5k2/8/5r2/1RRK4/6p1/8/5p2/8 w - - 0 1
Right. Or between these two: 8/8/k1K5/8/8/6p1/5p2/1R6 b - - 0 1 and 8/8/k1K5/8/8/6p1/5p2/1R6 w - - 0 1. Same position, Black's move and White's move.
Thanks for the correction. I stopped analyzing after 7. Rd1+ Kc7.
The ultimate difference between 1.Rb1 and 1.Rc1 is these two positions: 8/8/k1K5/8/8/6p1/5p2/1R6 b - - 0 1 and 8/8/k1K5/8/8/8/5pp1/1R6 b - - 0 1.
Umeshpn, the king doesn't truly escape in either variation, but when white has been forced to play 7.Rd1+ vs 8.Rc1+ this costs white exactly one tempo, which black uses to play g2, and that move means that when black has been forced to, for example, a6 followed by white's Kc6, black can now play f1(Q) or g1(Q) guarding the a1 square and the threatened mate. Other than that one nitpick of mine, well done!
We need to consider another line also after 1. Rb1. 1... g3 2. Rc3 g2 3. Rb7 Rf7 4. Rb8+ Kg7 5. Rg3 Kf6 6. Rf3+ Kg6 7. Rg3+ draws. 7... Kf5 8. Rg3+ Kg4 9. Rxf7 wins for White.
White to play draws, but with precise play only.
- Rc1? Rf5+ 2. Kc4 Rxb5 3. Kxb5 g3 wins for Black. 2. Ke4 Rxb5 3. Kf4 also doesn't work due to 3... Rf5+! 4. Kxf5 g3. Now, 5. Kf6 Ke8 6. Ke6 Kd8 7. Rd1+ Kc1 and the BK escapes checks.
White can get is a draw with 1. Rb1! Rf5+ 2. Ke4! Rxc5 3. Kf4 Rf5+ (3... Rc4+ 4. Kg3 draws.) 4. Kxf5 g3, and now the fact that the R is on the b-file changes things. 5. Kf6! Ke8 (5... Kg8 6. Kg6) 6. Ke6! Kd8 7. Kd6 Kc8 8. Rc1+ Kb7 9. Rb1+ Ka6 10. Kc6! draws due to the mate threat in the other direction. An attempt to escape by going down loses: 10... Ka5 11. Kc5 Ka4 12. Kc4 Ka3 13. Kc3 Ka2? (13... Ka4 draws.), and now 14. Rf1! and now both 14... Ka3 Ra1# and 14... g2 15. Rxf2+ followed by 16. Rxg2 win for White.
I have no idea how to solve this puzzle for now.