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An antimagic square of order n is an <math> n \times n <math> square arrangement of numbers 1 to <math> n^2 <math> where the totals of the n rows, n columns and the two long diagonals form a sequence of <math> 2n + 2 <math> consecutive integers. There are no antimagic squares of size <math> 2 \times 2 <math> or <math> 3 \times 3 <math>.
Examples
Order 4
<math> 29, 30, \cdots,38 <math> form the totals of 4 rows, 4 columns, and 2 long diagonals for the following antimagic squares.
| 2
| 15
| 5
| 13
|
| 16
| 3
| 7
| 12
|
| 9
| 8
| 14
| 1
|
| 6
| 4
| 11
| 10
|
|
| 1
| 13
| 3
| 12
|
| 15
| 9
| 4
| 10
|
| 7
| 2
| 16
| 8
|
| 14
| 6
| 11
| 5
|
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Some open problems
- How many antimagic squares of a given order exist?
- Do antimagic squares exist for all orders greater than 3?
- Is there a simple proof that no antimagic square of order 3 exists?
See also
External links
- [Antimagic Squares (http://www.geocities.com/~harveyh/anti_ms.htm)]
- [Mathworld (http://mathworld.wolfram.com/AntimagicSquare.html)]
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