There are some instructions on making a hexaflexagon but I have found them (sorry to anyone with an instructables on here) not very helpful. I figured out how to do this one on my own and will try to explain it best I can. So please enjoy my instructable. There are no protractors or crazy math, just simple fun origami.
Step 1: Materials
You need 3 things - ruler - paper - scissors - tape or glue - pencil Other than that there are not much things to use.
Step 2: 1st Cut and Measurement
Not really a measurement but just a straight line is what I meant. Now just mark a straight line that is about a 3/4 of an inch thick all the way across the paper. See pictures.
Step 3: Another Cut
Now you are going to just cut off one end, a half of a triangle so you just have a small slant. Only do this to one side!
Step 4: Initial Fold
Now you are going to start the folding. Now take that slanted edge that you just cut and line it up with the top edge of the rest of the paper. Like in the pictures. Now unfold that and continue on.
Step 5: Tedious Folding
Now repeat the last step but with lining up the longer part of the paper with the fold line you just made. Repeat until most of the paper is done. For more info, look at pictures. In the end it should look like you have a line of upside down and right side up triangles in a line.
Step 6: Hexagonish Folds
Now you really only need 10 triangles so cut off 10 triangles and take that for this part. Really you need to follow the pictures. Just make sure only two triangles are following from the same strand. That will make sense later in the pics.
Step 7: Tape/glue and Design
Now tape or glue that lonesome triangle at the top down onto the triangle right below it. Design how you want with marker or pen but only on the front and back. I'll teach you how to fold it next.
Step 8: How to Use and Fold
Fold do you have a triangle, tri, x, thingy with the middle able to open up. Really just look at the pictures. Decorate the inside.
Step 9: Implenting
Now show your friends and family, monkeys and neighbors, daleks and cybemen. Everyone loves these. I once used them as valentines. Hope this was helpful. Tell me how I did. Have fun.