Reading Nakseo Sagudo Annotations as Combinatorics: 20 Numbers, Nine Palaces, and Sum 42

by gg582 · 2026-07-10 12:04:28 · 24 views

Reading Nakseo Sagudo Annotations as Combinatorics: 20 Numbers, Nine Palaces, and Sum 42

Nakseo Sagudo is a 20-vertex, 24-edge graph. The natural modern reading is a 3x2 rectangular arrangement with four hexagonal faces sharing the central vertices 5, 16, 10, and 11.

sanhak_graph (1).png

01_original_graph (5).png

14_shared_vertex_symmetry.png

The Corrected Graph

Position Cycle Sum
NW 19, 2, 14, 5, 16, 7 63
NE 17, 4, 16, 10, 12, 9 68
SW 5, 18, 3, 13, 8, 11 58
SE 10, 6, 1, 20, 15, 11 63

The graph has an outer Hamiltonian 20-cycle and an inner 4-cycle:

5 -> 16 -> 10 -> 11 -> 5

The inner 4-cycle has sum 42.

04_cycle_analysis (5).png

Residues Modulo 5

Phase Numbers Sum
Water 1, 6, 11, 16 34
Fire 2, 7, 12, 17 38
Wood 3, 8, 13, 18 42
Metal 4, 9, 14, 19 46
Earth 5, 10, 15, 20 50

The sums form the arithmetic progression 34, 38, 42, 46, 50. The average is 42.

02_wuxing_decomposition (5).png

12_wuxing_block_matrix.png

Nine Four-Element Blocks

The annotation can be read as a recombination of five residue classes into nine four-element blocks, all with sum 42.

Palace Four numbers Sum
NW 19, 2, 14, 7 42
N 19, 17, 2, 4 42
NE 17, 4, 12, 9 42
W 14, 7, 18, 3 42
C 5, 16, 10, 11 42
E 9, 12, 6, 15 42
SW 18, 3, 13, 8 42
S 13, 8, 20, 1 42
SE 6, 1, 20, 15 42

origin_02_9palace_grid (1).png

The boundary blocks follow a clockwise order. The center block is the inner 4-cycle.

origin_translated_03_overlap (1).png

origin_03_right_rotation (1).png

Mutual Transformation

The value 1890 follows from a weighted incidence count.

  • five phase classes
  • nine palace blocks
  • 5 x 9 = 45 interactions
  • each interaction weighted by 42
  • 45 x 42 = 1890

origin_04_mutual_transformation_1890 (1).png

Graph Invariants

  • Vertices: 20
  • Edges: 24
  • Degree-4 vertices: 5, 10, 11, 16
  • Degree-2 vertices: 16
  • Girth: 4
  • No bridges
  • No articulation points
  • Diameter: 6
  • Radius: 4
  • Average shortest-path distance: 3.2

05_centrality_invariants (5).png

The graph is bipartite because all simple cycles have even length. The four length-8 cycles all have sum 84, which is 42 x 2.

08_laplacian_spectrum (1).png

09_distance_matrix (1).png

10_cycle_distributions (1).png

Dual Structure and Extension

The dual graph of the four faces is a 4-cycle whose shared-vertex weights are 5, 16, 10, and 11. Their sum is again 42.

11_dual_graph (1).png

The 120-node construction can be modeled as six cyclic copies of the 20-node graph, linked through corresponding shared vertices. It has 120 nodes and 168 edges.

13_extension_120 (1).png

The SW face contains all four Wood vertices, 3, 8, 13, and 18, showing a concentrated phase pattern.

15_sw_wood_concentration (1).png

Conclusion

The annotation is a computational rule: split 20 numbers into five residue classes, recombine neighboring subsets into nine four-element blocks, preserve the invariant sum 42, and count 45 weighted incidences to obtain 1890.

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