Reading Gujagakdeuk as Modern Mathematics: 45 Numbers, Five Palaces, and 92

Gujagakdeuk places the numbers 1 through 45 into five palaces, nine numbers per palace. Each palace is a full 3x3 grid.

        Top palace
            |
Left palace - Center palace - Right palace
            |
        Bottom palace

Palace	Nine numbers	Sum
Top	12, 44, 9, 19, 21, 29, 37, 2, 34	207
Left	13, 43, 8, 18, 25, 26, 38, 3, 33	207
Center	15, 41, 6, 16, 23, 30, 40, 5, 31	207
Right	14, 42, 7, 17, 24, 28, 39, 4, 32	207
Bottom	11, 45, 10, 20, 22, 27, 36, 1, 35	207

The sum of 1 through 45 is 1035, and 5 x 207 = 1035.

Grid Structure

Each palace is a 3x3 grid graph with 9 nodes, 12 internal edges, girth 4, and several 4-cycles. The full diagram has 72 edges and is connected through the center palace.

Phase Sums

Phase	Numbers	Sum
Water	1, 6, 11, 16, 21, 26, 31, 36, 41	189
Fire	2, 7, 12, 17, 22, 27, 32, 37, 42	198
Wood	3, 8, 13, 18, 23, 28, 33, 38, 43	207
Metal	4, 9, 14, 19, 24, 29, 34, 39, 44	216
Earth	5, 10, 15, 20, 25, 30, 35, 40, 45	225

The phase sums increase by 9. The common palace sum, 207, is the Wood-group sum.

The 92 Invariant

Palace	Corner sum	Edge-midpoint sum	Center	Total
Top	92	94	21	207
Left	92	90	25	207
Center	92	92	23	207
Right	92	91	24	207
Bottom	92	93	22	207

Every palace has corner sum 92. The edge-midpoint sums are 90 through 94, and the center values 21 through 25 cover all five phase residues.

Graph Summary

• Nodes: 45
• Internal palace edges: 60
• Full-grid edges: 72
• Internal connected components: 5
• Full-grid connected components: 1
• Girth: 4
• Highest betweenness: node 23, the center of the center palace

Generalization

M0	Palace sum
1	189
2	198
3	207
4	216
5	225

Gujagakdeuk corresponds to M0 = 3.

Conclusion

Gujagakdeuk is a five-palace 3x3-grid structure with common sum 207, corner invariant 92, center values 21 through 25, and a natural M0-indexed family.