Last night I reached this position after White played 54. Kd4. I eventually arrived at this position after my opponent missed a totally killer move and a few moves later hung a bishop. So instead of being down a piece and a pawn, I was up a piece.
There are similarities between this position and the one from last Friday. I have a passed pawn that the opponent's king is within striking distance of, and the opponent has a pawn majority on the king side. My king can't go wandering too far away unless I get the bishop on the a3-f8 diagonal at the right time. Remembering last week's position I came up with the plan to either promote the a pawn or get the bishop on the crucial diagonal to free up my king. There are a number of different ways to win this position. I could have simply put the bishop back on b4 but I didnt want it where white could harass it and still be within the box. I decided I wanted to utilize what I had learned from last week's game.
54... a4 55. Kc4 a3 56. Kb3 Bb4! This time I remembered that he can not take the bishop and now I'm covering f8. Now White's king is tied down to the a pawn and my king is free to have white pawns for supper. 57.Ka2 Ke5 58. Kb3 Bf8 59. Ka2 Kf4 60. f6 Kg3 61. f7 Kxh3 62. Kb3 Kxg4 63. Ka2 h5 64. Kb3 h4 65. Ka2 h3 66. Kb3 h2 67. Ka2 h1=Q 68. Kb3 Qf3+ 69. Ka2 Qb7 70. Ka1 Qb2# I had 59 seconds when I mated.
Since these two games occurred within a five day period I was able to remember the pattern from the previous game. Had the games been several months apart it's possible I may have forgotten about using the bishop to try to draw the king out of the box. I probably would have found a way to win even without that idea. However I liked my marauding king coming in to munch on the king side pawns, and the promotion coming on h1 instead of a1.