate position eight: eight leaves remainder zero. Spacing represents neither waiting time nor cost, and step size is the added residue amount, not arrow length in pixels.

List the full addition-by-three sequence first

Starting at 0, repeatedly add 3 and take the nonnegative remainder modulo 8: 0,3,6,1,4,7,2,5,0. Six plus three is nine, whose remainder is one, not an off-circle nine. Eight distinct positions occur before return; the final zero closes the route, not a ninth distinct position. Write all nine entries externally before drawing eight consecutive arrows. With many crossings, inspect each numbered start/end rather than treating a star-like appearance as proof of full coverage. Write unreduced cumulative totals 9=1×8+1, 12=1×8+4 and 15=1×8+7 beside transitions 6→1, 1→4 and 4→7. Quotients are not circle positions: only residues 0–7 are positions, and wrap counts add no nodes.

Do not turn chord crossings into positions

Join successive residues with thin directed paths, placing arrowheads near destinations without covering digits. Interior crossings are not positions or switching points unless explicitly defined as nodes, which they are not here. List edges such as 0→3 and 3→6 in the checking copy. Offset crowded labels without changing endpoints. Rotating the whole diagram is allowed if digits and relationships move together; rotating only chords against old labels may preserve a star shape while losing addition by three.

Addition by two from zero visits four even residues

In the second diagram, addition by 2 from zero gives 0,2,4,6,0: four distinct residues and four arrows. It misses 1,3,5,7; do not switch at chord crossings to force coverage. Mark visited even residues and leave odd ones hollow as unvisited, not nonexistent. Six plus two is eight, returning to zero rather than position eight. There are four moves but five written entries including repeated endpoint; label those counts separately.

Start at one to show the other cycle

Starting separately at 1 gives 1,3,5,7,1, another four-position cycle. It shares no position with the zero cycle; distinguish it by pattern or numbering as well as colour. Together they cover all eight, but repeated addition by two from zero cannot jump into the odd cycle. Separate small views with a shared residue table are acceptable. Do not add an invented 0→1 bridge: that would change the fixed-add-two rule.

Check first returns using divisibility

A return after k additions of three requires 3k divisible by 8; the smallest positive k is 8. For additions of two, 2k divisible by 8 first occurs at k=4. List positive k values to verify independently of shape. Coprimality of 3/8 versus common factor 2 for 2/8 explains differing coverage, not every similar-looking star. In an error copy, record 6+3 as 2; repair it to remainder 1 and regenerate subsequent steps rather than changing one label while leaving incorrect edges.

Check monochrome and exported edges against the list

Save and open a PNG, checking the eight addition-by-three edges and both four-edge addition-by-two cycles against their lists. Repair unclear arrowheads locally; a double-headed line is not equivalent to a directed step. Move obscured labels without changing residue order. Hide ordinary checking circles/angle references when unwanted and inspect native grids in the actual file. Numbering must distinguish cycles in monochrome; lost colour does not merge them. Open other formats separately: similar appearance alone does not validate endpoints.

Accept entries, moves and distinct positions separately

Retain addition-by-three counts of nine entries/eight moves/eight distinct positions and, for each addition-by-two cycle, five/four/four, with independent even/odd starts. Extra positions suggest counting the repeated endpoint; false full coverage requires checking the start; crossing-as-node errors require the legend. Recalculate every remainder after repair, not merely eight drawn dots. Results apply to modulus eight and these steps; changes require new sequences. This differs from Gray code’s one-bit adjacency rule. No timing, cryptographic security or automatic traversal capability is claimed.