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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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    yes , exactly, is it ok?2012-04-30

2 Answers 2

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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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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 :)2012-04-30