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Somehow there might be a case for Chinese Chess lacking or mostly lacking in arbitrariness, which could fit into Fergus' thesis about any Next Chess candidate, as Chinese Chess is hugely popular (if possibly less so than chess). I think I'd have to strain myself a bit harder to figure out a plausible argument for why the considerably popular shogi (Japanese Chess) would also be lacking [at least mostly] in arbitrariness (that goes double for Chu Shogi), in view of the way the generals move seemingly so oddly, alone. At least by coincidence, shogi seems far less popular than Chinese Chess (or chess), though the real reason would seem to be that China has long had a much larger population than Japan.

If it's accepted chess is by far the most popular chess variant globally, I repeat I'd think any Next Chess would likely be some sort of extention of chess, though that could mean plausible candidates for such could be among the larger-board-size, different sliders and leapers crowd that H.G. has in the past expressed his disdain for, if nothing else due to his preference for ground-breaking variants. The latter may be becoming harder to come up with if they are of any quality, though, aside from that they may prove too complex rulewise at times for the average western grandpa or young child.