05/10/21 tough puzzle from sudoku.com.au

This one is equivalent to 05/09/28. Proof :

Now applying the same "dictionary" (changing rows, columns, values) to proof of 05/09/28 you will mechanically get a proof of today's. E.g. first filled cell in 05/09/28 was h2=8%block; in today's it becomes i1=4%block. Try it ! Or see the method explained step by step on 10/08 example.

In case that you be lazy enough not to find it by yourself, here are hints for today's proof (automatically derived from 05/09/28's proof).

A) short hints

1) easy eliminations to 26 filled.
2) considering 1 in Bh2, 3 in Bh2, 5 in Be5, 16 in a4c4, 25 in R7, remove some possibles : that will get to 39 filled.
3) considering 68 in h8i8, 63 in g3i3, remove some possibles : will get to 54 filled.
4) Look at only possibles e9=3,b9=3 in their row, conclude that c1=3 anyway. Now to unique solution.

B) Detailed hints

1) First eliminations : [i1=4%block, a6=2%block, b1=2%block, b4=9%col, b5=5%col] lead to 26 filled cells.
2) Look at only possibles e6=5,e4=5 in their block. They forbid {e1=5, e3=5, e7=5, e9=5}.
Look at only possibles h1=1,h2=1 in their block. They forbid {h5=1, h8=1, h9=1}.
Look at only possibles g3=3,i3=3 in their block. They forbid {e3=3, f3=3, a3=3, c3=3}.
Look at only possibles c4=1,c4=6 in their cell. Whether c4=6 (in which case a4=1%cell) or c4=1, in both cases, we have no more {g4=1, i4=1, a5=1}.
Look at only possibles c4=6,c4=1 in their cell. Whether c4=1 (in which case a4=6%cell) or c4=6, in both cases, we have no more {c6=6, a5=6, e4=6, g4=6, i4=6}.
Look at only possibles d7=2,i7=2 in their row. Whether i7=2 (in which case d7=5%row) or d7=2, in both cases, we have no more {d7=8, d7=9, d7=1}.
Look at only possibles i7=2,d7=2 in their row. Whether d7=2 (in which case i7=5%row) or i7=2, in both cases, we have no more {i7=7, i7=8}.
From here, easy eliminations to 39 filled cells.
3) Look at only possibles h8=8,h8=6 in their cell. Whether h8=6 (in which case i8=8%cell) or h8=8, in both cases, we have no more {f8=8, d8=8}.
Look at only possibles i3=6,i3=3 in their cell. Whether i3=3 (in which case g3=6%cell) or i3=6, in both cases, we have no more {h2=6, a3=6, c3=6, e3=6, h1=6}.
Look at only possibles h8=6,h8=8 in their cell. Whether h8=8 (in which case i8=6%cell) or h8=6, in both cases, we have no more {b8=6, g8=6, g9=6}.
From here, easy eliminations up to 54 filled cells.
4) Look at only possibles e9=3,b9=3 in their row. Whether e9=3 (in which case c1=3%row) or b9=3 (in which case b2=6%col), in both cases, we have no more {b2=3, c1=6}.
From here, easy eliminations up to unique solution.

First eliminations : [i1=4%block, a6=2%block, b1=2%block, b4=9%col, b5=5%col] to 26 filled cells.
(e6=5)==(e4=5) forbids {e3=5, e7=5, e1=5, e9=5},
(h1=1)==(h2=1) forbids {h8=1, h9=1, h5=1},
(g3=3)==(i3=3) forbids {c3=3, e3=3, f3=3, a3=3},
(c4=1)==(c4=6)--(a4=6)==(a4=1) forbids {i4=1, g4=1, a5=1},
(c4=6)==(c4=1)--(a4=1)==(a4=6) forbids {i4=6, e4=6, a5=6, c6=6, g4=6}.
Now easy (i5=1%block, a5=3%row, c6=4%block, f5=4%block, g4=7%cell) to 31 filled cells.
(d7=2)==(i7=2)--(i7=5)==(d7=5) forbids {d7=9, d7=1, d7=8},
(i7=2)==(d7=2)--(d7=5)==(i7=5) forbids {i7=8, i7=7}.
Now easy (g7=9%row, h6=9%block, h9=5%col, i9=7%block, d7=5%block, i7=2%block, d9=2%block, h3=2%block) to 39 filled cells.
(i3=6)==(i3=3)--(g3=3)==(g3=6) forbids {e3=6, h2=6, h1=6, c3=6, a3=6},
(h8=8)==(h8=6)--(i8=6)==(i8=8) forbids {d8=8, f8=8},
(h8=6)==(h8=8)--(i8=8)==(i8=6) forbids {b8=6, g8=6, g9=6}.
Now easy (e3=8%cell, f9=8%block, d5=8%block, i4=8%block, h8=8%block, i8=6%block, e6=6%block, h5=6%block, g3=6%block, i6=5%block, e4=5%block, g6=3%block, i3=3%block, c3=5%cell, f1=5%block)to 54 filled cells.
(c1=3)==(e1=3)--(e9=3)==(b9=3)--(b9=6)==(b2=6) forbids {b2=3, c1=6}.
Now easy (c1=3%cell, f2=3%block, f3=7%block, d1=9%block, f8=9%block, a3=9%block, d2=6%block, a1=6%block, b9=6%block, c4=6%block, e9=3%block, e7=4%block, b8=3%block, a9=4%block, g8=4%block, b2=4%block, a2=7%block, c2=8%block, a7=8%block, b7=7%block, h1=7%block, g9=1%block, d8=1%block, c7=1%block, a4=1%block, h2=1%block, e1=1%block)to unique solution.

See more illustrated examples of equivalence between puzzles : 09/26,09/29, 10/02, 10/03, 10/04.