Mgic Square

[miamivince]

I know how to deal with a 4*4 square, this has 49 numbers.The magic total is 1 less than 6 times the greatest number in the sequence. What is the magic number ? Is there 1 set of formulae for an even set of numbers and a different set of formulae for when the there is an odd number ? Thanks.

 

[lookagain]

I know how to deal with a 4*4 square, this has 49 numbers.The magic total is 1 less than 6 times the greatest number in the sequence. What is the magic number ? Is there 1 set of formulae for an even set of numbers and a different set of formulae for when the there is an odd number ? Thanks.

miamivince, if the magic square, odd or even, consists solely of entries of positive integers from 1 to \(\displaystyle n^2, \ \) inclusive, then the magic sum is \(\displaystyle \dfrac{(n + 1)\sqrt{n}}{2}. \ \ \) This is a consequence of the average sum of all of the entries of the square, multiplied by the number of entries in any given row or main diagonal.

 

[soroban]

Hello, miamivince!

I know how to deal with a 4*4 square. .This one has 49 numbers.
The magic total is 1 less than 6 times the greatest number in the sequence.
What is the magic total?

With 49 numbers, it must be a 7*7 magic square.
It has the numbers: .\(\displaystyle a,\,a+d,\,a+2d,\cdots\,a+48d\)

The sum of these numbers is: .\(\displaystyle \frac{49}{2}\big[2a + 48(1)\big] \:=\:49(a+24)\)
The magic total is: .\(\displaystyle T \;=\;\dfrac{49(a+24)}{7} \:=\:7(a+24)\) .[1]

The maximum number is: \(\displaystyle a+48\)
The magic total is: .\(\displaystyle T \;=\; 6(a+48) - 1\) .[2]

Equate [1] and [2]: .\(\displaystyle 7(a+24) \:=\:6(a+48)-1 \)

. . . \(\displaystyle 7a + 168 \:=\:6a + 288-1 \quad\Rightarrow\quad a \:=\:119\)

Substitute into [1]: .\(\displaystyle T \;=\;7(119+24) \;=\;7(143) \;=\;1001\)