1Checkmate, 2Material, 3Force, 4Time, 5Space/Mobility. Order of importance. Checkmate is not within the search horizon for either side. To be precise you're trading a Bishop and Knight for a Rook and Pawn. So you're essentially equal on Material value alone.
However Look at the other factors of the trade: The Black Knight on g4 puts a Force of 1 on h6,f6,e5,e3,f2, and h2 The Black Bishop on c5 puts a Force of 1 on a7,b6,d4,e3,f2,a3,b4,d6,e7,&f8 Where as the White Pawn on f2 only puts a Force of one on e3 and g3, and the White Rook only puts a Force of one on f2, d1 and e1. Not a good trade.
Time? Without the trade Black has 3 minor pieced developed to White's 4, White is castled, and black is not. Just a little behind but its Black's move and he can develop a 4th piece or castle. WITH the trade, Black has ONE minor piece developed to White's 4. MUCH worse. Time says NO! (time also says you shouldn't have moved the g8 Knight twice)
Space/Mobility? Black has 47 moves availble to him prior to initiating the trade. White: 32 After the trade: Black 35, White 41. I'm not going to add up forces on each square to find out which side controls more squares pre-trade and post-trade, but you could. My intuition says white would gain ground.
Remember to compare to the other legal moves you could make that may give you better checkmate, material, force, time, or space/mobility. There's many better moves, and IN THIS situation, that trade is not justified. Now in the endgame or elsewhere the same criteria may yield different results. But at least I hope I've helped you see why that is a bad trade in your example.
Also on your first example, the position is NOT illegal, its black to move, he's in check and must get out, but white is not in check, I do not see anything "illegal" about the position.
If I recall correctly there are two dark squared bishops and no pawn could have promoted to them, thus, a position that cannot occur in a game and is therefore illegal.
A good point
A simple way of taking a bad opponnets rook in 4 moves, emphasis on bad.
In post #29 you're simply winning the f7-pawn (what in the world?).
your diagrams are illegal positions
but some good points although I question your most recent illustration which is a bit random
well, so are you
No, generally when you swap two minor pieces for a rook you have put yourself in a losing game.. In the diagram you swapped a rook and a pawn for two minor pieces and even with the extra pawn you put yourself in a losing game. Remember a knight is generally worth maybe 3.2 pawns and a bishop is generally worth 3.3 pawns compared to a rook worth about 5.0 pawns.
aha, more numerology!
yes we seem to have gone full circle