The New Sudoku Players' Forum

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OK for 1. and 2. (obvious)
But 3. is not proven.

OK, so: if 1r1c1 hasn't been asserted as a hidden single, then there are one or more additional cells in r1 (for example), with candidates for '1'.
From (2), T1-delete, allows each of those candidates to be eliminted, leaving a hidden single in r1.

[denis_berthier]

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OK. That's the missing piece.
So, finally, template-depth 1 isn't an exception and T1bis is as strong as T1.
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[denis_berthier]

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Happy New Year to all.

Generally speaking, 17-clue puzzles are among the easiest (in the mean) to solve with the usual resolution rules; but they are among the hardest (still in the mean) to solve with templates; they have easy puzzles with very large numbers of templates.

You might think that this is not unexpected because a small number of clues should imply a larger number of candidates (and therefore supposedly a larger number of templates *). And this is indeed true at the start, as I showed in [HCCS2, Table 3.1]. But, as also shown in [HCCS2, Table 3.2], the situation is more contrasted after BRT ("after Singles"). Excluding 17c puzzles solvable in BRT:
- there are more candidates in the mean (153.3) than for the controlled-bias collection (129.6);
- there are fewer candidates in the mean (153.3) than for the old T&E(2) collections (eleven = 205.7, ph2010 = 234.4);
- there are fewer candidates in the mean (153.3) than in mith's T&E(3) collection (169.7) (in which no puzzle has template-depth >3).

I even found a few puzzles with large numbers of templates, but with very low SER:

Code: Select all

.......719......6..2.........4.7.....3....4.....91....7..6....8...3..2..1........ ED=4.4/1.2/1.2
.......81.6.5.....7........2..6..5....8....2.1..3......4..21....5....6........... ED=2.6/1.2/1.2
......1.3.8..2.........45...3.....2....5...7....1.....1..3..6..7...8......5...... ED=3.4/1.2/1.2
......1.5..4.7..........3..15.3........6...8..2...........512....6....7.8........ ED=2.0/1.2/1.2
......1.94...5........2.3.....1.9...5....7......3......13.......8.....4...9....2. ED=4.5/1.2/1.2

(*) this supposition is indeed false, as previously mentioned: there's no general correlation between the number of candidates after BRT and the template-depth of a puzzle.
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