05/10/31 tough puzzle from sudoku.com.au

A complete proof

1) First eliminations : [i3=4%row, h1=2%block, e3=2%block, h4=4%block, a6=4%block, c7=4%row] lead to 27 filled cells.

2) Look at only possibles f6=5,d6=5 in their row. They forbid{f5=5, d5=5}. Look at only possibles h2=8,i2=8 in their block. They forbid{b2=8, a2=8, f2=8, e2=8}. Look at only possibles a3=5,c3=5 in their row. They forbid{a2=5}. Look at only possibles d2=4,f2=4 in their row. Whether f2=4 (in which case d2=5%block) or d2=4, in both cases, we have no more {d2=7, d2=9}. Look at only possibles f2=4,d2=4 in their row. Whether d2=4 (in which case f2=5%block) or f2=4, in both cases, we have no more {f2=9, f2=8}. Now easy fillings up to 29 filled cells. (If needed, f1=8%block, e4=8%block)

3) Look at only possibles f4=6,f5=6 in their block. They forbid{f8=6, f9=6}. Look at only possibles a7=9,e7=9 in their row. Whether e7=9 (in which case d1=9%block) or a7=9, in both cases, we have no more {a1=9}.

4) Here is the crux ! Look at only possibles c5=3,c3=3 in their col. Whether c5=3 (in which case a4=6%cell,a1=7%cell) or c3=3 (in which case g3=9%cell,d1=9%row), in both cases, we have no more {d1=7}. Now easy fillings up to 50 filled cells. (If needed, e2=7%block, a1=7%block, d1=9%block, e6=1%cell, d8=1%block, i4=1%row, f4=9%row, f5=6%block, a4=6%block, d4=3%row, i6=3%row, d6=7%row, f6=5%block, d5=2%block, b6=2%block, d2=5%block, f2=4%block, d9=4%block, h5=5%cell, g7=5%block, h7=1%block)

5)Look at only possibles f8=3,f8=2 in their cell. Whether f8=3 (in which case h8=8%cell) or f8=2 (in which case i9=2%row), in both cases, we have no more {i9=8}. Look at only possibles g3=3,g3=9 in their cell. Whether g3=3 (in which case c5=3%col) or g3=9 (in which case b3=8%cell,b5=1%cell), in both cases, we have no more {c5=1}. Look at only possibles h2=3,h8=3 in their col. Whether h2=3 (in which case a2=9%cell) or h8=3 (in which case f8=2%cell,a7=2%col), in both cases, we have no more {a7=9}. Now easy fillings up to 81 filled cells. (If needed, b5=1%block, c1=1%block, b2=6%block, g1=6%block, g2=1%block, e7=9%row, e8=6%block, c9=6%block, i7=6%block, b9=8%row, b8=7%block, a8=9%block, b3=9%block, i2=9%block, h2=8%block, i8=8%block, g5=9%block, i5=7%block, g9=7%block, i9=2%block, f8=2%block, a7=2%block, f9=3%block, g3=3%block, a2=3%block, c5=3%block, a5=8%block, c3=8%block, h8=3%block, a3=5%block, c8=5%block)

Short hints for a proof

The crux is at step 4 :

1) easy to 27 filled.
2) Looking at 5 in row R6, at 8 in block Bh2, at 5 in row R3, at 45 in d2f2, eliminate some possibles. Then easy to 29 filled.
3) Looking at 6 in Be5, eliminate some possibles. Looking at 9 in row R7, eliminate a1=9 (beware, pentagon).
4) Looking at 3 in column Cc, eliminate d1=7 (beware, 11-agon ! but what a reward to get it by yourself...). Then easy to 50 filled.
5) Looking at f8, eliminate i9=8 (heptagon). Looking at g3, eliminate c5=1 (nonagon). Looking at 3s in column Ch, eliminate a7=9(nonagon).Now easy to unique solution.

Forbidding-chain-like proof

The 11-agon is needed here :

around 27 filled
(f6=5)==(d6=5) forbids {f5=5, d5=5}
(h2=8)==(i2=8) forbids {a2=8, b2=8, f2=8, e2=8}
(a3=5)==(c3=5) forbids {a2=5}
(d2=4)==(f2=4)--(f2=5)==(d2=5) forbids {d2=7, d2=9}
(f2=4)==(d2=4)--(d2=5)==(f2=5) forbids {f2=9, f2=8}
around 29 filled
(f4=6)==(f5=6) forbids {f8=6, f9=6}
(a7=9)==(e7=9)--(e2=9)==(d1=9) forbids {a1=9}
(a1=7)==(a1=6)--(a4=6)==(a4=3)--(c5=3)==(c3=3)--(g3=3)==(g3=9)--(g1=9)==(d1=9) forbids {d1=7}
around 50 filled
(h8=8)==(h8=3)--(f8=3)==(f8=2)--(f9=2)==(i9=2) forbids {i9=8}
(c5=3)==(c3=3)--(g3=3)==(g3=9)--(b3=9)==(b3=8)--(b5=8)==(b5=1) forbids {c5=1}
(a2=9)==(a2=3)--(h2=3)==(h8=3)--(f8=3)==(f8=2)--(a8=2)==(a7=2) forbids {a7=9}

Equivalence with 09/20

Now applying the same "dictionary" (changing rows, columns, values) to proof of 05/09/20 you will mechanically get a proof of today's. E.g. first filled cell in 05/09/20 was a7=9%row; in today's it becomes i3=4%row. 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 05/09/20's proof).

Go : back to sandbox