How to Create a Magic Square With Negative Numbers

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Magic squares can be created using both negative numbers and fake numbers using the same methodology. Create a magic square with negative numbers with help from a longtime mathematics educator in this free video clip.

Part of the Video Series: College Math
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Video Transcript

Hi, I'm Jimmy Chang and we're here to talk about how to create a magic square with negative numbers. Now, usually you create magic squares with positive fake numbers. But you can also create magic squares with negative numbers, as long as you follow the same methodology, as you did with positive numbers. Now, the demonstration we're going to use today, is that of the staircase method. Which is a method guaranteed to develop odd order magic squares, which are three by three, five by five, seven by seven, etc. So, first an foremost, we're going to draw a nine by nine array, first. And if you are familiar with the staircase method, then this should be easy to follow. But if you're not, this might be a little bit of adjustment. But you should be able to pick it up pretty soon. Now, first and foremost, we're going to start off with the first negative number that we think of, which is negative one. And the first number is always going to be at the center of the top row. So, it's going to be over here. Now, the general pattern of the staircase method is, you're moving in an upward pattern, left to right, in a diagonal fashion. So, there's going to be some rules to consider, so let's see what they are. Now, if you're going from negative one, you're going to be over here, vertically you're outside the square. Now, when that happens, you're going to drop all the way down to the bottom of the row. And put the next number there, it's going to be negative two. Now, when you're moving here to here, you're outside the square, horizontally. So, you're going to the beginning of this row, and you're going to write negative three over here. Now, if you go from here to here, you are going to be in a situation where the number's already occupied. So, whenever that happens, you drop down to one, so this will be negative four. Now, this should be smooth sailing for the next couple of entries, because go from here to here. There's an empty number, so you have negative five, from here to here, it's going to be negative six. Now, when you are at the very top right hand corner of a square, in an odd order. You drop down one notch, and that will be negative seven. Now, from here, you're going to go diagonally over here. But again, horizontally you're outside the square. So, you're going to go to the beginning of that row and that's going to be negative eight. And lastly, if you move here, you're going to be vertically outside the square. So, you're dropping down, that'll be negative nine. Now, what you do on your own, is verify that the sum of each row, column and diagonal is going to give you the same sum, because that's what a magic square does. Now, as you can probably tell, the sum of every row, column and diagonal is going to give you the sum of negative 15. Which verifies that it's a magic square. So, you can create your own magic squares with negative numbers, you just need to follow the proper strategies, going forward. So, I'm Jimmy Chang and that's how to create a magic square with negative numbers.


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