Puzzles/Luoshu Squares

History

Luo Shu (or known also as Nine Hall Diagram) was considered to be a tool used for divination by ancient Chinese feng shui masters.

There are several slightly different stories about different masters, the more popular one is about the Emperor Yu walking along the river Lo (thus Lo Shu Square is translated as the Scroll of River Lo). This legend dates way back to 650 BC, which is the times of great floods in China. The turtle that emerged from the river had an unusual 3 x 3 pattern on its shell which later became the basis of the Lo Shu Square, a mathematical grid where the sum of numbers from each row, column or diagonal is the same. An ancient Chinese master derived the wisdom of the magic square from the patterns on the back of the turtle. See pics 1.
By reading those patterns he saw a deeper pattern of natural rhythms, or laws of the Universe as expressed in the Lo Shu square.The black circles depict even (feminine) numbers or yin; the white circles, odd (masculine) numbers or yang

Basically, no matter which direction you add the numbers – horizontal, vertical or diagonal – they always add up to 15. Also note that numbers highlighted in red are odd numbers meanwhile numbers highlighted in blue are even numbers.

Number 15 is considered a powerful number because it corresponds to the number of days in each of the 24 cycles of Chinese solar year. In other words, it is the number of days in the cycle of the new moon to the full moon. See pics 2

After several centuries, the 3x3 magic square made its way out of China and entered the Indian subcontinent. The 4x4 magic square was used as part of rituals in India from vedic times, and continues to be used till date. The Kubera-Kolam is a floor painting used in India which is in the form of a magic square of order three. It is essentially the same as the Lo Shu Square, but with 19 added to each number, giving a magic constant of 34. See pics 3

At the beginning of the fifteenth century, magic squares reached Europe . A magic square appears in one of Albrecht Durer's best engravings, "Melencolia" (1514).See pics 4

Pic 1: The pattern of magic squares
Pic 2: Chinese Loshu Square
Pic 3: 4X4 magic square (Typical magic squares in India)
Pic 4: Engraving details of magic square found in Melencolia (1514)

Puzzles

Could you figure out what is the magic square numbers for Luoshu ?
Here are some rules / hints

All the squares can be filled by each number (1 to 9) once only
The number of middle/center of the squares are 5
The numbers that are directly horizontal/vertical ( + ) to the center are odd numbers
Each number can be added horizontal, vertical and diagonal to be total of 15
- If still stuck, refer to pics 1

Solution

The answer are as below:

Bibliography