For the first two positions, the game is a dead draw even though Black is two pawns up. But for Position 3, Black can win easily.
Sounds a little crazy? Then let's take a step back and imagine all 3 positions without the extra pawn on the 4th rank. You should know by now that will mean a draw, so long as White shuffles his rook along the h-file, and keeps the king on a2 and b2.
Note those words in italics; that's the reason why White can draw our first two positions. As long as the extra pawn is on the a or b file, it cannot help to chase away the White king from either a2 or b2, and White maintains the draw by the old rook shuffle.
So for example in position 1, White draws after 1... a3 2. Ka2! (NOT 2. Kxa3?? Ra1+ 3. Kb2 h1=Q losing) and White sticks his king on a2 forever, while his rook dances along the h-file and kicks away the Black king with a check should he get too close to his h2 pawn.
How about position 3? Black wins because his c-pawn can pull the White king off the critical a2 and b2 squares:
There is no way the king can stop both pawns at once. After 3. Kxc3 White loses to 3... Rc1+ 4. Kd2 h1=Q, while after 3. Ka2 c2 the c pawn will march to promotion.
White's best try; after 4.Rxh2, 4... Ra2+ wins the White rook with a skewer.
5. Rh4+ Ke3
6. Rh3+ Kf4
Now if White continues to keep up the checks Black simply marches his king to the h-file to destroy White's checking distance.
7. Rxc3 h1=Q
One thing that's so beautiful about chess is how a small shift, no matter how imperceptible, can make a world of difference!
Before we end off Part 1, I will leave y'all with another rook endgame position:
|White to play and win|
Of course you know that if that Black pawn on c5 were to suddenly vaporize, then White will have a similar winning position that we saw in Position 3. So his main goal will be to try to get rid of that pawn... but how?
Accurate play is needed here, so you can challenge yourselves again to try to solve this problem. We'll go through it in Part 2; have fun! (:
The Vancura Position: http://nushsblackknights.blogspot.sg/2014/01/the-vancura-position.html
Passive rook + Pawn on 7th rank (Part 1): http://nushsblackknights.blogspot.com/2014/02/2nd-lunar-new-year-challenge.html
Passive rook + Pawn on 6th rank: http://nushsblackknights.blogspot.sg/2014/02/passive-rooks-shall-stay-passive.html
"Silman's Complete Endgame Course" by Jeremy Silman