What is the minimum effort the White king must put into arriving at this destination? His Majesty is not feeling productive today.
This position was composed by Gianni Donati and published in Phénix 2014.
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1. h4 a5 2. Nf3 Nf6 3. Nd4 Nd5 4. Nb3 Nc3 5. a3 Nxd1 6. Ra2 Ne3 7. Rh3 Nf5 8. Re3 a4 9. Re6 Ng3 10. Rb6 Nh1 11. Re6 axb3 12. Rd6 bxa2 13. Re6 axb1=B 14. Rd6 Ba2 15. Rc6 Bd5 16. Rd6 Be4 17. Rf6 Bg6 18. Re6 fxe6 19. h5 Bf7 20. h6 g6 21. Kd1 Bg7 22. Ke1 Be5 23. Kd1 Bh2 24. g3 Bg1 25. Bg2 Rf8 26. Bd5 Bg8 27. Ba2 Kf7 28. b3 Kf6 29. Bb2+ Kg5 30. Bd4 Ra5 31. Ba7 b6 32. Ke1 Rff5 33. Kf1 Kh5 34. Kg2 Qf8 35. Kh3 Qg7 36. hxg7 Bf7 37. g8=Q Rg5 38. Qh8 Raf5 39. Qd4 Bb7 40. Qe4 Bd5 41. Qg2 Bf3 42. Qh2 Bg4+ 43. Kg2+ Bh3+ 44. Qxh3#
The reason this is easy to miss is that one is tricked into being so obsessed with promoted Bishops that it doesn't occur to one that the Queen can be replaced, as well.
I count 8 plies with the white King.
Partial answer, still stuck.
Black king is in check, so last move was white. White queen has only h2 to retract to, capturing something in h3 in its last move. Black is only missing one pawn and one queen. It can't be pawn since black's f and g pawns are accounted for at e6 and g6, and black's a and b pawns can't reach h3. So it must have been black queen at h3 with a check.
Black pawn at e6 must have come from f7 via capture, so black pawn at g6 must have come from g7. If black pawn at b6 had come from a7 via capture, Ba7 couldn't have moved there normally, unless it was a promoted bishop at b8. But white only has one missing pawn, from column h, and there is not enough missing black pieces for white pawn to promote at b8. So black b6 came from b7, and black a-pawn is missing.
Assuming all the bishops didn't come from promotion, we notice a loop of causality:
Therefore our assumption was wrong, so at least one of those bishops were promoted.
Therefore Bf7 is a promoted piece (from black a-pawn), and it must have promoted at b1 with 3 captures (capture to move to column b, then b3 to a2 to pass white a-pawn, then a2 to b1). Since white rook can't move in and out of a1-a2-b1 area, the white pieces captured at a2 and b1 must be rook and knight respectively. The white piece captured by black a-pawn to go to b column can be the knight or the rook, and the piece captured by black f-pawn at e6 must be the other one, since white h-pawn can't go to e column.
But now I'm stuck. After black promotes the bishop at a2 and bring it to g6 in preparation to go to f7 (or e8), black must capture white's last free piece at e6, shown in this diagram:
(in the image above, I have made white capture black light squared bishop at c8 with a knight, since it should be captured at some point, and all the pawns are stuck so it could not move anywhere. This frees the queen and king to move to the respective places)
But now white's free piece is only the h-pawn, which only has 2 moves left. But we need at least 6 black moves to free another of white piece (unlocking white g2 by placing black bishop at h2 via: fxe6 Bf7 g6 Bg7 Be5 Bh2). So I must have gone wrong somewhere (unattributed cases, I suppose), but couldn't see it for now.