Reading Paljagakdeuk (八子各得) Through Modern Mathematics: 40 Numbers, 5 Palaces, and 164

When examining a historical mathematical diagram, the first step is to stop treating it as mere symbolism and instead count the numbers and analyze its structure. In this article, we reinterpret Paljagakdeuk (八子各得) using the language of modern graph theory and combinatorics.

1. What is Paljagakdeuk?

Paljagakdeuk arranges the numbers 1 through 40 into five palaces (궁, gung), with eight numbers assigned to each palace. Each palace occupies the eight surrounding cells of a 3×3 grid, leaving the center empty.

        Upper Palace
             │
Left Palace ─ Center Palace ─ Right Palace
             │
        Lower Palace

The first consistency check is immediate.

• The sum of the integers from 1 to 40 is 820.
• There are five palaces, each summing to 164, so

5 × 164 = 820.

This immediately tells us that the arrangement is not arbitrary, but a structured partition of 820 into five equal components.

2. Each Palace Forms an 8-Cycle

If we regard the eight numbers of a palace as vertices in a graph and connect orthogonally adjacent positions within the palace, each palace becomes a single 8-cycle.

In other words, Paljagakdeuk can be viewed as five 8-cycles arranged in a cross-shaped configuration.

Palace	8-Cycle	Sum
Upper	19 → 34 → 7 → 39 → 12 → 24 → 2 → 27 → 19	164
Left	18 → 33 → 8 → 38 → 13 → 23 → 3 → 28 → 18	164
Center	16 → 30 → 5 → 21 → 15 → 36 → 10 → 31 → 16	164
Right	4 → 22 → 14 → 37 → 9 → 32 → 17 → 29 → 4	164
Lower	1 → 25 → 11 → 40 → 6 → 35 → 20 → 26 → 1	164

If this were the whole story, the diagram would simply consist of five independent cycles.

However, the palaces are also connected to one another, allowing the entire figure to be interpreted as a single graph.

3. The Center Palace Connects the Four Directions

Looking at the complete grid, there are twelve inter-palace edges.

Every one of them connects the Center Palace with one of the four surrounding palaces.

The four corner vertices of the Center Palace—30, 21, 31, and 36—are adjacent to two neighboring palaces while also remaining connected within their own cycle, giving each of them degree 4.

For example, vertex 30 is adjacent to 24 in the Upper Palace and 28 in the Left Palace, while also connecting internally to 16 and 5.

These four vertices also achieve the highest betweenness centrality, meaning that the center of the diagram is not merely geometric—it is also the primary routing hub from the perspective of graph theory.

4. Where Does 164 Come From?

Now comes the central question.

Why does every palace sum to 164?

First, partition the integers from 1 to 40 according to their residue modulo 5.

Five Phases	Numbers	Sum
Water (水)	1, 6, 11, 16, 21, 26, 31, 36	148
Fire (火)	2, 7, 12, 17, 22, 27, 32, 37	156
Wood (木)	3, 8, 13, 18, 23, 28, 33, 38	164
Metal (金)	4, 9, 14, 19, 24, 29, 34, 39	172
Earth (土)	5, 10, 15, 20, 25, 30, 35, 40	180

The sums of the five residue classes are

148, 156, 164, 172, 180,

forming an arithmetic progression with common difference 8.

Notice that 164 is exactly the total of the Wood residue class.

Now consider the Five Phases composition of each palace.

• Upper Palace: 4 Metal + 4 Fire
• Left Palace: all 8 Wood numbers
• Center Palace: 4 Earth + 4 Water
• Right Palace: 4 Fire + 4 Metal
• Lower Palace: 4 Water + 4 Earth

Half Metal and half Fire yield

(172 + 156) / 2 = 164.

Likewise, half Earth and half Water give

(180 + 148) / 2 = 164.

Meanwhile, the Left Palace simply contains the entire Wood residue class, whose sum is already 164.

Thus, the constant sum 164 is obtained either by balancing two complementary residue classes or by taking the complete Wood class.

This is not an accident, but an explicit combinatorial construction.

5. The Secret of Corners and Edge Midpoints

Another pattern emerges if each palace is divided into its four corner positions and four edge-midpoint positions.

Palace	Corner Sum	Edge-Midpoint Sum
Upper	124	40
Left	122	42
Center	118	46
Right	120	44
Lower	126	38

The edge-midpoint sums

38, 40, 42, 44, 46

form an arithmetic progression with common difference 2.

The corner sums supply the remaining values needed to keep every palace total equal to 164.

6. A Family of Generalizations

The original source also records the following generalized family.

M0	Palace Sum
1	148
2	156
3	164
4	172
5	180

The present Paljagakdeuk corresponds to M0 = 3, giving every palace a sum of 164.

Each increment of M0 increases the palace sum by 8, producing another arithmetic progression.

From this viewpoint, the Left Palace containing the complete Wood residue class can be interpreted as representing the M0 = 3 member of this generalized family.

7. Graph-Theoretic Properties

• Number of vertices: 40
• Number of edges (within palaces only): 40 (five 8-cycles)
• Number of edges (entire grid): 52
• Connected components (within palaces only): 5
• Connected components (entire graph): 1
• Degree-4 vertices: 30, 21, 31, 36 (the four corners of the Center Palace)
• Degree-3 vertices: 16
• Degree-2 vertices: 20

If the Five Phases are interpreted as residue classes modulo 5, the graph contains edges connecting vertices of the same phase, as well as edges corresponding to both generating and overcoming relationships in the traditional Five Phases framework.

This indicates that Paljagakdeuk is not merely a simple Five Phases cycle, but a layered combinatorial structure in which multiple relationships coexist.

8. Conclusion

Paljagakdeuk is a compact combinatorial construction built from the numbers 1 through 40.

At first glance, it appears to be nothing more than a diagram dividing numbers among five palaces. A graph-theoretic reading, however, reveals multiple layers of mathematical structure.

• Every palace forms an 8-cycle.
• The four corner vertices of the Center Palace act as bridges connecting the four surrounding palaces.
• The sums of the Five Phases residue classes form an arithmetic progression with common difference 8.
• The palace sum 164 arises either as the average of two complementary residue classes or as the complete Wood residue class.
• The edge-midpoint sums form an arithmetic progression with common difference 2.
• A generalized family parameterized by M0 exists.

Viewed in this way, Paljagakdeuk is no longer simply an old mathematical diagram, but a carefully designed combinatorial structure built from forty integers.