Magic Squares
What the magic squares are It is easy to make you understand what the magic squares are by showing it. It is known well as a puzzle for kids. It is easy to how it works, and there is an interest how it is made. There is a simple rule to make it. The sum of one of any lines is always same to another sum. In this case, there are eight of equality.
8+1+6 =6+7+2 =4+9+2 =8+3+4 =3+5+7 =1+5+9 =8+5+2 =6+5+4
There are also many kinds in magic squares. This is simple, but also mysterious.
There is a legend that the magic square was written on the shell of turtle in 3000b.c. It shows that humans knew it from the far past. It was used as charms by fortunetellers in medieval Europe because of mystery. It was found in India and Egypt for a long time ago, too.
1 14 11 8
12 7 2 13
6 9 16 3
15 4 5 10 Here is a magic square of 4×4. There are 10 of equalities, but there are 880 solutions for a square of 4×4. You cannot feel interesting as you find one of them, so I want to introduce some special magic squares. n1 n2 n3 n4 1 12 15 6 n5 n6 n7 n8 14 7 4 9 n9 n10 n11 n12 8 13 10 3 n13 n14 n15 n16 11 2 3 16
The square on the left side is a special magic square. It looks same with other one. The difference is that it has more equality than others. There are 4 squares when it is divided at the center into 4. (a square below) The sum in each square is same.
1 12 15 6 14 7 4 9
8 13 10 3
11 2 3 16
There is much equality than others in this special square.
n1+n16=n2+n15=n3+n14=n4+n13=n5+n12=n6+n11=n7+n10=n8+n9 n1+n2+n5+n6=n3+n4+n7+n8=n9+n10+n13+n14=n11+n12+n15+n16 It is actually very rare; you need much time to find this kind of the magic square.
There is another special rare magic square.
This is an example of another special magic square. It is still a magic square if you move the top line to the below side or move the left line to the right side. This is really rare. The solutions of this kind of magic squares must
8+1+6 =6+7+2 =4+9+2 =8+3+4 =3+5+7 =1+5+9 =8+5+2 =6+5+4
There are also many kinds in magic squares. This is simple, but also mysterious.
There is a legend that the magic square was written on the shell of turtle in 3000b.c. It shows that humans knew it from the far past. It was used as charms by fortunetellers in medieval Europe because of mystery. It was found in India and Egypt for a long time ago, too.
1 14 11 8
12 7 2 13
6 9 16 3
15 4 5 10 Here is a magic square of 4×4. There are 10 of equalities, but there are 880 solutions for a square of 4×4. You cannot feel interesting as you find one of them, so I want to introduce some special magic squares. n1 n2 n3 n4 1 12 15 6 n5 n6 n7 n8 14 7 4 9 n9 n10 n11 n12 8 13 10 3 n13 n14 n15 n16 11 2 3 16
The square on the left side is a special magic square. It looks same with other one. The difference is that it has more equality than others. There are 4 squares when it is divided at the center into 4. (a square below) The sum in each square is same.
1 12 15 6 14 7 4 9
8 13 10 3
11 2 3 16
There is much equality than others in this special square.
n1+n16=n2+n15=n3+n14=n4+n13=n5+n12=n6+n11=n7+n10=n8+n9 n1+n2+n5+n6=n3+n4+n7+n8=n9+n10+n13+n14=n11+n12+n15+n16 It is actually very rare; you need much time to find this kind of the magic square.
There is another special rare magic square.
This is an example of another special magic square. It is still a magic square if you move the top line to the below side or move the left line to the right side. This is really rare. The solutions of this kind of magic squares must