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

Today's solution has been proposed by Steve From Ohio. It's very short, though involving two steps of depth 4. I don't know if there is a solution of depth 3.

A complete proof

1) First eliminations : c8=3%row, g4=3%row, h2=3%row, e8=9%row, a7=9%row, a6=3%row lead to 27 filled cells.

2)
Look at only possibles h1=2,h1=1 in their cell. Whether h1=1 (in which case i1=2%cell) or h1=2, in both cases, we have no more {h3=2, b1=2}.
Now easy fillings up to 37 filled cells. (If needed, h3=4%cell, b3=2%block, b5=7%col, b7=4%col, h9=7%col, i8=4%block, b1=9%cell, e2=9%block, b9=3%cell, a5=3%col)

3) :
Look at only possibles g4=5,g4=4 in their cell. Whether g4=5 (in which case i2=5%block) or g4=4 (in which case e6=4%col,a2=4%col), in both cases, we have no more {a2=5}.
Look at only possibles a6=1,a8=1 in their col. Whether a6=1 (in which case a9=9%col) or a8=1 (in which case e7=1%row,c7=5%row), in both cases, we have no more {a9=5}.
Now easy fillings up to 81 filled cells. (If needed, a3=5%col, a1=8%col, e1=6%cell, i2=5%row, c3=6%block, c2=7%col, a2=4%row, i7=3%col, h7=2%block, h5=6%col, i9=6%block, e9=8%row, h1=1%cell, i1=2%cell, a9=9%cell, a8=6%col, d6=6%col, g7=8%block, e7=5%block, d3=8%row, e3=7%row, d2=3%block, g3=3%row, f7=6%row, c9=5%row, c7=1%row, a6=1%block, e6=4%cell, c5=4%col, g4=4%row, f5=9%row, f6=8%col, i5=8%row, i6=7%row, i4=1%cell, d4=5%row, f4=7%block, f8=3%col, c6=9%cell, g5=5%row, d8=7%col, e8=1%row, d5=1%block, e4=3%cell)

Short hints for a proof

Two nonagons here, one is bolded.

1) easy to 27 filled.
2) eliminate h2=3 (2 sets), then easy to 37 filled.
3) eliminate a2=5 (gives pb with 4s in row 4, 4 sets), a9=5 (gives pb with 1s in Bb8, 4sets). Then easy to the end.

Total sets used :10, max depth :4.

Forbidding-chain-like proof

Two nonagons here, one is bolded

around 27 filled
(h1=2)==(h1=1)--(i1=1)==(i1=2) forbids {h3=2, b1=2}
around 37 filled
(i2=5)==(g3=5)--(g4=5)==(g4=4)--(e4=4)==(e6=4)--(a6=4)==(a2=4) forbids {a2=5}
(a9=9)==(a6=9)--(a6=1)==(a8=1)--(c7=1)==(e7=1)--(e7=5)==(c7=5) forbids {a9=5}

That's all for today, folks...