This is a simple prooblem. This is a case of what is popularly known
as "magic squares"
A magic square is defined as quare arrangement of numbers wherein
each row, column and diagonal add up to the same sum.
There are two sub classes of magic squares.
1. Odd squares 3 x 3, 5 x 5 , 7 x 7 .....
2. Doubly even squares: 4 x 4 , 8 x 8, 12 x 12, 16 x 16, 20 x 20 ...
Arrangement of natural numners 1 thru 16 into 4 x 4
to form a magic square:
First construct by filling natural numbers 1 thru 16 in order
in the square top left to bottom -right corners. We shall have:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
[ For a 12 x 12 for ex, we shall fill 1 at the top left thru 144
at the right most bottm corner ]
Draw diagonal lines through the square joining the corners.
We here have two diagonal lines
First: ( top left thru bot. right ) 1 6 11 16
Second ( top right thru bot. left ) 4 7 10 13
Rewrite each diag. line , reversing the order of numbers.
Now we have:
First diag: 16 11 6 1
Second diag: 13 10 7 4
A simple way to reversing the numbers is to subtract each number in these
diag. squares from 17 ( ie., 4 x 4 + 1 ).
First 1 6 11 16 becomes 17-1 17-6 17-11 17-16 = 16 11 6 1
Now we have the final square as:
16 2 3 13
5 11 10 8
9 7 6 12
4 14 14 1
If you are solving the problem of 12 x 12 or 20 x 20 magic square,
You have two draw diagonal lines in each 4 x 4 divisions. Some of
these lines join to form longer diag. lines, some remail short.
Numbers in each square that falls on a diag. line should be trated
for reversal as usual. For 20x20 , you subtract the numbers in these
squares from 401 ie 1 + 20 x 20.
If you are interested I shall write about odd-order magic squares
( 3 x 3, 5x5 , .... ).
- Syamala Rao.
--Boundary-13324636-0-0
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Sent: 15 May 1996 12:43:27
From:"Devaraju, Mohan" <mdevaraju@gi-link.dcrb.dla.mil>
To: Multiple,recipients,of,list,telusa@cs.wisc.edu
Subject: A math problem
Reply-to: telusa-scit@smartcad4.me.wisc.edu
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Can any one solve this good ol math problem ?
4 rows
4 columns
one number in each cell,
Numbers 1 through 16
Totals for each column must add up 34
Totals for each row must add up to 34
Totals for 2 diagonals must add up to 34
What is the arrangement of Numbers ?
--Boundary-13324636-0-0--