The New Sudoku Players' Forum

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[quote="denis_berthier"]
As your computation times are very low, could you compute all the T&E(3)-expands of coloin's T&E(3) collection and all the associated minimals? As I said before, considering the differences with mith's list, I think his list is far from being saturated.
BTW, which software do you use for these calculations?
[/quote]
For the last question: my own software.

[quote="coloin"]
I think when I "cleaned" up miths big file, removing the non-minimals but I found only a few more by expanding and minimizing.
max expanding is tricky enough and sometimes its "branches out" and these are easily missed [by me]
Perhaps you can check whether I completely max expanded my file ..... pretty sure I twinned many of them .....
[/quote]
First thing to mention: Colin's list contains 51 puzzles that are only T&E(2):

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1...5.7.9...1.9.3.6..37.51..615..3.7......69........513.6....75.48......7.29.....
1...5.7.9...1.9.366..37.51..615..3.7......69........513.6....75.48......7.29.....
...4.6...4.718..3.....73....4.3.1.78.....761..7.86...3.14.38....85....6.9.2......
...4.6...4.718..3.....73....4.3.1.78....4761..7.86...3.14.38....85....6.9.2......
...4.6...4.718..3.....73....4.8.1.73.....761..7.36...8.14.38....85....6.9.2......
...4.6...4.718..3.....73....4.8.1.73....4761..7.36...8.14.38....85....6.9.2......
...4.6...4.718..36....73....4.8.1.73....67.1....34.6.8.14.38....85....6.9.2......
...4.6...4.718..36....73....4.3.1.78....67.1....84.6.3.14.38....85....6.9.2......
...4.6...4.718..36....73....4.3.7.18....61.7....84.6.3.14.38....85....6.9.2......
....5.7.9...1.9.366..37.51..6159.3.7......69........513.6....75.48......7.29.....
...4.6...4.718..36....73....4.3.1.78....4761....86...3.14.38....85....6.9.2......
...4.6...4.718..36....73....4.8.1.73....4761....36...8.14.38....85....6.9.2......
...4.6...4.718..3.....73......8.1.73....4761..7.36.4.8.1..38....85....6.9.2......
1...567.9...1.9.36...37.51..615..3.7......69........513......75.48......7.29.....
...4.6...4.718..3.....73......3.1.78....4761..7.86.4.3.1..38....85....6.9.2......
...4.6...4.718..3.....73......8.1.73....6741..7.34.6.8.1..38....85....6.9.2......
...4.678.4..18.2.66...27.41.3.....7..95....1.8.4..1.......72......6.4.....281..6.
...4.6...4.718..3.....73......3.7.18....6147..7.84.6.3.1..38....85....6.9.2......
...4.6...4.718..3.....73......3.1.78....6741..7.84.6.3.1..38....85....6.9.2......
...4.6...4.718..36....73......8.1.73....4761....36.4.8.1..38....85....6.9.2......
...4.6...4.718..36....73......8.1.73....6741....34.6.8.1..38....85....6.9.2......
...4.6...4.718..36....73......3.1.78....4761....86.4.3.1..38....85....6.9.2......
...4.6...4.718..36....73......3.1.78....6741....84.6.3.1..38....85....6.9.2......
...4.678.4..18.2.66...27.41.3.....7..95....1.8.4..........72......6.4.2...281..6.
...4.6...4.718..36....73......3.7.18....6147....84.6.3.1..38....85....6.9.2......
....567.9...1.9.36...37.51..6159.3.7......69........513......75.48......7.29.....
...4.6...4.718..36....73....4.3.7.18....6147....84.6.3.14.38....85....6.9.2......
...4.678.4..18.2.66...27.41.3.....7.795....1.8.47.........72......6.4.2...281..6.
...4.678.4..18.2.66...27.41.3.....7.795....1.8.47..........2......6.4.2...281..6.
...4.678.4..18.2.66...27.41.3.....7.795....1.8.47.1........2......6.4.....281..6.
...4.678.4..18.2.66...27.41.3.....7.795....1.8.47.1.......72......6.4.2...281..6.
...4.6...4.718..3.....73....4.3.7.18....61.7..7.84.6.3.14.38....85....6.9.2......
...4.6...4.718..36....73....4.3.7.18....6147..7.84.6.3.14.38....85....6.9.2......
...4.6...4.718..36....73....4.3.1.78....6741....84.6.3.14.38....85....6.9.2......
...4.6...4.718..36....73....4.8.1.73....6741....34.6.8.14.38....85....6.9.2......
...4.6...4.718..3.....73....4.8.1.73....67.1..7.34.6.8.14.38....85....6.9.2......
...4.6...4.718..3.....73....4.3.1.78....67.1..7.84.6.3.14.38....85....6.9.2......
....567.9...1.9.366..37.51..6159.3.7......69........513.6....75.48......7.29.....
...4.6...4.718..36....73....4.3.1.78....4761....86.4.3.14.38....85....6.9.2......
...4.6...4.718..36....73....4.8.1.73....4761....36.4.8.14.38....85....6.9.2......
1...567.9...1.9.366..37.51..6159.3.7......69........513.6....75.48......7.29.....
...4.678.4..18.2.66....2.41.3.....7.795....1.8.47.1.......2.......6.4.....281..6.
...4.678.4..18.2.66...72.41.3.....7.795....1.8.47.1.......27......6.4.2...281..6.
...4.678.4..18.2.66...72.41.3.....7.795....1.8.47.........27......6.4.2...281..6.
...4.678.4..18.2.66...72.41.3.....7..95....1.8.4..1.......27......6.4.....281..6.
...4.678.4..18.2.66...72.41.3.....7.795....1.8.47.1.......2.......6.4.....281..6.
...4.678.4..18.2.66...72.41.3.....7..95....1.8.4..........27......6.4.2...281..6.
...4.678.4..18.2.66...72.41.3.....7.795....1.8.47.........2.......6.4.2...281..6.
123.....94..1.9....8......1..6.41.97.7.69.......5.7..6....159..5..76..14...9.46.5
123.....94.71.9....8......1..6.41..7.7.69.......5.7..6....159..5..76..14...9.46.5
123.....94.71.9....8......1..6.41..7.7.6........5.7..6....159..5..76..14...9.46.5


None of thier minimals, are T&E(3).

The remaining puzzles expand to 204336 minimals, on 9539 grids -- grid count down by 9.
The puzzles have 10069 max-te3-expands -- agreeing with Denis' result, if you subtract 51 for the non-TE3 puzzles.
The max-expands, produce 204336+136= 204472 minimals.
The 136 that are missing, can be gotten from these BRT-expands - 50 of them:

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1...5678........368..3.71...9..65...645..3...7.1...6...1.5...68...6.137......85.1
1....678.........68..3.71.5.857.136...15...78.....85.1..2..3....49..5...7.8...65.
1...5678.........68..3.71.5.857.136...15...78.....85.1..2..3....49..5...7.8...65.
12..5........89.......3..4.2.6...3.431.6...27.743..61.....6.47.7......63.6...31.2
12..5........89.......3..4.2.6...3.431.6...27.743..61..3..6.47.7......63.6...31.2
12.45........89.......3..4.2.6...3.431.....27.743..61.....6.47.7......63.6...31.2
12.45........89.......3..4.2.6...3.431.....27.743..61..3..6.47.7......63.6...31.2
12.45.7.9.571.9.......725...4.9.5...5..74..9.79..21.5....2.794..1....3...7....6.8
.234.6...4.678.2...8..............9..38...6.7...2.8315.4.6...7...2.......678.3..4
.234.6...4.678.2...8..32......3...9...8...6.7...2.8315.4.....7...2.......678.3..4
.234.6...4.678.2...8..32......3...9..38...6.7...2.8315.4.....7...2.......678.3..4
.234.67..4.678.2..78..............9..38...6.7...2.8315.4.6.......2.......678.3..4
.234.67..4.678.2..78..32......3...9...8...6.7...2.8315.4.6.......2.......678.3..4
..3.567.9..71.9.36..9...15.24....3...91.......36...59.....179...7.5.361........7.
..3.567.9..71.9.36..9...15.24....36..91.......36...59.....179...7.5.361........7.
12.4.67..4.718.2...86...1...7........642.8.7...176.....18...42..42....93......61.
12.4.67..4.718.2...86...1...7........642.8.7...176.....18...42...2....93......61.
...4..78.....8.1.3..8...56.2..5..9....9.2....84.96.2...84......5.264..9.96.8.2.5.
...4..78.....8.1.3..8...56.2..5..9....9.2....84.96.2...84......56.8.2.9.9.264..5.
...4.678....18.2.6....7..14......67.74..2.1.8.1.7.8.4239...7...5..8......6.....27
...4.678....18.2.6....7..14......67.74..2.1.8.1.7...4239...7...5..8......6.....27
...4.678....1..2.6.......14......67.74..2.1.8.1.7.8.4239...7...5..8......6.....27
...4.678....1..2.6.......142.....67.74..2.1.8.1.7.8.4239...7...5..8......6.....27
...4.678....1..2.6.......142.....67.74..2.1.8.1.7.8.4239...7...5..8......6.....2.
1...56......18...668.2.71.52.85.1.6....72....57..68.........9......75621.......73
1...56......18...668.2.71.52.85.1.6....72....57..68...7.....9......75621.......73
12..56.8..5718....6.82.7.1.......9.37.6....51......4....2...1..5.167...28...21...
12..56.8..57.8....6.82.7.........9.37.6....51......4....2...1..5.167...28...21...
.......894.7....3.6......51..4...9..3..9.48..9.8.673...3.7.8...7.6.43..884.69...3
.......894.7....3.6......51..4...9..3..9.48..9.8.673...3.7.8...7.6.43..884.69..73
.....6.894.7....3........51..4...9..3..9.48..9.8.673...3.7.8...7.6.43..884.69...3
.....6.894.7....3........51..4...9..3..9.48..9.8.673...3.7.8...7.6.43..884.69..73
...456..9......12........64.3594.6.....5.3.......6735.37..9....5.4.......693....7
...456..9......12........64.3594.67....5.3.......6735.37..9....5.4.......693.....
...4567.9......12........64.3.94.6.......3.......6735.37..9....5.4.......693....7
...4567.9......12........64.3.94.67......3.......6735.37..9....5.4.7.....693....7
....567.9.5.1.9.36.6.37.51.294.....7.1..97.....8....9.3...1597...5...........3.65
....567.9.5.1.9.36.6.37.51.294....57.1..97.....8....9.3...1597...5...........3.65
....567.9.5.1.9.36.6.37.51.294....57.1...7.....8....9.3...1597...5...........3.65
...4.6.8.4..18...6.8..72.41..481..67....27.....86.4.2.5......7.8.2.61...93....6..
...4.6.8.4..18...668..72.41..481..67....27.....86.4.2.5......7.8.2.61...93....6..
...4.6.8.4..18...668..72.41..481..67....27......6.4.2.5......7.8.2.61...93....6..
....5..89.5.18.2.68..........98.56.1...69.82...8.21.953.......2742....68..126....
....5..89.5.18.2.68..........98.56.1...69.82...8.21.953.......2742....6...126....
....5..89.5.18.2.68..........98.56.1...69.82...8.21.953.......2742....68..1.6....
....567.9.5.1.9.36.6.37.51.27869.....1....9.7..47.....3..9.76.1..6.........56.3..
....567.9.5.1.9.36.6.37.51.27869.....1....9....47.....3..9.76.1..6.........56.3..
....567.9.5.1.9.36.6.37.51.278.9.....1....9.7..47.....3..9.76.1..6.........56.3..
......78.4...8.2.66..72..54..1.7.....39..2...8.6...52.....6.4.2..42...65.6254.87.
......78.4...8.2.66..72..54..1.7.....39..2...8.6...52.....6.4.2..42...65.6.54.87.


The max-expands have 98 "twins" that aren't represented.
They produce another 1226 minimals, with 335 BRT-expands, over 97 solution grids -- 96 new, and one duplicate.

BRT-expands:

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.........4.7.8.2.68..7...542.6.74......392.65........2.72..5.48..4...6.768...752.
.........4.7.8.2.68..7...542.6.74......392.65........2.72..5.4858...762.6.4...5.7
.......8.4.7....3668......5...61..477..3.5.......72..33.67....854.8..67..785..3.4
......7.9...7...23798...65..17..5.9.86.9.7.159.51......795.8...5.167.9..68....5..
......7.9...7...23798...65..17..5.9.86.9.7.159.51......795.8...5.16..9..68....5..
......7.....18..36.......15.319.86.486.......9.4.613.8.4.69.....86.13.4...98.4.6.
......7.....18..36.......15.319.86.4.6.......9.4.613.8.4.69.....86.13.4...98.4.6.
......7.....18..36.9.....15.319.86.486.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....18..36.9.....15.319.86.486.......9.4.613.8.4.69.....86.13.4...98.4.6.
......7.....18..36.9.....15.319.86.4.6.........4.613.8.4.69......6.13.4...98.4.6.
......7.....18..36.9.....15.319.86.4.6.........4.613.8.4.69.....86.13.4...98.4.6.
......7.....18..36.9.....15.319.86.4.6.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....18..36.9.....15.319.86.4.6.......9.4.613.8.4.69.....86.13.4...98.4.6.
......7.....189.36.......15.31..86.486.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.......15.31...6.486.......9.4.613.8.4.69.....86.13.4......4.6.
......7.....189.36.......15.31..86.4.6.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.......15.31..86.4.6.......9.4.613.8.4.69.....86.13.4....8.4.6.
......7.....189.36.......15.319.86.486.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.......15.319.86.486.......9.4.613.8.4.69.....86.13.4...98.4.6.
......7.....189.36.......15.319.86.4.6.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.......15.319.86.4.6.......9.4.613.8.4.69.....86.13.4...98.4.6.
......7.....189.36.9.....15.31..86.486.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.9.....15.31..86.4.6.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.9.....15.319.86.486.......9.4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.9.....15.319.86.486.......9.4.613.8.4.69.....86.13.4...98.4.6.
......7.....189.36.9.....15.319.86.4.6.........4.613.8.4.69......6.13.4...98.4.6.
......7.....189.36.9.....15.319.86.4.6.........4.613.8.4.69.....86.13.4...98.4.6.
1.3.......5........9.372.....4.38.9...97.4.2....92.4........9..87..93.4294.2.783.
1.3.......5........9.372.....4.38.9...97.4.2....92.4........9..87..93.4294...783.
12...6.8..5718....6.8.72.........97.7.1....65...7..4.3..28.5...5........8.6.17..2
....5.78..5.18.2.6...7.......86.51......27.68...81.5.23.2.......7126....94.....2.
.2..5....4.7.....668..3......6.7541...134.6.7.......53.....416..6..1.3.4....63.75
.2..5....4.7....3668..3......6.7541...134.6.7.......53.....416..6..1.3.4....63.75
.23.5....4.7.....668.........6.7541...134.6.7.......53.....416..6..1.3.4....63.75
....56.....678..3278.2.3..52.5..8...37.6.2....685..2.....3.78.1...8.5.948......2.
....56.....678..327..2.3..52.5..8...37.6.2....685..2.....3.78.1...8.5.948......2.
1...56.8..5.18.......3.7.51.7.....943.......8...87.61.53..681..7.65.18.....73....
1...56.8....18.......3.7.51.7.....943.......8...87.61.53..681..7.65.18.....73....
1...56.8..5.18.......3.7.51.7.....943.......8...87.61.53..618..7.65.81.....73....
1...56.8....18.......3.7.51.7.....943.......8...87.61.53..618..7.65.81.....73....
....567......8.......2.15...75..389.83.......9.4...3..3.7.4.9.8.9....47..48.9..53
....567......8.......2.15...75..389..3.......9.4...3..3.7.4.9.8.9....47..48.9..53
1..45...9..........9..72.15..17.59.4..9.14.52..4.9.17..........832..7...915....27
1..45...9..........9..72.15..17.59.4..9.14.52..429.17.....2....832..7...91.....27
1..45...9..........9..72.15..17.59.4....14.52..429.17..4..2....832..7....1.....27
1..45...9..........9..72.15..17.59.4..9.14.52..4.9.17..4.......832..7...915....27
1..4..78....18.2.6....7..41..47.861.......8.2..162..74318......5..8......76...128
1..4..78....18.2.6....7..41..47.861.......8.2..162..7431.......5.2.......76...128
1..4..78....18.2.6....7..41..47.861.......8.2..162..7431.......5..8......76...12.
1..4..78....18.2.6....7..41..47.861.......8.2..162..7431.......5.2.......76...12.
1..4..78....18.2.6....7..41..47.861.......8.2..162..74318......5.........76...12.
1..4..78....18.2.6....7..41..47.861.......8.2..162..743.........76...12891.8.....
1..4..78....18.2.6....7..41..47.861.......8.2..162..743.........76...1289128.....
1..4..78....18.2.6....7..41..47.861.......8.2..162..743.8.......76...12891.8.....
1..4..78....18.2.6....7..41..47.861.......8.2..162..743.........76...12.91.8.....
1..4..78....18.2.6....7..41..47.861.......8.2..162..743.........76...12.912......
1..4..78....18.2.6....7..41..47.861.......8.2..162..743.8.......76...12.91.......
1.3..67.9.57..9...69...315...68.........6..1.93124....3......95......37.7.....6.1
1.3..67.9.57..9...69...315...68.1.......6..1.93124....3......95......37.7.....6.1
1.3..67.9.57..9...69...315..768.........6..1.93124....3......95......37.7.....6.1
1.3..67.9.57..9...69...315..768.1.......6..1.93124....3......95......37.7.....6.1
1.3..67.9.57..9...69...315..768.........6..1.9.124...........95......37.7.....6.1
1.3..67.9.57..9...69...315..768.........6..1.9.124....3......95......37.7.....6.1
1.3..67.9.57..9...69...315..768.1.......6..1.9.124...........95......37.7.....6.1
1.3..67.9.57..9...69...315..768.1.......6..1.9.124....3......95......37.7.....6.1
...4..78.45......3.......64.49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78.45....1.3.......64.49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78.456.....3.......64.49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
......78.456.....3.......64.49......53.....1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78.456...1.3.......64..9......53.9.4.1.6.13..9..3..19...6.1.6.3...96..45..1
...4..78.456...1.3.......64.49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
......78.456...1.3.......64.49......53.....1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78..56.....3.......6..49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78.456...1.3.......6..49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78.45......3.......6..49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78.45....1.3.......6..49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
...4..78.456.....3.......6..49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
1....6..9.5...9.36......51..35.6..4.64.......9.1.4..6.3.46.5......87........23..1
1....6..9.5...9.36......51..35.9..4.6.1.4..9.94.......3.49.5......87........23..1
...4.6.8.........686..72.542.4......39..2......5....27...76..42..2.486.5...2.587.
...4.6.8.........686..72.542.4......39..2....6.5....27...76..42..2.486.5..62.587.
...4.6.8.........68...72.542.4......39..2....6.5....27...76..42..2.486.5...2.587.
...4.6.8.....8...6....72.4523..4....5.6....72.71.2...........5.7...6482.8..2..6.7
...4.6.8.....8...6....72.4523..4....5.6....72.71.2...........5.7...6482.8.....6.7
...4.6.8.....8...6.8..72.4529.........8...572571.2....7...6582.8...4.6.7.......54
...4.6.8.....8...6.8..72.4529.........8...572571.2....7...6582.8..24.6.7.......54
...4.6.8....78....789.31....75...8.38......9.93..7..1539.1.5.785.....9.1.1.......
...4.6.8....78....789.31....75...8.38.1....9.93..7..1539...5.785.....9.1.1.......
...4.6.8....78....789.31....75.1.8.38.1....9.93..7..1539.1.5.785.....9.1.1.......
...4.6.8....78....789.31....7..1.8.38.1....9.93..7..1539.1.5.785.....9.1.1.......
...4.6.89...18.2.6.....2.4.....61.....42.8.91..894.62.571.2......6....1..3..14...
...4.6.89...18.2.6.....214.....61.....42.8.91..894.62.571.2...........12.3..14...
...4.6.89...18.2.6.....214.....61.....42.8.91..894.62.57..2......6.......3..14...
...4.6.89...18.2.6.....2.4.....61.....42.8.91..894.62.571.2......6....12.3..14...
...4.6.89...18.2.6.....214.....61.....42.8.91..894.62.57..2......6.....2.3..14...
...4.6.89....8.2.6.....214.....61.....42.8.91..894.62.57..2......6....1..3...4...
...4.6.89....8.2.6.....214.....61.....42.8.91..894.62.571.2......6....12.3...4...
...4.6.89...18.2.6.....2.4.....61.....49.8.21..824.69.571.2......6....1..3..14...
...4.6.89...18.2.6.....214.....61.....49.8.21..824.69.571.2...........12.3..14...
...4.6.89...18.2.6.....214.....61.....49.8.21..824.69.57..2......6.......3..14...
...4.6.89...18.2.6.....2.4.....61.....49.8.21..824.69.571.2......6....12.3..14...
...4.6.89...18.2.6.....214.....61.....49.8.21..824.69.57..2......6.....2.3..14...
...4.6.89....8.2.6.....214.....61.....49.8.21..824.69.57..2......6....1..3...4...
...4.6.89....8.2.6.....214.....61.....49.8.21..824.69.571.2......6....12.3...4...
1.3...78..57....366.....5.1.68......53.6.7...7.18.56.331...8......9.4.6....2.....
1.3...78..57....366.....5.1.68......53.6.7...7.18.56.331...8......91436....2.....
1.3...78..57....366.....5.1.68......53.6.7...7.18.56..31...8......91436....2.....
1.3...78..57.8..3668....5.1.68......53.6.7...7.18.56.331...8......9.4.6....2.....
1.3...78..57.8..3668....5.1.68......53.6.7...7.18.56.331...8......91436....2.....
1.3...78..57.8..3668....5.1.68......53.6.7...7.18.56..31...8......91436....2.....
1.3...78..57....366.....5.1.68......5.18.76..73.6.5...31...8......91436....2.....
1.3...78..57....366.....5.1.68......5.18.76.373.6.5...31...8......9.4.6....2.....
1.3...78..57....366.....5.1.68......5.18.76.373.6.5...31...8......91436....2.....
1.3...78..57.8..3668....5.1.68......5.18.76..73.6.5...31...8......91436....2.....
1.3...78..57.8..3668....5.1.68......5.18.76.373.6.5...31...8......9.4.6....2.....
1.3...78..57.8..3668....5.1.68......5.18.76.373.6.5...31...8......91436....2.....
.2........571.9...8......14...3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2........571.9...8.9....1....3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2........571.9...8.9....14...3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2.......4571.9...8......14...3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2.......4571.9...8.9....1....3.4.9539..15.7.54.97.3.1..47.3.5.....91.43........7
.2.......4571.9...8.9....1....3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2.......4571.9...8.9....14...3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2......9.571.....8......1....3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2......9.571.....8......14...3.4.9539..1547.54..7.3.1..47.3.5.....91.43........7
.2......9.571.9...8.9....1....3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2......9.571.9...8.9....14...3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2......94571.9...8.9....14...3.4.9539..1547.54.97.3.1..47.3.5.....91.43........7
.2......94571.9...8.9....1....3.4.9539..15.7.54.97.3.1..47.3.5.....91.43........7
.2......94.71.9...6.9...5....8.75.14..5...9.7..49.185.....94..884.71..95...5.8...
.2......94.71.9...6.9...5....8.71.54..5...9.7..49.581.....94..884.71..95...5.8...
..3.........1.9.36.68327....36..5.91.........8.1.9...53.58..96.61.9..5.8.89....13
..3...78..56.....3....3..64.49......53.9.4.1.6.13..94.3..19...6.1.6.3...96..45..1
...4.678.4.....2.6.....2.45..5.....8.3.8.....9.8..7.....6.4852...4...8.7..27.5.64
...4.678.4.....2.6.....2.45..5.....8.3.8.4...9.8..7.....6.4852...4...8.7..27.5.64
...4.678.4.....2.6.....2.45..5.....8.3.8.....9....7.....6.4852...4...8.7..27.5.64
...4.678.4.....2.6.....2.45..5.....8.3.8.....9.8..7.....2.4856...4...8.7..67.5.24
...4.678.4.....2.6.....2.45..5.....8.3.8.4...9.8..7.....2.4856...4...8.7..67.5.24
...4.678.4.....2.6.....2.45..5.....8.3.8.....9....7.....2.4856...4...8.7..67.5.24
..3..6.8.4.7.89..6....3.....7689....3.8.47...94.6.38.....3749.8.3......5.......1.
..3..6.8.4.7.89..6....3.....7689....3.8.47...94.6.38.....3749...3......5.......1.
..3..6.8.4.7.89.36....3.....7689....3.8.47...94.6.38.....3749.8.3......5.......1.
..3..6.8.4.7.89..6....3.....7689....3.8.74...94.6.38.....3479.8.3......5.......1.
..3..6.8.4.7.89..6....3.....7689....3.8.74...94.6.38.....3479...3......5.......1.
..3..6.8.4.7.89.36....3.....7689....3.8.74...94.6.38.....3479.8.3......5.......1.
1.3.5.....57.8..3668..7.1.5..8...6..3.68...175.....3.873.5.8......932..1...7....3
1.3.5.....57.8..3668..7.1.5..8...6..3.68...175.....3.8731..8......932..1...7....3
1.3.5.....57.8..3668..7.1.5..8...6..3.68...175.....3.87315........932..1...7....3
1.3.5.....57.8..3668..7.1.5..8...6..3.68...175.....3.8731.........932..1...7....3
...4.6.8.........668..72.542.4......31..2....8.5....27..87...42..2.486.5...2.587.
...4.6.894..1...3......31.424.8......39...84.7.86............68..49.83.1..136.49.
...4.6.894..1...3......31.424.8......39...84.7..6............68..49.83.1..136.49.
.2..5....4...89...........123........647...137.16..42.34.....62.72...1.4..6...37.
.2..5....4...89..........4123........647...13..16...2.34.....62.72...1.4..6...37.
.2..5....4...89......3....123........647...137.16..42.34.....62..2...1.4..6...37.
.2..5....4...89......3...4.23........647...137.16...2.34.....62..2...1.4..6...37.
...4.6...4.718..366.8.731.4239....415.......3..631.....6.8.741....64.3.8....31..7
...4.6...4.718..3.6.8.731.4239....415.......3..631.......8.741....64.3.8....31..7
...4.6...4.718...66.8.731.42.9....4.5..........631.....6.8.741....64.3.8....31..7
...4.6...4.718..366.8.731.4239....415..........6.1.....6.8.741....64.3.8....31..7
...4.6...4.718..3.6.8.731.42.9....4.5..........631.....6.8.741....64.3.8....31..7
...4.6...4.718..366.8.731.42.9....4.5..........631.....6.8.741....64.3.8....31..7
...4.6...4.718..366.8.73..4239....415.......3..631.....6.8.741....64.3.8....31..7
...4.6...4.718..3.6.8.73..4239....415.......3..631.......8.741....64.3.8....31..7
...4.6...4.718..366.8.73..4239....415..........6.1.....6.8.741....64.3.8....31..7
12..56.8..5718....8.62.7.5.21....47.......5..67....9.35..8..6...6..128.5.8.6.5...
12..56.8..5718....8.62.7.5.21....47....7..5..67....9.35..8..6..76..128.5.8.6.5...
12..56.8..5718....8.62.7.5.21....47....7.....67....9.35..8..6..76..12....8.6.5...
12..56.8..5718....8.62.7.5.21....47....7.....67....9.35..8..6...6..12....8.6.5...
12..56.8..5718....8.62.7.5.21....47.......5..67....9.35..8..6...6..128.5...6.5...
12..56.8..5718....8.62.7.5.21....47....7..5..67....9.35..8..6..76..128.5...6.5...
12..56.8..5718....8.62.7...21....47....7..5..67....9.35..8..6..76..128...8.6.5...
12..56.8..5718....8.62.7.5.21....47....7.....67....9.3...8..6..76..12....8.6.5...
12..56.8..5718....8.62.7...21....47....7..5..67....9.35..8..6..76..128.....6.5...
.2..5.......1.92......27.5.....3....36....19..48...32.5..7.291.7...935.2...51..73
.2..5.7...571.923.....27.5..7..3....36....19..48...32.5..7.291.7...935.2...51..73
.2..5.......1.923.....27.5.....3....36....19..48...32.5..7.291.7...935.2...51..73
.2..5.......1.923.....27.5..7.......3.....19..48....2.5..7.291.7...935.2...51..73
.2..5.....5.1.923.....27.5..7.......36....19..48....2.5..7.291.....935.2...51..73
.2..5.....571.923.....27.5..7.......36....19..48....2.5..7.291.7...935.2...51..73
.2..5.7...571.923.....27....7..3....36....19..48...32.5..7.291.7...935.2...51..73
.2..5.....5.1.923.....27....7.......36....19..48....2.5..7.291.7...935.2...51..73
.2..5.....5.1.923.....27....7.......36....19..48....2.5..7.291.....935.2...51..73
.2..5.....5.1.923.....27.5..7.......36....19..48....2.5..7.291.7...935.2...51..73
.2..5.7.....1.92......27.5.....3....36....19..48...32.5..7.291.7...935.2...51..73
.2..5.7.....1.923.....27.5.....3....36....19..48...32.5..7.291.7...935.2...51..73
.2..5.7...571.923.....27.5..7.......36....19..48....2.5..7.291.7...935.2...51..73
.2..5.7...571.923.....27....7.......36....19..48....2.5..7.291.7...935.2...51..73
.2..5.......1.92......27.5.....3....36....91..48...32.5..7.219.7...935.2...51..73
.2..5.7...571.923.....27.5..7..3....36....91..48...32.5..7.219.7...935.2...51..73
.2..5.......1.923.....27.5.....3....36....91..48...32.5..7.219.7...935.2...51..73
.2..5.......1.923.....27.5..7.......3.....91..48....2.5..7.219.7...935.2...51..73
.2..5.....5.1.923.....27.5..7.......36....91..48....2.5..7.219.....935.2...51..73
.2..5.....571.923.....27.5..7.......36....91..48....2.5..7.219.7...935.2...51..73
.2..5.7...571.923.....27....7..3....36....91..48...32.5..7.219.7...935.2...51..73
.2..5.....5.1.923.....27....7.......36....91..48....2.5..7.219.7...935.2...51..73
.2..5.....5.1.923.....27....7.......36....91..48....2.5..7.219.....935.2...51..73
.2..5.....5.1.923.....27.5..7.......36....91..48....2.5..7.219.7...935.2...51..73
.2..5.7.....1.92......27.5.....3....36....91..48...32.5..7.219.7...935.2...51..73
.2..5.7.....1.923.....27.5.....3....36....91..48...32.5..7.219.7...935.2...51..73
.2..5.7...571.923.....27.5..7.......36....91..48....2.5..7.219.7...935.2...51..73
.2..5.7...571.923.....27....7.......36....91..48....2.5..7.219.7...935.2...51..73
.2.4..78.......2.668.27..5424....6.8.76...54.5.8....72....9....86.315.........8.5
.2.4..78.......2.668.27..5424...76.8.76...54.5.8....72....9....86.315......7..8.5
.2.4..78.......2.668.27..5424...76.8.76...54.5.8....72....9....86.315.........8.5
.2.456....5..........37..5123..9.1.7.75.1..93......52.3.....91.59.....72.1.9..3.5
.2.456....5..........37...123..9.1.7.75.1..93.......2.3.....91.59.....72.1.9..3.5
.2.456....5..........37..5127..9.1.3.35.1..97......52.3.....91.59.....72.1.9..3.5
.2.456....5..........37...127..9.1.3.35.1..97.......2.3.....91.59.....72.1.9..3.5
....5678..5718....6..3.7......7.8.45...5.189...5...6.15.68....371.6.3..8.3...5...
....5678..5718....6..3.7......7.8.4....5.189...5...6.15.68....371.6.3..8.3...5...
....5678..5718....6..3.7......7.8.4....5.189.......6.15.68....371.6.3..8.3...5...
.23....894.7......68..........8..5.33.521.89........21....92.1....5.19.8...3...52
.23...7894.6......78.......2.78.395.3.5.9.8.7.......23.....739..3.9...75.7.3.52.8
.23...7894.6......78.........78.395...5.9.8.........2......739..3.9...75.7.3.52.8
1.3......45.789....89......2.49.8.7....54....9.5.274.8.4.29...7.9.8.52.4....749..
1.3......45.789....89......2..9.8.7....5.....9...274.8.4.29...7.9.8.52.4....749..
.2.4.6.......8..3..8.37....23..68...7.824...3.647.3..8.46....12......3...7....95.
.2.4.6.....7.8.....8.37....23..68...7.824...3.647.3..8.46....12......3...7....95.
.2.4.6.....7.8..3..8.37....23..68...7.824...3.647.3..8.46....12......3...7....95.
.234.6.....7.8..3..8.37....23..68...7.824...3.647.3..8.46....12......3...7....95.
.2.4.6.......8..3..8.37....23..68...7.824...3.467.3..8.64....12......3...7....95.
.2.4.6.....7.8.....8.37....23..68...7.824...3.467.3..8.64....12......3...7....95.
.2.4.6.....7.8..3..8.37....23..68...7.824...3.467.3..8.64....12......3...7....95.
.234.6.....7.8..3..8.37....23..68...7.824...3.467.3..8.64....12......3...7....95.
...4..7.94.......3.8....41.2...67.9..6.29....79.8.4.26.46.28...87.......9.274.6..
...4..7.94.......368....41.2...67.9..6.29.....9.8.4.26.46.28...87.......9.274.6..
...4..7.9........3.8....41.2...67.9..6.29....79.8.4.26.46.28...87.......9.274.6..
...4..7.9........368....41.2...67.9..6.29....79.8.4.26.46.28...87.......9.274.6..
12..567...571.92..6.927.....9....67.76.....43....6.5.2.76.95.2.....1....91...2...
..3.5678....18..36...3.75.1.41...65.3...15....9........3.8.1.658...63.7....57....
..3.5678....18..36...3.75.1241...65.3...15....9.......73.8.1.658...63.7....57....
....5678....18..36...3.75.124....6......15....9........3.8.1.658...63.7....57....
....5678....18..36...3.75.124....6......15....9.......73.8.1.658...63.7....57....
1....6.8..5.18....68..73......36.47....7..6.2.76....1....63....73..158...618.73.5
1....6.8..5.18....68..73....76....1....7..6.4...36.97273..15......63.....61..73.5
1....6.8..5.18....68..73....76....1....7..6.4...36.97273..15......63.....618.73.5
....5678..5.18.....6.3.71....68.137.3...756.....63.....82....1.67.518...9........
1..45...9..........9..72.15..9.14.57..4.9.12...12.59.4382..........2....915....72
1..45...9..........9..72.15..9.14.57..479.12...12.59.4382..7.......2....91.....72
1..45...9..........9..72.15....14.57..479.12....2.59.4382..7....4..2......5....72
1..45...9..........9..72.15....14.57..479.12...12.59.4382..7....4..2.....1.....72
1..45...9..........9..72.15..9.14.57..4.9.12...12.59.4382.......4..2....915....72
....56......7.9...78921....2.7.41.98.48.9..1.91......4..19..82..921...4787.......
...4.6....57....3..9..........6.14.8....4832.8..32..61..186...2...2.3....82.146.3
...4.6....57....3..9..........6.14.8....4832....32..61..186...2...2.3....82.146.3
...4.6...457....36.9..........6.14.8....4832.8..32..61..186...2...2.3....82.146.3
...4.6...457....36.9..........6.14.8....4832.8..32..61..186...2...2.3....82.14..3
...4.6...457....36.9..........6.14.8....4832....32..61..186...2...2.3....82.146.3
...4.6...457....36.9..........6.14.8....4832....32..61..186...2...2.3....82.14..3
1.3.56..945.1.9.36...3......35..49..61.9.34..9.45........6.157454.....9......5..2
12.4..7.94.71.923...9......2..87.......9..37.7.43.5...3.1.9.4.7..2....939...3.12.
12.4..7.94.71.923...9......2..87.......9..37.7.13.5...3.4.9.1.7..2....939...3.42.
1.3...789.56...1.27.91.....2...7849....9.12..9..24..1..9....8.1..481.92........74
1.3...789.56...1.27.91.....2...7849....9.12..9..24..1..9....8.1..481.9.........74
1.3...789.56.....27.9......2...7849....9.12..9..24..1..9....8.1..481.92........74
1.3...789.56.....27.9......2...7849....9.12..9..24..1..9....8.1..481.9.........74
1.3...789.56......7.9......2...7849....9.1...9..24..1..9....8.1..481.92........74
1.3...789.56...1..7.91.....2...7849....9.1...9..24..1..9....8.1..481.92........74
1.3.5678..5.......68...715....6735..........6......42.3..7.1..557..68..1.1.53....
1.3.5678..5.......68.3.715....67.5.8........6......42.3..7.1..557..68..1.1.53....
1.3.5678..5.......68.3.715....6735..........6......42.3..7.1..557..68..1.1.53....
1.3.5678..57......68....15....6735..........6......42.3....1..557..68..1.1.53....
1.3.5678..57......68...715....6735..........6......42.3..7.1..55...68..1.1.53....
1.3.5678..57......68.3.715....67.5.8........6......42.3..7.1..557..68..1.1.53....
1.3.5678..57......68.3.715....6735..........6......42.3..7.1..557..68..1.1.53....
1.3.5678..57......68.3.715....6735..........6......42.3..7.1..55...68..1.1.53....
......7.....189.36.......51.6894...3...8.1...91..63..4...3.4....46.1839..3.69..4.
1.3.56..94..1.9.....9.......31.649..6.591.4..94.5.36.....6.51.7.....1.26.16...39.
1.3.56..94..1.9.....9.......31.649..6.5.1.4...4.5.36.....6.51.7.....1.26.16...39.
1.3.56..94..1.9.............31.649..6.591.4..94.5.36.....6.51.7.....1.26.16...39.
1.3.567...5718....68....5...35.7....7.1..5..386..31.57316...8.5.............1.642
12..5.78..57.8.2.66.8.2..15...8.45.1...91.8.78.1.65...........878.....5..165.8.72
12.4..7.94.....2...8.....14.....497..94..81.271...2.48...8.1.97.7.6.....94..37...
12.4..7.9......2...8.....14......97..9...81.271...2.48...8.1.97.7.6.....94..37...
12.4..7.94.....2...8.....14......97..94..81.271...2.48...8.1.97.7.6.....9...37...
12.4..7.94.....2...8.....14.....497..94..81.271...2.48...8.1.97.7.6.....9...37...
1.3.5.78........366...7.5.1........75.....81.7.1..86.33....4...8.693........1536.
1.3.5.78........3668..7.5.1..8.....75.....81.7.1..86.33....4.....693........1536.
1.3.5.78.....8..3668..7.5.1..8.....75.....81.7.1..86.33....4...8.693........1536.
1.3.5.78.....8..3668..7.5.1..8.....75.....81.7.1..86.33....4.....693........1536.
1.3.567.9.57..9..669....51.......3..36.9...5..71...96.7..3.5......7.41......28.7.
1.3.567.9.57..9.3669....51.......3..36.9...5..71...96.7..3.5......7.41......2..7.
1.3.567.9.57..9.3669....51.......3..36.9...5..71...96.7..3.5......7.41......28...
1.3.567.9.57..9.3669....51.......3..36.9...5..71...96.7..3.5......7.41......2....
1..45...9..........9..72.15..17.59.4..9.14.52..4.9.17..........832.47...9.5....27
1..45...9..........9..72.15..17.59.4..9.14.52..429.17.....2....832.47...9.5....27
1.345..8945.....3..98.......84....5131...594.5.9..48.3.31.48......27139.....6....
1.345..8945.....3..98.......84....5131...594.5.9..48.3.31.48......2.1.9.....6....
1.345..8945.....3..98.......84....5131...594.5.9...8.3.31..8......27139.....6....
1.345..8945.....3..98.......84....5131...594.5.9..48.3..1.48......27139.....6....
.2....78....1.9236.......5.2.19.48...9871.4..74..28....12.97.488.4..1...97.......
12....78....1.9236.......5.2.19.48...9871.4..74..28....12.97.488.4..1...97.......
12....78......92.6.......5.2.19.48...9871.4..74..28....12.97.488.4..1...97.......
1...567.9...1.9.366..73.51...156...7...9.3..1.........384....95..2....7371.3.5...
1...567.9...1.9.366..73.51...156...7...9.3..1.........384....95..2....737..3.5...
.2....78....1.9236.......542.8.6..1..619.8...79..12.68.768..........76..9.26.18..
12...6......78........32...26..4.9.13.49..6...19......6....349..4....3.693...4.12
12.4.6......78........32...26..4.9.13.49..6...19......6....349..4....3.693...4.12
123..6......78........32...26..4.9.13.49..6...19......6....349..4....3.693...4.12
1234.6......78........32...26..4.9.13.49..6...19......6....349..4....3.693...4.12
1..4.6.8.4..18....6...72.4123.7.......962....8.1....27.1.8.7.62...2..1.4..2...87.
12...6....571.92.66.9...1..2.57.1.6.79..62.1....59.........5..4572....91....1.35.
1.3...7...571....368.37.1.5...9..8.....2.8..68367.5...3.1..75.857...16...68......
12..5.78..5.1..2.3.........27...513.8.1..35.2.35..1.78......8.....9683.778.5.2...
.....678.4.7...2.668...2.452.856.47.5......62..62..5.8....18...8..92.......6.4827
12..56....567.9..27.912....21......7.....182....2.53.1.7.......56..1..9.9..5..6..
12..56....567.9..27.912....21......7.....182....2.53.1.7.......56..1..9.9.....6..
12..56....5678....7.81.25.....8.564....6...35.65...2....2.......87.6...1.1...8.7.
12..56....5678....7.81.25.....8.564....6...35.65...2....2.......87.6...1.1.....7.
12..567...5678....7.81.25.....8.564....6...35..5...2....2.......87.6...1.1...8.7.
12..567...5678....7.81.25.....8.564....6...35..5...2....2.......87.6...1.1.....7.
12..56....57..9..66.9......2.6.1597..917.256.57....12.712...........1.4.....2.3..
12..56....57..9..66.9......2.6.1597..917.256.57....1..712...........1.4.....2.3..
12..56....57..9..66.9.......71...........1.4.....2.3...927.516.5.6.1297.71....52.
12..56....57..9..66.9......271...........1.4.....2.3...927.516.5.6.1297.71....52.
12..56....57..9..66.9......271...........1.4.....2.3...927.516.5.6.1297.71....5..
....5..8....1....6689......29...4.7..74......8.6....42.4279..6876.84.92.9.826....
....5..8....1....6689......29...4.7..74......8.6....42.4279..6876.84.92.9.82.....
12..56..9.57......6.9.7...52..6.5.9..65.91...9..7..65........4.......83.....12...
12..56..9.57......6.9.7...52..6.5.9..65.91...9.172..5........4.......83.....12...
12..56..9.57......6.9.7...52..6.519..65.91...9..7..65........4.......83.....12...
12..56..9.57......6.9.7...52..6.519..65.91...9..72..5........4.......83.....12...
........9...189.36......54..7..13...3..7.8.1.81.96.37..68.37......89...393.6.1..7
12..5.78..57.8.2.66.8.2..15....136.8.16.72........41..5........7.1..586...2...5.1
1.3...78....18.2.36.8....412.5.31..8.368.5..281.62...5..251....5..........1.68...
......78....18.2.36.8....412.5.31..8.368.5..2...62...5..251....5..........1.68...
1...567.9...1.9.366..73.51...156...7............9.3..1384....95..2.....3.163.5...
1...567.9...1.9.366..73.51...156...7............9.3..1.84....95..2.....3.16..5...
1...567.9...1.9.366..73.51...156...7............9.3..1384....95..2.....3..63.5...
1...567.9...1.9.366..73.51...156...7............9.3..1.84....95..2.....3..6..5...
12.456......1.9..6....37.1.28...514..15.42.68..........415..62.5.26..8..86.....5.
12.456......1.9..6....37...28...514..15.42.68..........415..62.5.26..8..86.....5.
.....6..94.....23..8.....5..9..41..7.4876...1...9.8....7481....8.6.97...91.6.4...
...4.6..94.....23..8.....5..9..41..7.4876...1...9.8....7481....8.6.97...91.6.4...


The final expanded list has: 205698 minimals with 9635 solution grids, 32571 BRT expansions, and 10167 max-te3 expansions.

Added: It looks like its really 48846 BRT expansions. I'm gussing that some are subsets of others (?)

Added2: Confirmed that. BRT expansions are a mess, aren't they ?
14708 or the 48846, are not subsets of any other(s), and are sufficient for producing the 205698 minimals.
Also noticed: Colin's list contains several entries that are not BRT expansions of minimal puzzles.

(edit: spelling )

[denis_berthier]

Hi Blue
Thanks for your calculations.
Indeed, I hadn't checked coloin's list of min-expands for T&E-depth.

[Added: It looks like its really 48846 BRT expansions. I'm gussing that some are subsets of others (?)

A BRT-expand can be a strict sub-puzzle of another BRT-expand. I've analysed this situation in great detail in [HCCS3]. One can even have long strictly descending chains of BRT-expands. See examples there, for any T&E-depth.
On the set of self-expands and the subset of min-expands, one can define an obvious inclusion relation, turning them into posets PE and ME.
Minimal elements wrt to this order (not minimal puzzles in our usual sense) are isolated puzzles in the associated dual Alexandrov topology, BRTinc in my notation.
Any such puzzle is characterised by a very useful property: it is the BRT-expand of all its minimals.
It is not true that you can recover all the min-expands from only such isolated puzzles - unless you use (1+BRT)-expansions.

color=blue]Added2[/color]: Conformed that. BRT expansions are a mess, aren't they ?

Not if you are careful about them.
BRT-equivalence classes of puzzles (or equivalently self-BRT-expand puzzles, self-expands in short) are fundamental for doing any analysis beyond minimals.
Inclusion of puzzles is useful only modulo BRT-equivalence.
Monotonicity of ratings/classifications wrt puzzle inclusion is true only modulo BRT-equivalence.

When I deal with a new collection, I start from its minimals and I discard anything else.
Usually, I check the T&E-depth.
Then, I put them all in gsf-solution-minlex form and I discard isomorphs (a single step in gsf's solver).
Then, I apply (1+BRT)-expansion until I reach the T&E(n)-expands. Min-expands and T&E-expands are only a by-product of the whole process.

Note that I have no step to saturate the minimals wrt the min-expands or wrt the T&E(n)-expands. I have a very good reason for this:
- in T&E(3), the only large collection has been mith's for a long time. Mith said it was saturated wrt min-expands (he said it in his own words but that's what it meant) and I checked a small part of it.
. in T&E(2) or worse in T&E(1), computing the BRT-expands of the minimals is not a problem, but then computing their minimals is extremely time-consuming. As a result, in these situations, it is impossible to compute the isolated puzzles, let alone to re-generate all the minimals from them.


In case you start from minimals and you want to saturate the minimals wrt min-expands, the procedure would be:
- compute the BRT-expands
- find all their minimals
- compute again the BRT-expands. Note that they can only be (not necessarily strict) sub-puzzles of the first BRT-expands ==> no need to iterate minimisation.


In case you start from T&E(n)-expands and you want to saturate the minimals wrt min-expands, the procedure would be:
- find all their minimals
- compute the BRT-expands. Note that their minimals can only be (not necessarily strict) sub-puzzles of the original T&E(n)-expands ==> no need to iterate minimisation.
(But you need to iterate it if you want saturation of minimals wrt T&E(n)-expansion.)

.

[blue]

Added2: Conf(i)rmed that. BRT expansions are a mess, aren't they ?

Not if you are careful about them.
BRT-equivalence classes of puzzles (or equivalently self-BRT-expand puzzles, self-expands in short) are fundamental for doing any analysis beyond minimals.
Inclusion of puzzles is useful only modulo BRT-equivalence.
(...)

It's just a shame that if you're keeping them around as a stand in for a large (and increasing) set of minimals, then as new minimals are added, old (BRT) expansions can become redundant ... superceded by new ("accidental") superset expansions.

Monotonicity of ratings/classifications wrt puzzle inclusion is true only modulo BRT-equivalence.

It's off topic, I know, but I don't quite understand that.
Valid sub-puzzles are always "at least as difficult" as the originals, right (?) ... expansions aside.

[denis_berthier]

It's just a shame that if you're keeping them around as a stand in for a large (and increasing) set of minimals, then as new minimals are added, old (BRT) expansions can become redundant ... superceded by new ("accidental") superset expansions.

Not sure what you mean. I'll take "them" to refer to isolated puzzles.
In order to compute the isolated puzzles of a collection, one must make sure its minimals and its min-expands are cross-saturated (finally, I prefer to say it that way).
If this condition isn't satisfied, nothing worth can be obtained.
Long ago, I had a talk with mith about isolated puzzles (I didn't have a name for them at that time; mith called them min-expands). My position is unchanged: if one keeps only isolated puzzles instead of all the min-expands in my sense (the sense that has prevailed), reconstructing the minimals from them is very time consuming. It involves (1+BRT)-expansions, minimisations, ...

Monotonicity of ratings/classifications wrt puzzle inclusion is true only modulo BRT-equivalence.

It's off topic, I know, but I don't quite understand that.
Valid sub-puzzles are always "at least as difficult" as the originals, right (?) ... expansions aside.

Right. I should have written "natural" instead of "true".
Indeed, more fundamentally, any definition of an order between puzzles has to be modulo BRT-equivalence.
Let me copy section 4.6.2 of HCCS3, where P is the set of consistent puzzles and M the set of minimals:

Notice that it would also make sense on P (or M) to define a partial order relation between two puzzles P and Q by: P ≤ Q if and only if BRT-expand(P) is a sub-puzzle of BRT-expand(Q). But beware that this definition, which doesn’t make any difference between puzzles with identical BRT-expands wouldn’t mean that P is a sub-puzzle of Q; indeed, Q could well be a strict sub-puzzle of P, as long as they have the same BRT-expand. We shall therefore avoid to write this order relation on P (or M). This is a clear example of the necessity of taking BRT-equivalence seriously.

My purpose here is only to avoid confusions.
.

[coloin]

Sorry about the 51 TE2 inclusions...
Certainly the variable expansion [plus manual additions !] to a maximum is a very gray area.
I only missed a few expanded puzzles and a few twins....

taking a snippet from one grid

Code: Select all

1......8....18...66..2.7...2.4...8.3.95...1.27...2.....36.7.......61.3...713.8...#expands
1......8....18...66..2.7...2.4...8.3.95...1.27...2.....36.72......61.3...713.8...#expands
1......8....18...66..2.7...2.4...8.3.95...1.27...2.....36.72......61.3...713.862.#expands
1......8....18...66..2.7..12.4...8.3.95...1.27...2.....36.7..1....61.3...713.8...#expands
1......8....18...66..2.7..12.4...8.3.95...1.27...2.....36.7..18...61.3.7.713.8...#expands
1......8....18...66..2.7..12.4...8.3.95...1.27...2.....36.72.1....61.3...713.8...#expands
1......8....18...66..2.7..12.4...8.3.95...1.27...2.....36.72.1....61.3...713.862.#expands
1......8....18...66..2.7..12.4...8.3.95...1.27...2.....36.72.18...61.3.7.713.8...#expands
1......8....18...66..2.7..12.4...8.3.95...1.27...2.....36.72.18...61.3.7.713.862.#expands [max]

1......8....18...66..2.7..12.4...8.3.95...1.27...2.....36.72.18...61.3.7.713.862.# max-expand

I maximally expanded [ manually added clues and retested] the rest.
On minimization I got 204419 minimals, then I rexpanded [ with gsf -E] and got 48429 expands

There were indeed 1226 new minimals from the unaccounted twins with 335 expands.

In total I will re publish these 48429 plus 335 = 48764 expands

This should then be comparable to mith's file ....TE3nonmithsreloaded

[denis_berthier]

then I rexpanded [ with gsf -E] and got 48429 expands

gsf -E doesn't give min-expands;
In order to get min-expands, you need to use gsf -qFN -E
.

[coloin]

then I rexpanded [ with gsf -E] and got 48429 expands

gsf -E doesn't give min-expands;
In order to get min-expands, you need to use gsf -qFN -E
.

Yes I do use that now ...its significantly quicker !!! but I thought the output was the same though.

[denis_berthier]

then I rexpanded [ with gsf -E] and got 48429 expands

gsf -E doesn't give min-expands;
In order to get min-expands, you need to use gsf -qFN -E
.

Yes I do use that now ...its significantly quicker !!! but I thought the output was the same though.

It's different.
The output is the expansion wrt to the rules defined by the -q option. The option with only Singles is -qFN. By default (if you don't put any -q something), more rules are used.
.

[denis_berthier]

First thing to mention: Colin's list contains 51 puzzles that are only T&E(2)
[...]
None of thier minimals, are T&E(3).
[...]
The remaining puzzles expand to 204336 minimals, on 9539 grids -- grid count down by 9.
The puzzles have 10069 max-te3-expands -- agreeing with Denis' result, if you subtract 51 for the non-TE3 puzzles.
The max-expands, produce 204336+136= 204472 minimals.
The 136 that are missing, can be gotten from these BRT-expands - 50 of them:
[...]
The max-expands have 98 "twins" that aren't represented.
They produce another 1226 minimals, with 335 BRT-expands, over 97 solution grids -- 96 new, and one duplicate.
[...]
The final expanded list has: 205698 minimals with 9635 solution grids, 32571 BRT expansions, and 10167 max-te3 expansions.

I decided to redo all my calculations.
I started from the coloin's original list of min-expands, I deleted the 51 that are T&E(2) and I added Blue's new 335, Ie. I started from the same 32571 min-expands as Blue.
I took 20 hours or so to re-generate all the minimals with gsf's solver. The good news is, I get the same number as Blue (205,698), with the same number of solutions: 9,635.
I'll now:
- take those 205,698 minimals as my new starting point,
- restart all the expansion process in order to answer the questions about the number of min-expands and T&E(3)-expands generated by them.
At this point, I don't expect results significantly different from those I reported before. But at least, they will be cleaner.

The question of why there's such a difference between the two T&E(3) collections will therefore remain unchanged.


[Edit]: I find 48,846 min-expands and 10,167 T&E(3)-expands (1.055 per solution).
.

[blue]

It's just a shame that if you're keeping them around as a stand in for a large (and increasing) set of minimals, then as new minimals are added, old (BRT) expansions can become redundant ... superceded by new ("accidental") superset expansions.

Not sure what you mean. I'll take "them" to refer to isolated puzzles.
In order to compute the isolated puzzles of a collection, one must make sure its minimals and its min-expands are cross-saturated (finally, I prefer to say it that way).
If this condition isn't satisfied, nothing worth can be obtained.
Long ago, I had a talk with mith about isolated puzzles (I didn't have a name for them at that time; mith called them min-expands). My position is unchanged: if one keeps only isolated puzzles instead of all the min-expands in my sense (the sense that has prevailed), reconstructing the minimals from them is very time consuming. It involves (1+BRT)-expansions, minimisations, ...

I didn't mean "isolated puzzles".
I meant BRT expansions of minimal puzzles ... some or all ... used to represent (all of) the minimals in what you've called a "cross-saturated" list (or pair of lists).
I should have kept quiet about them "being a mess".

[denis_berthier]

It's just a shame that if you're keeping them around as a stand in for a large (and increasing) set of minimals, then as new minimals are added, old (BRT) expansions can become redundant ... superceded by new ("accidental") superset expansions.

Not sure what you mean. I'll take "them" to refer to isolated puzzles.
In order to compute the isolated puzzles of a collection, one must make sure its minimals and its min-expands are cross-saturated (finally, I prefer to say it that way).
If this condition isn't satisfied, nothing worth can be obtained.
Long ago, I had a talk with mith about isolated puzzles (I didn't have a name for them at that time; mith called them min-expands). My position is unchanged: if one keeps only isolated puzzles instead of all the min-expands in my sense (the sense that has prevailed), reconstructing the minimals from them is very time consuming. It involves (1+BRT)-expansions, minimisations, ...

I didn't mean "isolated puzzles".
I meant BRT expansions of minimal puzzles ... some or all ... used to represent (all of) the minimals in what you've called a "cross-saturated" list (or pair of lists).
I should have kept quiet about them "being a mess".

OK. But then min-expands (= BRT-expansions of minimal puzzles) never become redundant.
Once a minimal has been found, it always remains a minimal. Once a min-expand has been found, it always remains a min-expand.
As new min-expands are found, there may appear more inclusion relations with the new ones. But that's only revealing more of the intrinsic poset structure of the set of min-expands. The existence of such a structure on the set of min-expands may come as a surprise, but we can only take it as a fact.

Note 1: when you try to cross-saturate minimals and min-expands by the procedure I mentioned earlier: (minimals1 -> their) min-expands1 -> their minimals2 -> their min-expands2, the new minimals that may appear in minimals2 can only be sub-puzzles of the first ones (never strict super puzzles that would "supersede" the old ones in your sense).
As a result, the new min-expands2 can only be sub-puzzles of the min-expands1 and re-minimising them wouldn't produce new minimals.
This means that the cross-saturation process is finished - and that the isolated puzzles of the collection may be calculated after doing it.

Back to coloin's collection, revised by you: in the results of my previous post (same as yours), in the collection we get, the minimals and min-expands are cross-saturated. One can therefore compute the isolated puzzles: there are 20,114, i.e. 41 % of the min-expands.

I haven't tried to cross-saturate minimals and T&E(3)-expands - because the mean number of T&E(3)-expands per grid is already very close to mith's collection and I don't think that would help explain the difference between the two.
.