06/02/05 tough puzzle from sudoku.com.au

Today's puzzle is considerably simpler than yesterday's, however, beginners might find it a bit tough.

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 ?

A complete proof

1) First eliminations : f4=6%row, g8=3%block, e5=2%block, g2=9%cell lead to 26 filled cells.

2)
Look at only possibles d5=3,d6=3 in their block. They forbid{d1=3, d2=3}.

3) :
Look at only possibles i9=5,f9=5 in their row. Whether i9=5 (in which case b5=5%row) or f9=5 (in which case d2=5%col), in both cases, we have no more {b2=5}.
Now easy fillings up to 39 filled cells. (If needed, b2=7%cell, a7=7%col, h6=7%col, a9=1%block, h9=2%col, h2=6%cell, d4=7%row, i7=6%col, c8=6%row, d1=4%cell, g1=2%cell, g3=4%cell, b1=9%cell)

4) (needs depth 4) :
Look at only possibles i6=4,c6=4 in their row. Whether c6=4 (in which case h7=4%row) or i6=4, in both cases, we have no more {h5=4, i9=4}.
Look at only possibles f8=4,f9=4 in their col. Whether f8=4 (in which case b5=4%col) or f9=4 (in which case i9=5%row,b5=5%row), in both cases, we have no more {b5=8, b5=1}.
Now easy fillings up to 43 filled cells. (If needed, b4=1%col, g6=1%block, f5=1%col, f6=9%cell)

5) Two last nonagons.
Look at only possibles b5=5,i5=5 in their row. Whether b5=5 (in which case b8=4%col) or i5=5 (in which case f9=5%row,f8=4%col), in both cases, we have no more {h8=4}.
Look at only possibles d7=5,f9=5 in their block. Whether d7=5 (in which case g7=8%cell) or f9=5 (in which case f8=4%col,b8=8%cell), in both cases, we have no more {c7=8, h8=8}.
Now easy fillings up to 81 filled cells. (If needed, h8=9%cell, i9=5%cell, f9=4%cell, b8=4%row, c6=4%col, b3=8%col, b5=5%col, i6=2%cell, c7=9%cell, e9=9%row, d7=5%row, d2=1%col, f3=5%block, c2=5%block, c3=3%col, i1=3%col, i3=7%cell, a3=2%row, e3=6%cell, f8=7%col, e2=3%col, e8=8%cell, h7=4%row, g7=8%row, g4=5%cell, i5=4%row, c4=2%block, e1=7%row, e7=1%col, c9=8%cell, a4=8%row, a6=3%cell, d5=3%col, a5=9%block, d6=8%row, i4=9%row, a1=6%block, h5=8%col)

Short hints for a proof

Two nonagons here


1) easy to 26 filled.
2) eliminate d12=3 (1 set).
3) with 5s in row 9, eliminate b2=5 (3 sets), then easy to 39 filled.
4) eliminate {h5=4, i9=4} (pbm with 4s in col c, 2 sets), with f9, eliminate b5=1 (pbm with f9, 4 sets), then easy to 43 filled.
5) eliminate h8=4 (pbm with 5s in col i, 4 sets) , eliminate {c7=8, h8=8}
(pbm with f9, 4 sets). Then easy to the end.

Total sets used :18, max depth :4.

Forbidding-chain-like proof

Two nonagons here


around 26 filled
(d5=3)==(d6=3) forbids {d1=3, d2=3}
(b5=5)==(i5=5)--(i9=5)==(f9=5)--(d7=5)==(d2=5) forbids {b2=5}
around 39 filled
(i6=4)==(c6=4)--(c7=4)==(h7=4) forbids {h5=4, i9=4}
(b5=4)==(b8=4)--(f8=4)==(f9=4)--(f9=5)==(i9=5)--(i5=5)==(b5=5) forbids {b5=8, b5=1}
around 43 filled
(b8=4)==(b5=4)--(b5=5)==(i5=5)--(i9=5)==(f9=5)--(f9=4)==(f8=4) forbids {h8=4}
(g7=8)==(g7=5)--(d7=5)==(f9=5)--(f9=4)==(f8=4)--(b8=4)==(b8=8) forbids {h8=8, c7=8}

That's all for today, folks...