White can always win against any pawn on the seventh except a bishop's pawn or a rook's pawn because there is no stalemate defence. The technique is to force the king either to abandon the pawn or to assume a position in front of the pawn thereby allowing the white king to approach.
1.Qd4+
Kc2
2.Qe3
Kd1
3.Qd3+
Ke1
4.Kb7
Kf2
5.Qd4+
Kf1
6.Qf4+
Kg2
7.Qe3
Kf1
8.Qf3+
Ke1
9.Kc6
Kd2
10.Qf4+
Kd1
11.Qd4+
Kc2
12.Qe3
Kd1
13.Qd3+
Ke1
14.Kd5
Kf2
15.Qd4+
Kf1
16.Qf4+
Kg1
17.Qg3+
Kf1
18.Qf3+
Ke1
19.Kd4
Kd2
20.Qd3+
Ke1
21.Ke3
Kf1
22.Qxe2+
Kg1
23.Kf3
Kh1
24.Qg2#
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