06/02/13 tough puzzle from sudoku.com.au

A puzzle dedicated to those who wish working on 3sets-deep moves. Steve found a solution, probably shorter, but involving a 4sets-deep move.

A complete proof

1) First eliminations : h7=6%col, h1=2%block, h9=8%block, b3=6%block, a2=1%block lead to 27 filled cells.

2)
Look at only possibles f6=3,f4=3 in their block. They forbid{f3=3}.
Look at only possibles i2=4,i2=3 in their cell. Whether i2=3 (in which case g2=4%cell) or i2=4, in both cases, we have no more {g3=4, e2=4, i1=4, c2=4, h3=4}.
Look at only possibles g2=3,g2=4 in their cell. Whether g2=4 (in which case i2=3%cell) or g2=3, in both cases, we have no more {e2=3, g3=3, h3=3, i1=3}.
Now easy fillings up to 31 filled cells. (If needed, e3=3%row, c2=8%cell, d1=8%block, c8=2%cell)

3)
Look at only possibles a9=4,a9=9 in their cell. Whether a9=9 (in which case c7=4%cell) or a9=4, in both cases, we have no more {b9=4, b7=4}.
Now easy fillings up to 32 filled cells. (If needed, b9=1%cell)

4)
Look at only possibles c3=4,f3=4 in their row. Whether c3=4 (in which case c7=9%cell) or f3=4 (in which case e1=9%cell), in both cases, we have no more {e7=9}.
Now easy fillings up to 33 filled cells. (If needed, e7=1%cell)

5)
Look at only possibles d3=9,d3=5 in their cell. Whether d3=5 (in which case d7=9%cell) or d3=9, in both cases, we have no more {d4=9, d6=9, d5=9}.

6)
Look at only possibles c7=4,c7=9 in their cell. Whether c7=4 (in which case f3=4%row) or c7=9 (in which case d3=9%col), in both cases, we have no more {f3=9}.
Look at only possibles d7=9,d3=9 in their col. Whether d7=9 (in which case c6=9%col) or d3=9 (in which case g4=9%col), in both cases, we have no more {a4=9, i6=9}.

7)
Look at only possibles i5=9,g4=9 in their block. Whether g4=9 (in which case i5=5%block) or i5=9, in both cases, we have no more {i5=7, i5=4, i5=1}.
Look at only possibles g4=5,i5=5 in their block. Whether i5=5 (in which case g4=9%block) or g4=5, in both cases, we have no more {g4=7, g4=3, g4=4}.

8)
Look at only possibles d3=9,g3=9 in their row. Whether d3=9 (in which case d7=5%cell) or g3=9 (in which case g4=5%cell), in both cases, we have no more {g7=5}.

9)
Look at only possibles g7=4,g7=3 in their cell. Whether g7=3 (in which case g2=4%cell) or g7=4, in both cases, we have no more {g9=4}.

10)
Look at only possibles i9=5,i5=5 in their col. Whether i9=5 (in which case a9=4%row) or i5=5 (in which case a4=5%block), in both cases, we have no more {a4=4}.

Now easy fillings up to 81 filled cells. (If needed, a4=5%cell, i5=5%block, g4=9%cell, c1=5%row, i1=9%block, d3=9%block, d7=5%col, f3=5%col, g9=5%block, e1=4%block, a1=3%cell, b1=7%cell, c6=7%block, a5=9%block, b7=3%col, a8=8%cell, f7=8%row, c3=4%block, g7=4%row, g2=3%cell, i2=4%cell, i8=3%row, f8=6%cell, g8=1%block, h3=1%row, e8=7%row, d5=1%row, d2=6%col, d6=2%cell, f9=2%col, e9=9%block, e2=2%row, i6=1%block, b4=2%block, i9=7%row, d4=7%col, g3=7%row, h5=7%col, a9=4%col, b5=8%col, b6=4%block, h4=4%col, h6=3%block, f4=3%col, c7=9%block, f6=9%block, e5=6%block, f5=4%block)

Short hints for a proof

The last 3sets-deep move.

1) easy fillings to 27.
2) with f46=3, g2i2=34, eliminate some possibles, then easy fillings to 31 filled cells.
3) with a9c7=49, eliminate some possibles, then easy fillings to 32 filled cells.
4) with 4s in row 3 eliminate e7=9, then easy fillings to 33 filled cells.
5) with d3d7=59, eliminate some possibles.
6) with c7 eliminate f3=1. With 9s in col d, eliminate {i6=9, a4=9}.
7) with 5s and 9s in Bg5, eliminate some possibles.
8) with 9s in row 3 eliminate G7=5.
9) with g2g7=34, eliminate some possibles.
10) with 5s in col i, eliminate a4=4, then easy fillings to the end.

Total sets to examine : 26, max depth : 3 (at steps 4,6,8,10)

Forbidding-chain-like proof

The last 3sets-deep move.

Here are, ordered, the eliminations needed. The rest is only easy fillings.
(f6=3)==(f4=3) forbids {f3=3}
(i2=4)==(i2=3)--(g2=3)==(g2=4) forbids {h3=4, c2=4, e2=4, g3=4, i1=4}
(g2=3)==(g2=4)--(i2=4)==(i2=3) forbids {g3=3, h3=3, i1=3, e2=3}
(a9=4)==(a9=9)--(c7=9)==(c7=4) forbids {b7=4, b9=4}
(c7=9)==(c7=4)--(c3=4)==(f3=4)--(e1=4)==(e1=9) forbids {e7=9}
(d3=9)==(d3=5)--(d7=5)==(d7=9) forbids {d5=9, d4=9, d6=9}
(f3=4)==(c3=4)--(c7=4)==(c7=9)--(d7=9)==(d3=9) forbids {f3=9}
(c6=9)==(c7=9)--(d7=9)==(d3=9)--(g3=9)==(g4=9) forbids {i6=9, a4=9}
(i5=9)==(g4=9)--(g4=5)==(i5=5) forbids {i5=7, i5=4, i5=1}
(g4=5)==(i5=5)--(i5=9)==(g4=9) forbids {g4=3, g4=4, g4=7}
(d7=5)==(d7=9)--(d3=9)==(g3=9)--(g4=9)==(g4=5) forbids {g7=5}
(g7=4)==(g7=3)--(g2=3)==(g2=4) forbids {g9=4}
(a9=4)==(i9=4)--(i9=5)==(i5=5)--(a5=5)==(a4=5) forbids {a4=4}