magic square

[abby]

on a four by four grid using the numbers 1 through 8 and the numbers -1 through -8 distribute the numbers so that all horizontal,vertical, and main diagonals equal 0.

 

[Denis]

Interesting...how far did you get?
Did you TRY, like top line: 1 2 3 -6 ?

 

[soroban]

Hello, abby!

On a 4-by-4 grid, place the numbers 1 through 8 and the numbers -1 through -8
so that all rows, columns, and main diagonals total 0.

First I made a magic square with the number 1 through 16.

. . \(\displaystyle \L\begin{array}{ccccccccc}\hline \\ | & 1 & | & 15 & | & 14 & | & 4 & |\\ \hline \\ | & 12 & | & 6 & | & 7 & | & 9 & | \\ \hline \\| & 8 & | & 10 & | & 11 & | & 5 & | \\ \hline \\| & 13 & | & 3 & | & 2 & | & 16 & | \\ \hline\end{array}\)

From the bottom eight numbers (1 to 8), subtract 9.
From the top eight numbers (9 to 16), subtract 8.

And we have:

. . \(\displaystyle \L\begin{array}{ccccccccc}\hline \\ | & -8 & | & 7 & | & 6 & | & -5 & |\\ \hline \\ | & 4 & | & -3 & | & -2 & | & 1 & | \\ \hline \\| & -1 & | & 2 & | & 3 & | & -4 & | \\ \hline \\| & 5 & | & -6 & | & -7 & | & 8 & | \\ \hline\end{array}\;\) . . . There!


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

To construct a magic square with the numbers 1 - 16,
. . there is an easily remembered routine.

Starting at the upper-left, write the numbers 1 to 16 in the cells,
. . but only in the cells on the main diagonal.

. . \(\displaystyle \L\begin{array}{ccccccccc}\hline \\ | & \fbox{1} & | & - & | & - & | & \fbox{4} & |\\ \hline \\ | & - & | & \fbox{6} & | & \fbox{7} & | & - & | \\ \hline \\| & - & | & \fbox{10} & | & \fbox{11} & | & - & | \\ \hline \\| & \fbox{13} & | & - & | & - & | & \fbox{16} & | \\ \hline\end{array}\)


Starting at the lower-right, write the numbers 1 to 16 in the cells,
. . moving left and up, but only in the unoccupied cells.

. . \(\displaystyle \L\begin{array}{ccccccccc}\hline \\ | & 1 & | & \fbox{15} & | & \fbox{14} & | & 4 & |\\ \hline \\ | & \fbox{12} & | & 6 & | & 7 & | & \fbox{9} & | \\ \hline \\| & \fbox{8} & | & 10 & | & 11 & | & \fbox{5} & | \\ \hline \\| & 13 & | & \fbox{3} & | & \fbox{2} & | & 16 & | \\ \hline\end{array}\)

And we have a 4-by-4 magic square with a magic sum of 34.

 

[abby]

Hello Soroban!


Thank you soooo much for helping me. That was exactly what I needed. I looked everywhere to find out how to do this problem. Now I know. Once again, thank you.

Yours Truly,

Abby
:!: