The document discusses the history of magic squares. It states that the oldest known magic square is the Lo Shu, a 3x3 grid seen on a turtle's back by Emperor Yu in China over 4000 years ago. Each row, column and diagonal of numbers in a magic square adds up to the same sum, and the Lo Shu uses consecutive integers from 1 to 9. The document also provides instructions for building magic squares of odd numbers by placing the digits in order while moving up and to the right at each step.

1. MAGIC SQUARES
VIVIAN P. CASTRO,
PhD

2. BRIEF HISTORY
 The lo – shu is the oldest known
example of magic square.
The myth claims that it was first seen by
emperor Yu decorated upon by the back
of the turtle along the bank of the yellow
river about four thousand years ago.

3. BRIEF HISTORY
Draw a square and divide it into nine
smaller squares or cells.
In each cell, write the numeral that
indicates the number of dots in the
corresponding position on the turtle’s
back.

4. BRIEF HISTORY
The figure formed is a “magic square” of the
third order.
It is called “magic” because each column, each
row, and each diagonal has equal sums.
It is “of the third order” because it has three
cells along each side of the square.

5. BRIEF HISTORY
The lo – shu is a pure magic square of the third
order because the numbers on it are consecutive
integers starting with one.
The digits in a pure magic square of the third
order can be arranged in eight ways in the center
and the even numbers are in the four corners.

6. 8 1 6
3 5 7
4 9 2
15 15 15
15
15
15
15
15

7. 6 7 2
1 5 9
8 3 4
2 9 4
7 5 3
6 1 8
6 1 8
7 5 3
2 9 4

8. 2 7 6
9 5 1
4 3 8
2 7 6
9 5 1
4 3 8 2 7 6
9 5 1
4 3 8

9. BUILDING A MAGIC SQUARE OF ODD
NUMBERS
Write the digits in order by putting each digit in a
separate cell. Begin by writing 1 in the middle
cell of the top row.
Move from one cell to another cell by going up
and to the right one step at a time . If you are
going up of the square, or getting into the cell
that is already filled, make the following moves:

10. a. When a move takes you out of the top
of the square, move down to the lowest
cell in that column. When a move takes
you out of the right of the squares, move
to the farthest left cell in that row.
b. When a move takes you to a cell that
is already filled up, drp down one arrow
instead. When a move takes you out at
the upper right hand corner, drop one row
instead.

11. 1
2
1

12. 1
3
2
1
2
3
4

13. 1
2
3
4
5
6 1 6
3 5
4 2
7

14. 1
2
3
4
5
6
7
8 8 1 6
3 5 7
4 2
9