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I read on the wiki talk page for Tenjiku about the idea that jumping generals could have been intended to do more than just jump-capture.

The start position shows the generals in front of the Fire Demons, both straight ahead and diagonally. There is no opportunity for jumping generals to capture Fire Demons in the start position, which is intentional.

It doesn't seem right that there is so much protection for Fire Demons, yet in some versions of the game, the King could be threatened and captured without even being able to evade the attack, as it's boxed in.

(In fact, the Bishop General could just mate the opponent's King on the first move if there were no restrictions on jumping.)

If the generals could jump whenever they wanted as far as they liked, the game would become even more tactically sharp than it already is. However, it doesn't break the game.

The Great General can't jump two squares diagonally to threaten one of the Fire Demons as the Rook General would capture it.

The best the Bishop Generals can do is to manoeuvre and attack a Horned Falcon or Soaring Eagle.

The idea that jumping generals could capture all of the opposing pieces they jumped over in one turn is plausible. I don't know if the game would break, but the inventor(s) of Tenjiku weren't sentimental about pieces — allowing Fire Demons to punch large holes into positions — so having more pieces do the same seems logical.

The Great General could just capture the Vice General; Free Eagle; Queen; Drunk Elephant; and a Pawn on the first move (with check). This doesn't seem intentional.

One of the Bishop Generals could move to the edge of the board and threaten to capture a Soaring Eagle; Water Buffalo; Phoenix; Drunk Elephant; and a Pawn. The Soaring Eagle can leap out of the way, though.

Otherwise, everything is sufficiently well defended that jumping generals couldn't capture a total of anything worth more than themselves, as they are even stronger than their current form.

That could be why Fire Demons are so powerful — with burning and sliding and an area move — because they would be ecliped otherwise by the jumping generals.

The board size (16×16) and large number of pieces that do very little makes more sense if there were even more crazy pieces. The game might devolve into a capture-fest, but that might have been the intention.

Whoever had the imagination to create this game has to be admired. I consider this to be the most ambitious of all the historical variants as its construction is so delicately balanced, even with the ambiguity in the rules themselves.