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The natural 1.axb5 wins.
1…b6 is a try, but there is no blockade: 2.c6 Kd6 3.Kd6 and the WK can’t be denied the e6 square (sacking the c pawn if necessary). Once the WK gets there, b6 must fall, and Kb6 and Pb5 always win.
1.Kxb5? is drawn, because there’s no way to sac the a-pawn without allowing …Kc8.
Only two material based moves are permitted white because of a check by a pawn on the white king.
1. King takes the pawn – or –
2. Pawn takes the pawn
#1 permits an endgame for white.
ab5 kc7
kd5 kd7
b6 kc8
kd6 kd8
c6 kc8 draws
ab5 kc7
kd5 kd7
b6 kc8
kd6 kd8
c6 kc8 draws
Instead of 5.c6?? in the above line, try 5.Ke6. Black can’t wander away from the b-pawn, but White can then squeeze Black into the a8 corner & sac the c-pawn to free the b-pawn (and avoid stalemate).
1. axb5
Now white has connected pawns. Black king cannot enter a6-b6-c6-d6.
If black pushes pawn to b6, white simply plays c6 and c-pawn cannot be stopped from queening. One possible line could be
1. axb5 Kc7
2. Kd5 Kd7
3. Ke5 Ke7
4. Kf5 Kf7?
5. c6 bxc6
6. b6 and b-pawn becomes queen.
1. axb5 Kc7
2. Kd5 Kd7
3. Ke5 Ke7
4. Kf5 Kd7
5. Kf6 Kd8
6. Ke6 Kc7
7. Ke7 and so on.
Black can not play b6, as white can reply with c6, which gives even more powerful position for white (connected pawns with one of the pawn is passed pawn and that too no other pieces on the board).
— P. Anandh