Posted 29 Jul 2013 Classic magic number square problem but hopefully with a slight twist. If all the rows, columns, and diagonals add up to 33 using any number you desire. Is the center uniquely determined? 0 Share this post Link to post Share on other sites

0 Posted 29 Jul 2013 (edited) Here's a simple magic square that adds up to 15: 4 9 2 3 5 7 8 1 6 We can add (33 - 15) / 3 = 6 to each cell to raise the total up to 33, turning our original square into: 10 15 8 9 11 13 14 7 12 We also have the option of doubling each cell and then incrementing each cell by one to raise the sum to 33, and that gives us: 9 19 5 7 11 15 17 3 13 which I honestly thought would give us a different center, but I guess not... Are we allowed to have negative numbers or fractions? Edited 29 Jul 2013 by gavinksong 0 Share this post Link to post Share on other sites

0 Posted 29 Jul 2013 Here's a simple magic square that adds up to 15: 4 9 2 3 5 7 8 1 6 We can add (33 - 15) / 3 = 6 to each cell to raise the total up to 33, turning our original square into: 10 15 8 9 11 13 14 7 12 We also have the option of doubling each cell and then incrementing each cell by one to raise the sum to 33, and that gives us: 9 19 5 7 11 15 17 3 13 which I honestly thought would give us a different center, but I guess not... Are we allowed to have negative numbers or fractions? yes. any type of real number (no complex) 0 Share this post Link to post Share on other sites

0 Posted 29 Jul 2013 Yes. Let's label the squares as follows: ABC DEF GHI Now if we add them row by row we get 3*33 = (A+B+C)+(D+E+F)+(G+H+I). However, if we add the four lines that go through the center, we get 4*33 = (A+E+I)+(B+E+H)+(C+E+G)+(D+E+F) = (A+B+C+D+E+F+G+H+I)+3*E. Subtract the first equation from the second to get 33=3*E, which fixes E at 11. 0 Share this post Link to post Share on other sites

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Classic magic number square problem but hopefully with a slight twist.

If all the rows, columns, and diagonals add up to 33 using any number you desire. Is the center uniquely determined?

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