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Activity II: Magic Squares and Stars (Grade Levels: 4-8)
About Math Concepts |
Proving the Pythagorean Theorem |
Magic Squares and Stars |
The Tower of Hanoi |
More Math Concepts
Standards:
Standard 1: Number and Operations
Standard 9: Connections
Objectives:
The activity is about the history and creation of a mathematical puzzle or recreation called magic squares.
Students will have the opportunity to:
- use number sense and arithmetic facts to find missing numbers in a magicsquare and/or magic star.
- follow directions given in words and diagrams to create a magic square of odd order.
Activity 2
This elementary activity can be used at higher levels by having students determine all possible combinations of 4 digits between 1 and 12 inclusive whose sum is 26 (there are 32) and then creating as many magic stars as they can using their list.
Magic Squares and Stars
For
centuries, mathematicians and individuals interested
in recreational mathematics have been interested in
magic squares. Practically all historians agree that
the magic square had is origin in China centuries before
the birth of Christ. The exact origin has been lost,
but Oriental tradition holds that the Emperor Yu (c.2200
B.C.) was standing on the bank of the Yellow river when
a tortoise appeared with a mystic symbol on its back.
This figure came to be know as the lo-shu, and
is shown below.
The
lo-shu consists of a 3 x 3 square of numbers,
indicated by knots tied in a string, and so arranged
that the sum of the number of knots in any row, column,
or diagonal is fifteen. In decimal representation, it
appears as the following figure:
The
square is "magic" because the sum of any row, column,
or diagonal is the same.
The
magic square is still common in China today. It is found
on buildings and in artistic designs, and fortune tellers
uses them in their trade.
After
many centuries, the magic square found its way out of
China. In the ninth century, magic squares were used
by Arabian astrologers to read horoscopes. The magic
square appeared in India in the eleventh and twelfth
centuries. The figure below shows a magic square found
in the ancient town of Khajuraho.
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7
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12
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1
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14
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2
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13
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8
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11
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16
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3
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10
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5
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9
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6
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15
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4
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Constructing
Magic Squares
The
method for constructing magic squares of any order is
called the La Loubre method. As a demonstration, here
are the steps for constructing a magic square of order
three.
1.
Place the successive numbers in the cells in their natural
order in a diagonal line that slopes upward to the right.
Begin with 1 in the center of the top row.
2.
Any time you reach the top row, write the next number
in the lowest cell of the bottom row of the next column
on the right.
3.
When you reach the right-hand column, write the next
number in the left-hand column as if it immediately
succeeded the right-hand column.
4.
When you reach a cell that is already filled, or when
you reach the right-hand upper cell, write the next
number in the cell directly below the last number written.
Example
of Rules
;;;;;;;;;;;;;;;;;;;;
2
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
9
You
can only use the La Loubre method to construct odd order
magic squares.
(3 X 3, 5 X 5....) For this activity, your work is limited
to odd order magic squares. Check the resources for
books to further study magic squares of either order.
1. Below is a magic
square of order 7. The sum of the rows, columns, and diagonals is 175.
Find the missing number from the rows or columns.
2. Construct a magic
square of order five.
Magic
Stars
Today we have other
puzzles that are related to magic squares. Mutsumi Suzuki is a Professor
of Engineering in the Laboratory for Process Systems Engineering, Tohoku
University, Sendai, Japan. His research interests include reduced gravity
and chemical engineering and process system engineering. Magic Stars
are one of his hobbies.
A magic star is
a six-sided star with numbers at each of its vertices. The sum of the
numbers along each segment is a constant. Consider the following diagram.
Edge A + C + F +
H = Edge A + D + G + K = Edge B + C + D + E = Edge B + F + I + L
= Edge E + G + J + L = Edge H + I + J + K = N (constant)
Each Letter represents
a number from 1 - 12.
;;;;;;;;;;;;;;1
+ 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78 therefore A + B+
C + D + E + F + G + H + I + J + K + L = 78
now ;;;;;;;;;;
edge A + C + F + H = N
;;;;;;;;;;;;;;;;;
edge A + D + G + K = N
;;;;;;;;;;;;;;;;;
edge B + C + D + E = N
;;;;;;;;;;;;;;;;;
edge B + F + I + L = N
;;;;;;;;;;;;;;;;;
edge E + G + J + L = N
;;;;;;;;;;; +
;; edge H + I + J + K = N
;;;;;;;;;;; ____________________________
2A + 2B+ 2C + 2D
+ 2E + 2F + 2G + 2H + 2I + 2J + 2K + 2L = 6N
2(A + B+ C + D + E + F + G + H + I + J + K + L) = 6N
2(78) = 6N
;;156 = 6N
156/6 = N
26 = N
3. Below is a magic
star with numbers missing. Fill in the missing numbers.
4. Complete the
following the magic stars below by supplying the missing numbers.
a.
b.
c.
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