The Math Behind The Magic

To unravel the magic take a look at the magic array below. Select one number from each row and column.

6	12	9
5	11	8
10	16	13

Your choices would be:

6 + 11 + 13 = 30
6 + 16 + 8 = 30
5 + 16 + 9 = 30
5 + 12 + 13 = 30
10 + 12 + 8 = 30
10 + 11 + 9 = 30

Notice that in each case the sum is 30.
This magic array was created from the following addition table.

+	2	8	5
4	6	12	9
3	5	11	8
8	10	16	13

The sum of the first row and first column in the addition table is:

2 + 8 + 5 + 4 + 3 + 8 = 30.
In the more general case:

a + g	a + h	a + i	a + j	a + k	a + l
b + g	b + h	b + i	b + j	b + k	b + l
c + g	c + h	c + i	c + j	c + k	c + l
d + g	d + h	d + i	d + j	d + k	d + l
e + g	e + h	e + i	e + j	e + k	e + l
f + g	f + h	f + i	f + j	f + k	f + l

If you select one number from each row and column, the sum will always be:

a + b + c + d + e + f + g + h + i + j + k + l
Like the New Year's Magic Array it was developed using an addition table like the one below.

+	g	h	i	j	k	l
a	a + g	a + h	a + i	a + j	a + k	a + l
b	b + g	b + h	b + i	b + j	b + k	b + l
c	c + g	c + h	c + i	c + j	c + k	c + l
d	d + g	d + h	d + i	d + j	d + k	d + l
e	e + g	e + h	e + i	e + j	e + k	e + l
f	f + g	f + h	f + i	f + j	f + k	f + l

Although this may remind you of a Magic Square Puzzle, there are significant mathematical differences. For more information on magic squares I suggest:

Eric's Treasure Trove of Mathematics Magic Squares.
Suzanne Alejandre's Web Unit Magic Squares

Return to The New Year's Magic Lesson Plan

For more information, please send mail to Judy Ann Brown, judyann@forum.swarthmore.edu
Web page created Wednesday, December 10, 1997
updated Friday, December 10, 1999