I think I have invented the ancient Japanese art of origami fic challenges.
Or, at least, I found out how to make a cube out of paper (if by figure out, you really mean "found where someone else explains how to do it") and, as a cube has six sides, I think it would be great for KAT-TUN fic pairing challenges. Have a prompt for a general idea, then roll two paper dice to see who the pairing is.
If you are curious about how to make your own paper cubes of awesomey goodness, check this out:
http://michal.kosmulski.org/origami/bbu/index.html For each cube/die you will need:
6 of the A1 tiles and put it together like is says under "Cubic connection method"
6 pieces to be the external faces (for the purpose of this, I find B8 to be great as there's a stripe to write names on).
Now, if you're not familiar with crease pattern diagrams, you're in pretty much the same boat as me. But I could probably describe how to do the A1 and B8 parts. I figured out a bunch of the more simple ones. I figured out the As and Bs and most of the Cs and a handful of early Ds.
Oh, and if you haven't seen Joker Game yet, go watch it now. I'll wait. It's insanely good.