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EPub supports coding that could make the randomness possible, in maybe a couple of different ways, but not all ePub readers would support that code. Mobi doesn't support this; there's no way to build scripts into a mobi file. (At least, not for end users. Who knows what Amazon might come up with in the future?) An app would definitely be easiest, and most consistent; there is just no way to force a particular page order (not even "random") on all the various ebook devices currently available.
If word placement is *vitally* important, it should probably be done as a set of image slides--which would have to be arranged to display on a 6" or 5" screen. CSS can do a lot to maintain original formatting, but it tends to get strange when the reader increases or decreases the font size.
It's possible that non-image poems could be done, as suggested, as chapters, with a link at the bottom of each for "random next poem." This wouldn't get the guaranteed "no repeats until you've seen them all," but is probably more do-able through CSS that's more widely supported.
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