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It's even full of duplicate tiles, so plenty of room for expansion.
I don't imagine you'll be anywhere near the limit for sprites. Unlike background tiles, NES can flip sprite tiles horizontally and vertically. So if you've got a left facing tank in the set, you also have a right facing tank for free.
NES is no fun.
Actually, you don't have more than 256.
I checked the lazy man's way which ignores possible same tiles that share a palette.
If the HUD pushes it over, you might want to make it not occupy the same lines as the main level. (Right now it's on the same row as what looks like a door from the main map.) There's a way to swap to another set of 256 tiles mid screen, (Kirby's Adventure does it), but it would affect everything on the row.
Seems like you've got it except perhaps for the HUD. Great work!
It's technically possible to do things not tile aligned, but uh... I wouldn't.
The HUD also breaks lots and lots of rules assuming you haven't intended to use sprites there. Like I said in an earlier post, I recommend you post the actual palettes you've chosen with the image so it's easier for everyone to check this stuff. Here are my guesses as to your intended background palettes:
But sprite overlays should be used VERY sparingly, because of the eight sprite per scanline limitation.