ending 3 KxQ vs KxH, how to mate

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It seems that Q vs H is a general win, like Q vs R. (A bit surprising, as H is more valuable than R, and I thought R would be about the maximum that a Queen can still beat.) The given position seems to be a mate in 26 through 1. Qb5 or 1. Qe4 (all other moves take longer). Fairygen says the maximum DTM is 40 moves:

KQ_KH

WON.wtm 3802839
K capture 1574104
other 2228735
0. 844026
1. 3910
2. 3546
3. 4786
4. 6289
5. 8756
6. 15798
7. 28804
8. 56721
9. 92248
10. 126990
11. 131452
12. 133442
13. 118033
14. 91090
15. 72918
16. 66682
17. 66063
18. 58365
19. 51080
20. 58017
21. 73202
22. 85702
23. 86628
24. 88079
25. 85054
26. 86353
27. 84030
28. 74766
29. 64858
30. 59275
31. 54152
32. 50689
33. 46972
34. 35890
35. 17000
36. 3466
37. 812
38. 310
39. 76
40. 50
WON.btm 2192354
stalemate 2
W check 775874
LEGAL 3036382
TOTAL 3812256

Maximum distance to conversion is 36 moves, however, so the 50-move rule still gives you a fair leeway.

Well, it is certainly true that a piece that controls all adjacent squares, and thus cannot be approached by a King, will be able to draw against a Queen. So the King + Queen vs King + non-royal King (Commoner) is a draw, provided there is no initial tactics that gains the Commoner. (Which in practice is only the case if the Commoner is in reach of its own King, as it is defenseless against distant attack, and very easily forked by a Queen.) So a Commoner manages what the (approximatey equally strong) Knight and Bishop, and even the much stronger Rook cannot do.

Nominally the Hawk is stronger than a Rook, but as mentioned by many different people, the Hawk’s strength lies in its jumping abilities that can launch powerful attacks in the opening and middlegame. The more open the position is, ie fewer pawns, the weaker the Hawk is. Considering that this is a four-man ending, the Hawk is probably weaker than a Rook here, suggesting a win for White.