Enigma 1301 - Magicless square

Enigma 1301 - Magicless square New Scientist magazine, 7 August 2004. by Susan Denham.

You are probably very familiar with magic squares such as:

8 1 6 in which you have to place the numbers 1 to 9 so 3 5 7 that each row, each column and each of the two 4 9 2 main diagonals add up to the same total.

Today your ask is to do the opposite. Place the numbers 1 to 9 into the grid so that the three row totals, the three column totals, the two main diagonals and the total of the four corner entries are all different.

There are a few ways of doing this but, in order to retain some magic, do it so that the three-figure numbers formed by reading across the first row, by reading across some other row, and by reading down some column are perfect squares. Please send in this magicless square.

Ciao,

If you say so

Three-digit perfect squares with distinct digits, and their digit sums:

169 196 256 289 324 361 529 576 625 729 784 841 16 16 13 19 9 10 16 18 13 18 19 13

Bearing in mind that some column and some other row are from this list, consider what the first row could be.

row1=169(16) => col2=625(13) => rowX no options row1=196(16) => col3=625(13) => rowX no options row1=256(13) => col1=289(19) => row2=841(13) not allowed row1=289(19) => col1=256(13) => rowX no options or => col2=841(13) => rowX no options row1=324(9) => col1=361(10) => rowX no options or => col2=256(13) => row3=169(16) => corners = maindiag not allowed or => col2=289(19) => row3=196(16) => filling in 7 and 5 gives

3 2 4 7 8 5 1 9 6

with these totals:

row1 9 row2 20 row3 16 col1 11 col2 19 col3 15 maindiag 17 skewdiag 13 corners 14

Computer search confirms this is the only such square

You forgot 31*31 = 961

5 7 8 3 2 4 9 6 1 row1 = 20, row2 = 9, row3 = 16 col1 = 17, col2 = 15, col3 = 13 diag1 = 8, diag2 = 19, corners = 23 324 is square 961 is square 841 is square

Yes, but the problem states that the number reading across the *first* row must be square.

Ciao,

You're quite right, I did. Were it required in the solution, I might have noticed this myself.

But you have forgotten that the *first* row, as well as some other row and aome column, must give a square.

Note to self: Read problem description carefully.