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The Magic Cafe Forum Index » » Magical equations » » A half & half Magic Square (8 Likes) Printer Friendly Version

Mr. Mindbender
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Has anyone run across a magic square style grid where all of the rows total one specific number and all the columns total one specific number, but those two specific numbers are different? For example, if you add any of the rows it comes to 49, but when you add any of the column it comes to, say...53?

Just curious if any work has been done on this.
saxonia
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Well, the sum of all row totals has to be equal to the sum of all column totals (and to the sum of all numbers in the square). Therefore, a square as described cannot exist.
FrankFindley
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Depending on what effect you are trying to produce, an alternative might be a square where the rows and columns all sum to one number, while the two diagonals sum to another. As a simple example, adding the same number to each diagonal cell of a magic square will produce such a square.
FrankFindley
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Oh, and it is possible to make every other row and column be a different sum as well as one diagonal each too. So it truly would be half and half, just not it manner originally laid out.

This is very easy to do. Just produce magic square as per normal for lower desired number. And then modify four squares so that every other row/column is brought up to the higher desired number. Can give more details if interested.
Mr. Mindbender
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Thanks, Frank!
DragonLore
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Quote:
On Apr 1, 2022, FrankFindley wrote:
Oh, and it is possible to make every other row and column be a different sum as well as one diagonal each too. So it truly would be half and half, just not it manner originally laid out.

This is very easy to do. Just produce magic square as per normal for lower desired number. And then modify four squares so that every other row/column is brought up to the higher desired number.


You just need to add the difference to two cells that are in 2 (not 4) different rows & columns, no?
Michael Daniels
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Can someone show an example of such a 4x4 square?

Mike
DragonLore
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Something like
11 12 28 2
29 1 12 7
4 30 10 9
5 10 3 35

This one suffers from the flaw of using a duplicate number, which I leave as an exercise to the reader to do away with. Smile
FrankFindley
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On Apr 3, 2022, DragonLore wrote:
You just need to add the difference to two cells that are in 2 (not 4) different rows & columns, no?


You are right. But I am not sure that they will alternate all around that way. That is, won't one of the columns be side-by-side? But using two is definitely easier and so is superior. And you also get half of the four corner squares too!

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On Apr 3, 2022, Michael Daniels wrote:
Can someone show an example of such a 4x4 square?


Here it is step-by-step.

Let's say the numbers are 45 and 52.

1) Generate a Magic Square via your favorite method based on the lowest number. In this case it is 45.

Image


Each of the rows, columns, and diagonals (also four corner squares) will sum to 45.

Image


2) Take the difference between the high and low number. In this case: 52 - 45 = 7

3) Add that number to the two highlighted squares:

Image


This will result in a square where half the rows, columns, diagonals, and four corner squares each add to one of the two numbers.

Image


Now, how might we use a magic square modified like this?

It might work well as trick to perform for couples. For example, have a couple each think of a number. Again, let's say 45 and 52. Have them write this on a pocket pad (I********n P**) by drawing a line across the sheet where one number is above the line and another is below, while your back is turned. Have them tear out the sheet and tear it in half, each keeping their piece. Take back the pad and while getting another sheet ready to write on get a p***. Then create a magic square on the table using a half sheet of the pad for each number. For the two numbers which are switch numbers: on one side write the sum of the number it should be for a magic square plus the difference of the two numbers (in this case +7). Then say, "That doesn't seem write, let me try again." Turn over the half sheet and right in the proper number for the magic square. Thus you have a magic square on the table but if you flip over two of the pieces it becomes the half and half square. Tell the person that you believe this is their square. Have them show the number on their piece of paper. Then show how all the rows, columns, and diagonals add to it. Then ask second person what their number was. And say, "That's curious, look what happens when we use the other numbers I sensed." Turn over the two numbers revealing how the new numbers add. Then state, "This is highly unusual. You two must be a perfect match to have exactly 50-50. Have them take a picture on their phones of the two squares for them to share with others.

I think I will give this a try and see how it plays. Now where did I put my John Riggs Beautiful Butterfly?...
DragonLore
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On Apr 3, 2022, FrankFindley wrote:
You are right. But I am not sure that they will alternate all around that way. That is, won't one of the columns be side-by-side? But using two is definitely easier and so is superior. And you also get half of the four corner squares too!


Well, as usual, one must choose the trade off. Yes, with 2 numbers 2 columns will be side by side.

When the difference between the two numbers is even, one can simply add half to the four square at the intersection of the rows and columns one wants to have add up to the higher number. This also increments one diagonal.

An example with 49 & 53 is:
13 8 30 2
29 1 12 7
6 30 8 9
5 10 3 35

When the difference is an odd number, I cannot see an easy solution to keeping the diagonals intact using four extra numbers.
FrankFindley
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On Apr 3, 2022, DragonLore wrote
When the difference is an odd number, I cannot see an easy solution to keeping the diagonals intact using four extra numbers.


I did not restrict the mathematical solution to integers. So the ones with odd differences include half numbers like this:

Image


Image


I've found that in doing routines with repeated squares, it can actually be interesting to audience to throw in fractions. In the routine I posted earlier, the patter includes how the transition numbers seemed strange. So it flows. In other routines where the magic square is associate with money, it can be turned into dollars. This adds decimals that people are used to seeing. So, for example two digit numbers like 42 become $4.20 or $42.00. Thus the decimal is natural. I've even done a magic square with (fake) money was given out to the audience. It was for a casino night charity gig. Pretty fun application, though in that one fractions weren't needed.

One could also just require both numbers to be the either odd or even. For example, have first person choose a number, ask them what it is, and if it is odd say something like "That's great because I am going to use a lot of even numbers. Would you please also choose an odd number." Same for even but in reverse. Thus the difference will always be even. Another approach would be to just drop the diagonals from the explanation and divide the difference so it balances the rows and columns. So it would look like this:

Image


Really it just depends on what type of routine one wants to do as to how to address. If it is to be constrained to integers and also a purely free choice, than that would be another benefit of the two number approach.
Mr. Mindbender
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This is all super helpful - thanks Frank!!

For my purposes, I actually don't need the diagonals to be involved. Instead, I want to have a 4x4 or 5x5 grid that adds up to two different totals, either adding across the row or down the column. And I need the two totals to alternate, meaning, rows 1 & 3 equal one total, while rows 2 & 4 equal the other total. Same for the columns.

Since I don't have to worry about the diagonals, all I need to do is change two numbers. Using the above example, I'd change the #11 and #10.

The reason I'm interested in this is that I want to force a number on someone. I have several grids available, each on its own slip of paper. It will look like there are so many possibilities as each grid is made up of different numbers, but secretly, they all total to only two different numbers. Once the participant selects one of the slips, I'll have then pick any row or column and add up the numbers. After they've done that, I'll be able to say something along the lines of "You now have a total, that, if you had picked a different card or different row or column, would have generated a different number - for example, don't tell me if you picked a row or column, but go ahead and add up the row or column right next to the one you selected. Add those numbers up...it's a different number, right? They respond "Yeah" and then you continue "So, let's stick with your first total..." And now, I've forced one of two numbers on the spectator. From there - I have a few ideas of where to take this...

So again, thanks Fred!
FrankFindley
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On Apr 4, 2022, Mr. Mindbender wrote:
The reason I'm interested in this is that I want to force a number on someone. I have several grids available, each on its own slip of paper. It will look like there are so many possibilities as each grid is made up of different numbers, but secretly, they all total to only two different numbers. Once the participant selects one of the slips, I'll have then pick any row or column and add up the numbers. After they've done that, I'll be able to say something along the lines of "You now have a total, that, if you had picked a different card or different row or column, would have generated a different number - for example, don't tell me if you picked a row or column, but go ahead and add up the row or column right next to the one you selected. Add those numbers up...it's a different number, right? They respond "Yeah" and then you continue "So, let's stick with your first total..." And now, I've forced one of two numbers on the spectator. From there - I have a few ideas of where to take this...


That is so sneaky! Very nice idea!

Quote:
On Apr 4, 2022, Mr. Mindbender wrote:
Since I don't have to worry about the diagonals, all I need to do is change two numbers. Using the above example, I'd change the #11 and #10.


Yep. And since you don't care about diagonals, it will work the same basic way for a five-by-five square. Here is an example of a 5X5 magic square summing to 65 modified to alternate with 77:

Image


Image
Mr. Mindbender
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That's exactly what I need - thank you!
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