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The Magic Cafe Forum Index » » Rectangles and pentagons and squares, oh my! » » Magic Square paper fold? (1 Likes) Printer Friendly Version

Scott Cram
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Up in "Penny for your thoughts", there's an interesting and inspiring discussion related to origami and/or kirigami:

The basic question is this: How do you go about folding a single piece of paper into sixteen squares so as to be able to display and write on only one square at a time, yet easily access other squares?

I include several links to possible booklet-type folds that may work:

In an Apocalypse routine, Terry LaGerould apparently had a way of doing this, but it has never been explained. Here's how Harry Lorayne wrote it up in Apocalypse:
Terry has another presentation - for this one you DO have to memorize the entire basic magic square - that's unique. He's holding a pen and a FOLDED (into a small square) paper as he talks. He asks for a number.

He immediately writes a number on the upper surface of the folded paper. Then he folds that surface under, bringing another blank surface to top. He writes another number there.

He keeps doing this, always folding a blank surface to the top. He has it worked out so that no numbers show at any time as he folds up a blank surface.

Finally, he opens out the paper. There are 16 numbers, forming a magic square for the selected number!

What a good idea. It looks completely haphazard - and miraculous. Play with it. If you're interested enough, let me know. Maybe I can pry the exact method and handling out of Terry.

We may never find out the exact method, but does anyone have any ideas how it could be done?
Scott Cram
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Just for reference, I tried this fold and cut, and it didn't work:

I numbered the pages on the outside, and they wound up on 2 different sides of the paper, which definitely won't work for a magic square.

To make sure all 16 squares of one side are available after folding down to a single square, it seems that the paper must be cut in a way that each square is connected to 2 other square at the most. In some of the folds linked below, there are end piece which are linked to only 1 other square. This makes the front and back of a book fold on the other side, which I'd prefer not to have.

This one seems to be the best bet, and the stability for and end display seems better:
Vertical/horizontal/vertical cut:

These may still be worth exploring, though:
Zig-zag cut:
Spiral cut:
Scott Cram
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Having actually tried many of these out, the vertical/horizontal/vertical cut does have its charm, but when I made zig-zag cut as shown below, I found that it would probably work better for a real performance.


If you number the individual squares on the paper in the following way, before folding it into a booklet...

<br>01  02  03  04
<br>05  06  07  08
<br>09  10  11  12
<br>13  14  15  16

...then the zig-zag cut requires a rotation (keeping the same number face up, but rotating it 180 degrees so that face's top and bottom are flipped) after filling in every 4 squares. You'd be writing the squares down in the following order: 1-2-3-4-(rotate)-8-7-6-5-(rotate)-9-10-11-12-(rotate)-16-15-14-13

The rotation ensures all numbers have the same orientation when the booklet is opened to show all the pages.

Between the regular rotation every 4 squares, and the easy beat of sequential squares (even if the sequence is occasionally backwards), this one definitely has its strong points. It's trickier to unfold to a flat square than some of the others, but it's not too difficult.

I haven't tried the spiral cut, but I'm definitely sold on the zig-zag cut.
Geoff Pfeiffer
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Scott, this is great information. I also tried the same fold and it works well. This may not be the same fold Terry used, but it works well. I really like the random idea of placeing number on paper, then opening the paper up to show the Magic Square. I'm adding this to my personal bag of tricks.

Thanks again Scott for all of your help!
Scott Cram
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Here's another possibility, this time with no cutting: (I chose this particular video because you get to see the square opened, so you get an idea of the various orientations required.)

In origami, it's known as a "Face Changer" (watch the video, and you'll see why). Writing your numbers on the flipped sections and the back, in the correct orientations could make for an amusing presentation. Granted, you will see 4 numbers at a time instead of 1. On the upside, though, there's no cutting so the square is in one piece at the end. The nature of the fold is simple, which also helps make the correct orientations easier to work out.
federico luduena
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Wonderful find, Scott. If one cuts the rectangular paper as shown above, but then changes the orientation to landscape, it might be better for the final display. The numbering starts with the 1 in the upper left corner, the 2 to right of it, and so on. Then it is easier to hold the rectangle from the two slit openings and keep it more or less firm. Thicker paper would also help.
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