06/02/12 tough puzzle from sudoku.com.au

A progressive puzzle. The hardest step is the last one (4sets-deep).

A complete proof

1) First eliminations : h8=9%block, f9=3%block, b2=2%block, i8=3%row, b8=5%block lead to 27 filled cells.

2)
Look at only possibles b3=8,c3=8 in their block. They forbid{g3=8, e3=8}.
Look at only possibles g7=7,h7=7 in their block. They forbid{d7=7, a7=7, e7=7}.
Look at only possibles c8=6,c8=4 in their cell. Whether c8=4 (in which case c7=6%cell) or c8=6, in both cases, we have no more {a7=6, c6=6, c4=6, a8=6}.
Look at only possibles c7=4,c7=6 in their cell. Whether c7=6 (in which case c8=4%cell) or c7=4, in both cases, we have no more {c9=4, a8=4, c3=4, c4=4, a7=4, b9=4}.
Now easy fillings up to 40 filled cells. (If needed, a7=1%cell, a8=7%cell, g7=7%cell, h1=7%block, h7=4%cell, c7=6%cell, i9=1%cell, d9=4%row, e9=7%row, f3=7%block, c8=4%col, d8=6%cell, e8=1%cell)

3)
Look at only possibles h2=5,i2=5 in their block. They forbid{e2=5, f2=5}.

4)
Look at only possibles f2=6,f2=4 in their cell. Whether f2=6 (in which case e3=9%cell) or f2=4 (in which case a2=9%cell), in both cases, we have no more {e2=9, b3=9, c3=9}.
Now easy fillings up to 42 filled cells. (If needed, a2=9%block, e3=9%block)

5)
Look at only possibles f2=4,f1=4 in their col. Whether f2=4 (in which case e2=6%block) or f1=4 (in which case a4=4%col,a5=6%col), in both cases, we have no more {e5=6}.

Now easy fillings up to 81 filled cells. (If needed, e2=6%col, f2=4%cell, a1=4%row, a4=6%cell, c3=3%block, b3=8%row, a5=3%cell, c9=8%col, b4=4%block, g1=3%block, i2=5%cell, b9=9%row, d5=9%row, d6=7%col, b5=7%col, b6=1%col, h6=3%row, f6=5%row, f1=2%cell, f4=1%cell, h5=1%block, g2=1%row, e5=8%row, d1=8%block, e1=5%cell, d7=5%block, h4=5%row, h2=8%col, g4=8%block, c4=9%row, c6=2%block, d4=2%row, f5=6%cell, i5=2%cell, e7=2%cell, i6=6%cell, g3=6%col, g6=9%block, i3=4%block)

Short hints for a proof

Elimination of e5=6 at 42 filled.

1) easy to 27 filled.
2) Basic 1/2sets-deep eliminations, then easy to 40 filled.
3) looking at f2, eliminate {c3=9, e2=9, b3=9} (3 sets), then easy to 42 filled.
4) eliminate e5=6 (causes trouble for 4s in row 1, 4 sets), then easy fillings to the end.

Total sets to examine : 12, max depth : 4 (at step 4)

Forbidding-chain-like proof

4-sets move at 42 filled.

Here are, ordered, the eliminations needed. The rest is only easy fillings.
(b3=8)==(c3=8) forbids {e3=8, g3=8}
(g7=7)==(h7=7) forbids {e7=7, d7=7, a7=7}
(c8=6)==(c8=4)--(c7=4)==(c7=6) forbids {c6=6, a7=6, c4=6, a8=6}
(c7=4)==(c7=6)--(c8=6)==(c8=4) forbids {b9=4, c3=4, c9=4, a7=4, c4=4, a8=4}
(h2=5)==(i2=5) forbids {f2=5, e2=5}
(e3=9)==(e3=6)--(f2=6)==(f2=4)--(a2=4)==(a2=9) forbids {b3=9, c3=9, e2=9}
(e2=6)==(f2=6)--(f2=4)==(f1=4)--(a1=4)==(a4=4)--(a4=6)==(a5=6) forbids {e5=6}

That's all for today, folks...