The solution to the first puzzle could be that we are actually looking at board upside down and that the rook is actually on h8 instead of a1 and on white's last move he captured some black pieces on h8 with a pawn from g7 and promoted to a rook... that's the only solution I can come up with.
Two strange chess problems

If I'm reading this properly, the first problem doesn't have a solution. Of course this is assuming it is white to move, which is what I see. So unless black breaks the rules and willingly places his king into a double-check, it isn't happening. No en-passant possibilities either (i think?).
I don't think 1. Rc6 leads to anything but checkmate, correct me if I'm wrong. White resigns.

If you liked the second one, check out my awesome thread of "don't mate puzzles" that's sweeping the nation.

jdcannon, you got the two correct answers.
lkraft91, by moving your rook to c6 the black rook is no more pinned and can capture the bishop( notice that there are two light-squared bishops, one is a promoted pawn)

The solution to the first puzzle could be that we are actually looking at board upside down and that the rook is actually on h8 instead of a1 and on white's last move he captured some black pieces on h8 with a pawn from g7 and promoted to a rook... that's the only solution I can come up with.
But then pawn on g7 would have been pinned by white's Bishop. This position cannot happen in a game and seems to be an "error" and not strange.

The solution to the first puzzle could be that we are actually looking at board upside down and that the rook is actually on h8 instead of a1 and on white's last move he captured some black pieces on h8 with a pawn from g7 and promoted to a rook... that's the only solution I can come up with.
But then pawn on g7 would have been pinned by white's Bishop. This position cannot happen in a game and seems to be an "error" and not strange.
??? Pawn on g7 was white.

The first problem has no solution under the conditions you specified. Unless, of course, one is allowed to change the rules of chess.
The second is an old problem. I forget the composer's name - why don't you give it - instead of letting the readers assume it is your own problem (here, we call it plagarism or cheating).
Barefoot_Player

the white pawns are going up = the 2nd problem, so it was a bit of a fail misunderstanding from the OP, got me for a while too until i looked at the 2nd puzzle "aaah".

The solution to the first puzzle could be that we are actually looking at board upside down and that the rook is actually on h8 instead of a1 and on white's last move he captured some black pieces on h8 with a pawn from g7 and promoted to a rook... that's the only solution I can come up with.
But then pawn on g7 would have been pinned by white's Bishop. This position cannot happen in a game and seems to be an "error" and not strange.
made me think for a bit then, but then of course, its a discovery check, not a pin!
In the first position you have to look for a way for white to get in this position. In the second one, you have to find the only move that doesn't lead
white into checkmate (white pawns are going up, I know the position is strange)