# Anti Magic Square

Are the two examples of $4\times 4$ anti-magic squares currently on Wikipedia actually anti-magic squares under the definition given there?

The examples are:

$$\left[ \begin {array}{cccc} 2&15&5&13\\16&3&7&12\\ 9&8&14&1\\ 6&4&11&10 \end {array} \right]$$

and $$\left[ \begin {array}{cccc} 1&13&3&12\\15&9&4&10\\ 7&2&16&8\\ 14&6&11&5 \end {array} \right]$$

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Do you mean, check if the examples of $4\times 4$ anti-magic squares in that page are really anti-magic squares? Do you think one of them is wrong? – Thomas Andrews Apr 30 '12 at 18:23
yes , exactly, is it ok? – La Belle Noiseuse Apr 30 '12 at 18:32

For example, for the first one: $$\left[ \begin {array}{cccc} 2&15&5&13\\16&3&7&12\\ 9&8&14&1\\ 6&4&11&10 \end {array} \right]$$

The entries are the integers $1$ to $16$.

The rows sum to 35, 38, 32, 31 respectively, the columns to 33, 30, 37, 36, and the diagonals to 29, 34. These are the 10 consecutive integers 29 to 38.

So yes, it satisfies the definition.

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thank you dear sir. – La Belle Noiseuse Apr 30 '12 at 18:36

I wrote a program to evaluate the sums, and got, for the first example:

29 Left/right diagonal
30 col 2
31 row 4
32 row 3
33 col 1
34 Right/left diagonal
35 row 1
36 col 4
37 col 3
38 row 2


For the second example:

29 row 1
30 col 2
31 Left/right diagonal
32 Right/left diagonal
33 row 3
34 col 3
35 col 4
36 row 4
37 col 1
38 row 2


So it looks fine.

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nice, dear sir, i got it :) – La Belle Noiseuse Apr 30 '12 at 19:06