Putting the “Wow” into Magic Squares!

I’m sure that at some time in your life you’ve come across a magic square; usually a 4 x 4 table filled in with numbers where every row and column adds to give the same total. Here’s an example where the total is 34 and, as a bonus, the two leading diagonals add to give 34 as well.

Well, that’s quite nice, but hasn’t really got the wow factor, has it? The next one has, though: same numbers, filled in differently.

It’s quite clear that the rows and columns all add to give different totals, so where’s the magic gone? That comes because you can choose the numbers, and they will still add to give 34! Let’s start by choosing 9; we then cross out the row and column containing 9 so we can’t choose it again.

Now let’s choose 14, and cross out the row and column again.

Now choose 3…

And finally we must choose 8, being the only number we haven’t crossed out. So what have we got: 9, 14, 3, 8. Add them together and, abracadabra, we have 34. But this would have happened if we’d chosen completely different numbers, as long as we cross out the relevant row and column each time.

Now, you can amaze your family and stun your friends by asking them to select any number larger than 30, and you will instantly create a magic square where they can choose the numbers and get the total they selected. Here’s how (it will need a bit of practice if you want to look slick doing it):

Take 30 away from the number, then divide the result by 4. Remember the quotient and the remainder. (For example, they choose 108. 108 – 30 = 78, 78 divided by 4 gives 19 remainder 2.)

Select any row in the square and, starting with the quotient (19 in my example), fill in numbers consecutively, but selecting the columns in random order. It could look like this:

Now carry on with consecutive numbers, choosing the next two rows at random, but filling each row up in the same order as your first row.

For the last row, instead of the number you were going to start with (31 in my example), add on the remainder – so 33. And that’s it! If you want to check it’s OK, just add the numbers up on the leading diagonal (good for your mental arithmetic)!

25 + 33 + 22 + 28? Yes, that’s 108 – so, that means that this mother of magic squares will work successfully.

Check out some other maths games:

https://oxfordstudycourses.com/games-with-dots/

https://oxfordstudycourses.com/a-mathematical-card-trick/