05/10/28 tough puzzle from sudoku.com.au

Chainlike steps.

1) Around 51 filled cells, (c1=7)==(c1=6)--(c8=6)==(c8=7) forbids {c2=7, c6=7}
2) Around 53 filled cells, (i4=3)==(i3=3)--(g2=3)==(e2=3) forbids {e4=3}

Short hints for a proof.

1) easy to 51 filled.
2) Looking at 67 in c1c8, eliminate c2=7,c6=7 ; then easy to 53 filled.
3) Looking at only possibles 3 in row Ri, eliminate e4=3 (pentagon). Then easy to unique solution.

Proof.

1) First eliminations : [g9=2%row, i5=2%block, h9=3%row, g8=8%block, a9=8%block, c4=8%block, h3=8%block, g3=6%block, i8=9%row, e8=4%row, f7=6%block, d7=9%block, a2=4%row, g7=1%cell, e1=5%cell, e9=1%cell, d1=8%cell, f5=8%block, b5=3%row, c7=3%block, b7=2%block, a7=5%block, d5=5%row, f9=5%block, d9=7%block, h5=4%row, i7=4%block, h7=7%block, a5=6%cell, h2=9%cell, g4=9%row, f1=9%cell] lead to 51 filled cells.

2) Look at only possibles c1=7,c1=6 in their cell. Whether c1=6 (in which case c8=7%cell) or c1=7, in both cases, we have no more {c2=7, c6=7}. Now easy fillings up to 53 filled cells. (If needed, i2=7%row, i1=1%cell)

3) Look at only possibles i4=3,i3=3 in their col. Whether i3=3 (in which case e2=3%row) or i4=3, in both cases, we have no more {e4=3}.

Now easy fillings up to 81 filled cells. (If needed, e4=6%cell, h6=6%block, h4=1%block, d6=1%block, e6=2%block, d3=2%block, f3=4%block, d4=4%block, b6=4%block, e2=3%block, i3=3%block, g2=5%block, i4=5%block, c6=5%block, a6=9%block, c3=9%block, b4=7%block, f6=7%block, c1=7%block, a8=7%block, b1=6%block, c8=6%block, b3=5%block, g6=3%block, f4=3%block, c2=2%block, b8=1%block, a3=1%block)