I owned a Paperboy cartridge for the NES and something odd I noticed was that the random order of houses in the game stayed the same between pressing the Reset button, but were different when turning the console off and on. The cartridge didn't have any kind of a battery for saving, which is what I would expect would be necessary to store even that small amount of information, which means there must be some technical quirk to resetting that accidentally keeps a little saved data around, but not everything.
Any hints what's going on at a hardware level during this interaction?