# What type of magic square is this?

What type of magic square is this?

• Shouldn't the entry "23" (row 4, column 4) be "33"? – Gamow Oct 20 '15 at 16:04
• What exactly is the question? Is it "is there a name for a magic square where some lines add up to one thing and some to another with a 'missing' number"? – Set Big O Oct 20 '15 at 16:21

## 1 Answer

The given square $X$ results from the following classical magic square (which arranges the number $1,2,\ldots,25$ in a $5\times5$ square so that every row, every column, and each of the two main diagonals adds up to the magical sum 65)

$\displaystyle\left[\begin{array}{rrrrr} 15&16&22& 3& 9\\ 2& 8&14&20&21\\ 19&25& 1& 7&13\\ 6&12&18&24& 5\\ 23& 4&10&11&17 \end{array}\right]$

by subtracting 1 from every entry, adding 10 times a permutation matrix, and finally supressing the resulting 0-entry in the center of the square:

$\displaystyle X= \left[\begin{array}{rrrrr} 15&16&22& 3& 9\\ 2& 8&14&20&21\\ 19&25& 1& 7&13\\ 6&12&18&24& 5\\ 23& 4&10&11&17 \end{array}\right] - \left[\begin{array}{rrrrr} 1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1\\1&1&1&1&1 \end{array}\right] + \left[\begin{array}{rrrrr} 0&0&10&0&0\\ 0&0&0&0&10\\ 0&10&0&0&0\\ 0&0&0&10&0\\ 10&0&0&0&0 \end{array}\right]$

• +1 for this, but you should probably add a note that it assumes the 23 in the question's square to be wrong. – Set Big O Oct 20 '15 at 16:40
• Thanks , but i didn't understand the part of adding 10 times a permutation matrix – mehmet Oct 21 '15 at 22:29