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Andrei's notation

05/12/12 tough puzzle from sudoku.com.au

5 6
7
3 1 7
2 1 9
8 3
6 4 5
2 4 8
1
9 5

Want to see the whole thing? A complete proof
Just stuck somewhere and willing to have still work to do ? Short hints for a proof
Studied enough forbidding chains to appreciate this Forbidding-chain-like proof ?
Understood the equivalent puzzles stuff? Equivalence with 09/24

A complete proof

Filled cells are left blank on diagrams for easier reading.
First eliminations : [g3=5%col, c7=5%col, a5=5%col] lead to 23 filled cells.

148 1248 9 49 2347 8 2378 9 2347 89 289
1468 1246 89 2345 68 2356 89 2345 89 289 1234 9 1234 89
468 2468 2489 249 2489
467 3578 3457 8 68 67 78
4679 1247 27 1247 1267 9 1279
37 79 39 1237 8 2378 1279
1367 167 1367 1379 379
3467 8 4678 346 2356 78 2356 789 2357 89 2346 79 2347 9
346 1236 78 2367 8 1237 8 26 2346 7 2347

Look at only possibles c2=6,c1=6 in their col. They forbid{a3=6, b3=6, a2=6, b2=6}.
Look at only possibles g8=9,g9=9 in their col. They forbid{h8=9, h7=9, i8=9, i7=9}.
Now easy fillings up to 46 filled cells. (If needed, f7=9%row, e2=9%col, i3=9%row, d3=6%row, e8=6%col, a7=6%col, e6=5%col, f6=3%row, a3=3%row, c4=3%col, c9=9%col, g8=9%col, a4=7%cell, b4=9%cell, h5=9%col, b3=1%block, a9=1%row, b2=7%col, f3=7%cell, b5=6%row, b6=4%cell, a2=8%block, a8=4%col)

28 2347 8 2378 248 28
28 2358 258 123 1238
248 24 248
68 67 78
1247 27 124 127
128 28 12
46 235 25 2346 234
46 1238 238 128 26 2346 7 2347

Look at only possibles b9=2,b9=8 in their cell. Whether b9=8 (in which case g9=2%cell) or b9=2, in both cases, we have no more {f9=2, e9=2, d9=2}.
Look at only possibles b9=8,b9=2 in their cell. Whether b9=2 (in which case g9=8%cell) or b9=8, in both cases, we have no more {e9=8, d9=8, f9=8}.
Now easy fillings up to 51 filled cells. (If needed, f9=4%cell, d5=4%col, d9=7%col, e5=7%col, e9=3%cell)

28 28
28 258 258 123 1238
28 24 248
68 67 78
12 12
128 28 12
46 235 25 2346 234
46 1238 28 128 26 2346 7 2347

Look at only possibles f1=8,f8=8 in their col. Whether f1=8 (in which case e1=2%cell) or f8=8 (in which case d7=2%cell), in both cases, we have no more {d1=2, d2=2}.
Look at only possibles f1=8,f8=8 in their col. Whether f1=8 (in which case e1=2%cell) or f8=8 (in which case f2=5%col), in both cases, we have no more {f2=2}.
Now easy fillings up to 54 filled cells. (If needed, f2=5%cell, d8=5%col, d2=3%cell)

28 28
28 28 123 1238
28 24 248
68 67 78
12 12
128 28 12
46 246 24
46 18 28 128 26 2346 7 2347

Now, Look at only possibles e1=2,f1=2 in their block. They forbid{h1=2, g1=2, i1=2}.
Now easy fillings up to 81 filled cells. (If needed, g9=2%col, g6=8%col, b8=2%col, d7=2%row, b9=8%col, g1=6%col, c2=6%col, c1=4%col, h6=6%col, h1=7%col, i1=3%row, h8=3%col, h4=1%col, i8=1%col, i7=8%col, d1=1%col, f5=1%col, i5=2%row, e4=2%row, e1=8%col, f8=8%col, h2=2%row, f1=2%row, i2=4%col, h7=4%col, d4=8%col, i6=7%col)

1 8 9 7 3 4 2 5 6
4 2 7 5 6 8 9 3 1
6 3 5 2 1 9 7 4 8
2 4 1 9 5 3 8 6 7
5 6 8 4 7 1 3 9 2
7 9 3 8 2 6 4 1 5
3 1 2 6 4 7 5 8 9
8 7 6 3 9 5 1 2 4
9 5 4 1 8 2 6 7 3

Short hints for a proof

One of the two heptagons is highlighted.

28 28
28 258 258 123 1238
28 24 248
68 67 78
12 12
128 28 12
46 235 25 2346 234
46 1238 28 128 26 2346 7 2347


1) First eliminations to 23 filled cells.
2) Look at only possibles 6 in Cc. Look at only possibles 9 in Cg. Remove some possibles. Now easy eliminations take place again, leading to 46 filled cells.
3) Look at only possibles 28 in b9g9. Remove some possibles. Then simple eliminations take place again, up to 51 filled cells.
4) Looking at only possibles 8 in Cf, remove d1=2, d2=2, f2=2 (beware, heptagons). Now easy fillings up to 54 filled cells.
5) Look then at only possibles 2 in Be2. Remove some possibles. From here, simple eliminations lead to unique solution.

Forbidding-chain-like proof

One of the two heptagons is highlighted.

28 28
28 258 258 123 1238
28 24 248
68 67 78
12 12
128 28 12
46 235 25 2346 234
46 1238 28 128 26 2346 7 2347


around 23 filled
(c2=6)==(c1=6) forbids {b3=6, a2=6, a3=6, b2=6}
(g8=9)==(g9=9) forbids {h8=9, i8=9, i7=9, h7=9}
around 46 filled
(b9=2)==(b9=8)--(g9=8)==(g9=2) forbids {d9=2, f9=2, e9=2}
around 51 filled
(e1=2)==(e1=8)--(f1=8)==(f8=8)--(d7=8)==(d7=2) forbids {d2=2, d1=2}
(e1=2)==(e1=8)--(f1=8)==(f8=8)--(f8=5)==(f2=5) forbids {f2=2}
around 54 filled
(e1=2)==(f1=2) forbids {g1=2, h1=2, i1=2}

Equivalence with 09/24

1) Take 05/09/24 grid 2) Renumber as shown below :
replace
by		1
5		2
3		3
9		4
2		5
8		6
1		7
6		8
7		9
4

3) Replace cells as shown below		4) You get 05/12/12's grid.
1 7
8
8 6 2
9 1 7
2 5
3 4 6
5 9 4
6
3 1
g9 h9 i9 d9 e9 f9 a9 b9 c9
g8 h8 i8 d8 e8 f8 a8 b8 c8
g7 h7 i7 d7 e7 f7 a7 b7 c7
g4 h4 i4 d4 e4 f4 a4 b4 c4
g5 h5 i5 d5 e5 f5 a5 b5 c5
g6 h6 i6 d6 e6 f6 a6 b6 c6
g3 h3 i3 d3 e3 f3 a3 b3 c3
g2 h2 i2 d2 e2 f2 a2 b2 c2
g1 h1 i1 d1 e1 f1 a1 b1 c1
5 6
7
3 1 7
2 1 9
8 3
6 4 5
2 4 8
1
9 5

Now applying the same "dictionary" (renaming numbers, replacing cells) to proof of 05/09/24 you will mechanically get a proof of today's. E.g. first filled cell in 05/09/24 was i7=1%col; in today's it becomes c7=5%col. 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, see hints for today's proof (automatically derived from 05/09/24's proof).

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Andrei's notation