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Those three statements are all true.  The cannon must jump over exactly one piece (of either color) in order to capture.

In the picture above the black cannon at f12 is putting the white king in check.  Is that correct for a standard cannon movement? If no, I have to fix some code.  I went to piece-o-pedia and the cannon must jump a screen for the attack.  Am I correct with the following statements:

1) the white king is not in check

2) the black cannon at f12 can not attack anything due to a mess of black pieces in front of it.

3) the white cannon at g1 can take cannon g12

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I think I am correct, but I want to check with my fellow chess players to confirm that I understand the cannon capture move.