Why does the below process for a 4×4 grid with consecutive numbers 1-16 yield the magic constant 34

magic square

I have encountered a procedure that produces the magic constant 34 for a 4×4 grid with the numbers 1-16 arranged in consecutive order, and I can't figure out why it works. It is as follows:

  1. Arrange the numbers 1 to 16 into a 4 by 4 grid.
  2. Choose a number and circle it.
  3. Cross out the numbers which are in the same row as your chosen number and the
    numbers in the same column as your chosen number. (An image may make this
    instruction more clear.)
  4. Repeat this process twice more, choosing from the remaining numbers.
  5. Circle the last remaining number.
  6. Add the four circled numbers.

I have explored this site and the web and found much information about magic squares, but nothing that explains this. I can see that no circled number will be in the same row or the same column as another, which seems like it would have something to do with it, but I haven't been able to get beyond that.

I am a schoolteacher trying to help a smart middle-schooler work through this (and learn about magic squares in general), so an ideal explanation would not require math beyond algebra.

Best Answer

The number in row $r$ and column $c$ is $4r + c - 4$.

You pick out four numbers, one in each row and one in each column. Say those are the numbers in row $r_i$ and column $c_i$, for i = 1, 2, 3, 4.

So their sum is $4(r_1+r_2+r_3+r_4) + (c_1+c_2+c_3+c_4) - 4 \times 4$.

Now $r_1 + r_2 + r_3 + r_4 = 1 + 2 + 3 + 4 = 10$ because the row numbers are 1 through 4 in some order; similarly for the $c_i$. So the sum is $4 \times 10 + 10 - 16 = 34$.

Related Question