From Early Modern Puzzles to Modern Combinatorics: The Story of Saodo and Chiljagakdeuk

Code & Data

Have you ever solved a Sudoku puzzle? The simple rule of filling a 9×9 grid with numbers 1 through 9 captured the world's attention. Today, I introduce two puzzles found in traditional Korean mathematics (Sanhak) texts that possess a different charm from Sudoku: Saodo and Chiljagakdeuk. Both are number-placement puzzles, and surprisingly, they share a similar structural backbone.

---

1. Saodo: 5 Colors Hidden Within 20 Circles

Saodo consists of 20 circles arranged in a symmetrical cross shape, with each circle containing a single number from 1 to 20. At first glance, it looks like an arbitrary arrangement of numbers.

However, looking at the original text reveals something unusual.

The text borrows from the Five Elements theory to classify numbers into five systems, showing traces of numerical relationships.

Let’s apply 5 colors to this diagram. The rule is simple: color the circles that leave the same remainder when divided by 5 with the same color.

一六水二七火三八木四九金金五十土此圖水土相滲金火相守兩木相摩傳曰木與木相摩則然金與火相守則流

A primary, direct translation yields the following:

一六水 二七火 三八木 四九金 五十土

1 and 6 are Water. 2 and 7 are Fire. 3 and 8 are Wood. 4 and 9 are Metal. 5 and 10 are Earth.

此圖水土相滲

In this diagram, Water and Earth seep into each other.

金火相守

Metal and Fire stay together (guard each other).

兩木相摩

The two Woods rub against each other.

傳曰

It is said,

木與木相摩則然

When Wood rubs against Wood, it ignites (burns).

金與火相守則流

When Metal and Fire stay together, it flows (melts).

Note: Here, 然 is likely a phonetic loan character or a printing error for 燃, meaning to burn.

Reinterpreting this:

The original text uses the Five Elements to divide numbers into five categories:

• 1, 6: Water
• 2, 7: Fire
• 3, 8: Wood
• 4, 9: Metal
• 5, 10: Earth

Since Saodo uses numbers from 1 to 20, this classification applies repeatedly using the same rule.

From a modern perspective, this is identical to classification by modulo 5 (mod 5).

Since using traditional color names might not be intuitive, let's assign clearer colors:

• Water (Blue): 1, 6, 11, 16
• Fire (Red): 2, 7, 12, 17
• Wood (Green): 3, 8, 13, 18
• Metal (Gray): 4, 9, 14, 19
• Earth (Ochre): 5, 10, 15, 20

The remarkable thing is that each color group contains exactly 4 elements. Dividing 20 by 5 yields exactly 4 each. This is an intentional design, not a coincidence. Furthermore, the intention to group and classify numbers based on the Five Elements is clearly preserved in the text.

Additionally, the sum of numbers from 1 to 20 is 210. A note stating "共積二百十" remains in the original text. It feels like an initial validation condition, similar to Sudoku's requirement that "each row, column, and 3×3 box must contain 1 to 9 once."

---

2. Chiljagakdeuk: 5-Directional Clusters of 35 Numbers

A sibling puzzle to Saodo is Chiljagakdeuk. This puzzle features 35 numbers divided into 5 directions (Up, Down, Left, Right, Center). Each direction consists of 1 center slot and 6 surrounding slots, totaling 7 numbers. The name literally translates to "Each (각) Obtaining (득) Seven (칠) Characters."

The true interest of Chiljagakdeuk lies in the sum of each direction. Adding the 7 numbers in any given direction always equals 120, much like how each row in Sudoku sums to 45. Furthermore, the center numbers are exactly 1, 2, 3, 4, and 5, perfectly covering the remainders when divided by 5.

---

3. Finding Patterns: How Do They Resemble Each Other?

A puzzle enthusiast would naturally ask:

"Does Saodo have a sum rule like Chiljagakdeuk?"

Regrettably, Saodo only performs a primitive 5-group classification in the original text and lacks the clean, direction-specific sum rules found in Chiljagakdeuk. Let’s look at the sums of each color group:

• Water: 34
• Fire: 38
• Wood: 42
• Metal: 46
• Earth: 50

There is an interesting pattern increasing by 4, but the sums are not identical. Moreover, Saodo lacks explicit "center-surrounding" connecting lines like Chiljagakdeuk. Thus, Saodo can be viewed as a weak version or a different parameter setting of Chiljagakdeuk.

Nonetheless, it is fascinating that both puzzles share the underlying framework of 5 groups, mod 5 remainder classification, and total sum invariants. This is similar to how Sudoku and Kakuro share the core philosophy of "arranging numbers 1 to N regularly" while employing different specific rules.

---

4. Generalization: Viewing Puzzles as Parameters

We can take this a step further and unify both puzzles under a single formula:

Π(p, q, T)

• p: Number of directions (or groups)
• q: Number of surrounding slots per direction
• T: Target sum for each direction (cluster)

Chiljagakdeuk corresponds to Π(5, 6, 120): 5 directions, 6 surrounding slots, and a target sum of 120. It fits perfectly.

What about Saodo? It features 5 color groups with 4 elements each, which can be viewed as 1 center + 3 surrounding slots. Therefore, it fits Π(5, 3, T_d), where T_d varies by direction (34, 38, 42, 46, 50). It represents an inhomogeneous example within the same family.

Viewing them through parameters streamlines puzzle design, akin to categorizing sliding puzzles, logic puzzles, or number-placement puzzles.

---

5. Potential for Expansion: Can We Create More Π(p, q, T) Puzzles?

Let's use our imagination. With the Π(p, q, T) framework, can we design new puzzles?

for example:

• Π(4, 5, 80): 4 directions, 5 surrounding slots, target sum 80
• Π(6, 4, 100): 6 directions, 4 surrounding slots, target sum 100
• Π(5, 5, 90): 5 directions, 5 surrounding slots, target sum 90

Of course, not every combination is mathematically viable. Consistency is required between the total sum of the number set and the individual cluster sums. However, determining this constitutes a mathematical problem in itself.

For a given p, q, T, what are the conditions for the puzzle to exist?

This question mirrors problems in Sudoku regarding "uniqueness conditions" or the "minimum number of clues," serving as an excellent topic for puzzle designers.

---

6. Strengths of These Puzzles

6.1 Visual Pleasure

Coloring the circles reveals the hidden structure. Observing how the same colors scatter within Saodo’s symmetrical cross layout is visually engaging.

6.2 A Bridge Between the Ancient and the Modern

The author (Choi Seok-jeong) notes that Saodo originated from the Chinese Hetu (Yellow River Map) and the Five Elements theory. Chiljagakdeuk also appears within the context of East Asian Sanhak texts. The connection of these diagrams to modern concepts like graph coloring, block design, and invariants adds substantial depth.

6.3 Open Interpretation

The absence of a single "correct answer" or fixed "rule" allows for diverse mathematical interpretations. One can approach it as a graph problem, an algebraic structure, or a combinatorial design.

---

7. Limitations of These Puzzles

7.1 Ambiguous Rules

Arabic numbers have been written in pen over the original text to solve the puzzle. For reference, see page 317 of the National Library of Korea facsimile edition, or open the scanned copy of Gusu-ryak.

Unlike Sudoku, which has explicit rules, the original intentions behind Saodo and Chiljagakdeuk remain somewhat ambiguous. The significance of the number tilts or directional indicators is not yet fully understood.

7.2 Difficult to Judge Completeness from the Source Alone

While Chiljagakdeuk feels like a complete puzzle due to its strong sum invariant condition, Saodo does not. It remains unclear whether this represents intentional incompleteness or an early conceptual stage.

7.3 Excessive Interpretability

While open interpretation is a strength, it also means there is no definitive correct answer. This is engaging for researchers and puzzle enthusiasts but can be confusing for general readers.

---

8. Value as Educational Tools

Integrating these puzzles into mathematics education offers several benefits:

8.1 Visualization of Modular Arithmetic

Instead of treating mod 5 purely as a calculation, students can experience it visually through color. The rule "same color for numbers with the same remainder" is easily accessible to elementary students.

8.2 Sums and Invariants

The fact that the sum of 1 through 20 is 210, or that the directional sums in Chiljagakdeuk equal 120, introduces the foundational mathematical concept of invariants naturally.

8.3 Mathematical Thinking via Puzzles

Questions like "Can we arrange numbers to satisfy this rule?" foster advanced mathematical thinking regarding existence, uniqueness, and necessary and sufficient conditions, similar to the role Sudoku plays in education.

8.4 Connecting Math History and Culture

Adding historical context, such as the Hetu, the Five Elements, and Joseon mathematics texts, demonstrates that mathematics develops in tandem with culture rather than operating as isolated notation.

---

Conclusion

Saodo and Chiljagakdeuk remain relatively obscure puzzles. However, their simple conditions and understudied nature make them highly engaging to explore. They allow us to participate in various mathematical activities, including coloring, summation, pattern recognition, and generalization.

If reading this has sparked ideas for alternative rules, you are already a contributor to this puzzle research. I will return with more intriguing puzzles in the future.