What is the result with best play (in the diagram above) if it's white to move but there's a twist: White is allowed to place any white piece(s) anywhere on the board before his first move as long as the resulting position is legal (the position can be reached in any possible combination of moves).
The best move is:
Kxh2 to eventually force a draw by insufficient material or stalemate. (You capture the pawns (moving to f7 to force the h7 pawn to move and become capturable, if necessary), forcing a K-K endgame.)
The only way for all of these pawns to get to column h is for there to have already been 15 pawn captures by Black total: 1 for g7-h6, 2 for f7-g6-h5, 3 for e7-f6-g5-h4...
Of course, White only had 16 pieces at the start of the game. 15 have been captured, meaning no more white pieces can exist. So White's only option is to cause a stalemate.
The black pawns
must have captured at least 15 pieces to reach that position;
white can't place any piece on the board to arrive at a position that's legal (as white can have at most 16-15=1 piece on the board and there's already a king there.
If that wasn't true,
white could place a queen on f7 say and play Qf8#