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Modular Problem

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In college, our classics professor brought a box of Andes mints to class every day, which the 20 of us shared while discussing Plato and John Locke. One kid, however collected all of the wrappers and would come back with the weirdest, coolest constructions of origami and modular origami. I never knew how that worked, and (in the dark, ancient days before quality search engines) I didn’t know how to look for it.

Yesterday, I was looking through @Mythagon‘s stuff because, well, she’s awesome. And there it is! With tons of links for my little nerd-heart! So of course, I get obsessed with it. “Just one small structure! I’ll stop!” Neon Post-It notes were sitting in front of me.

Here’s my 30-piece ball to which she linked:
20131204-164855.jpg

Which is pretty. But there’s a mistake. Two, actually:
20131204-164838.jpg
see them?

Now, I’m pretty sure that the 30 pieces (10 each of yellow, orange, and pink) could be put together so that all colors only touch other colors. I wasn’t very careful with the construction, and just kinda jumped into it. But…
…is there a way I could figure out whether it’s possible?
AND
How can I make sure, during construction, that I get what I want?

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