The New Sudoku Players' Forum

by **JPF** » Tue Jun 16, 2020 3:35 pm

In the good old days on this forum, we used to have challenges.

Here is a new one:
What is the minimum number of solutions of a pattern having atmost one clue in each box B2B3B4B6B7B8B9?

There is possibly an exhaustive search, but here my first proposal after a quick review:

Code: Select all: +---+---+---+ |..1|...|...| |2.3|.4.|...| |567|...|8..| +---+---+---+ |.9.|628|...| |...|1.5|.7.| |...|.94|...| +---+---+---+ |..8|...|...| |...|...|..4| |...|5..|...| +---+---+---+ 550 sol.

I am sure one can get something better.
Any taker?

JPF

by **tarek** » Tue Jun 16, 2020 3:51 pm

I may have had a tool from 14 years ago that may be worth reviving as it may suit this challenge. I'll try to see if I can find it

T

by **Mathimagics** » Thu Jun 18, 2020 8:46 am

It's tough going ...

This is another case of 550 on the same B1/B5 combo:

Code: Select all: +-------+-------+-------+ | 9 4 1 | . . . | . . . | | 2 8 3 | . 4 . | . . . | | 5 6 7 | . . . | 1 . . | +-------+-------+-------+ | . 9 . | 6 2 8 | . . . | | . . . | 1 3 5 | . 2 . | | . . . | 7 9 4 | . . . | +-------+-------+-------+ | . . 5 | . . . | . . . | | . . . | 2 . . | . . . | | . . . | . . . | . . 9 | +-------+-------+-------+

by **tarek** » Thu Jun 18, 2020 1:41 pm

My best result is:

Code: Select all: . . 1 | . . . | . . . 2 . 3 | . 8 . | . . . 5 6 7 | . . . | 4 . . -------+-------+------ . 9 . | 6 2 8 | . . . . . . | 1 . 5 | . 7 . . . . | . 9 4 | . . . -------+-------+------ . . 4 | . . . | . . . . . . | . . . | . . 8 . . . | 5 . . | . . . 626 solutions

by **Mathimagics** » Thu Jun 18, 2020 3:10 pm

I rather suspect 550 is not going to be beaten ... I have found a grid that is ED to JPF's original, but it has the same solution count.

Here is a "normalised" version of JPF's original grid. Box 1 is fixed, and box 5 is in reduced form:

Code: Select all: +-------+-------+-------+ | 1 2 3 | . . . | . . . | | 4 5 6 | 2 . . | . . . | | 7 8 9 | . . . | 3 . . | +-------+-------+-------+ | . . . | 1 2 9 | . . . | | . 1 . | 4 5 8 | . . . | | . . . | 6 7 3 | . 4 . | +-------+-------+-------+ | . . 7 | . . . | . . . | | . . . | . . 4 | . . . | | . . . | . . . | . . 1 | +-------+-------+-------+

There are (at least) 36 different forms that give the same result, ie 550:

36 x 550: Show

Code: Select all: 123......4562.....789...3.....129....1.458......673.4...7...........4...........1 123......4562.....789...3.....129....1.458......673.4...7..............1.....4... 123......4562.....789...3.....129....1.458......673.4......4.....7..............1 123......4562.....789...3.....129....1.458......673.4.........1..7...........4... 123......4562.....789...3.....129....1.458......673.4......4...........1..7...... 123......4562.....789...3.....129....1.458......673.4.........1.....4.....7...... 123......4562.....789...3.....129....1.458......673..4..7...........4..........1. 123......4562.....789...3.....129....1.458......673..4..7.............1......4... 123......4562.....789...3.....129....1.458......673..4.....4.....7.............1. 123......4562.....789...3.....129....1.458......673..4.......1...7...........4... 123......4562.....789...3.....129....1.458......673..4.....4..........1...7...... 123......4562.....789...3.....129....1.458......673..4.......1......4.....7...... 123......4562.....789....3....129....1.458......6734....7...........4...........1 123......4562.....789....3....129....1.458......6734....7..............1.....4... 123......4562.....789....3....129....1.458......6734.......4.....7..............1 123......4562.....789....3....129....1.458......6734..........1..7...........4... 123......4562.....789....3....129....1.458......6734.......4...........1..7...... 123......4562.....789....3....129....1.458......6734..........1.....4.....7...... 123......4562.....789....3....129....1.458......673..4..7...........4.........1.. 123......4562.....789....3....129....1.458......673..4..7............1.......4... 123......4562.....789....3....129....1.458......673..4.....4.....7............1.. 123......4562.....789....3....129....1.458......673..4......1....7...........4... 123......4562.....789....3....129....1.458......673..4.....4.........1....7...... 123......4562.....789....3....129....1.458......673..4......1.......4.....7...... 123......4562.....789.....3...129....1.458......6734....7...........4..........1. 123......4562.....789.....3...129....1.458......6734....7.............1......4... 123......4562.....789.....3...129....1.458......6734.......4.....7.............1. 123......4562.....789.....3...129....1.458......6734.........1...7...........4... 123......4562.....789.....3...129....1.458......6734.......4..........1...7...... 123......4562.....789.....3...129....1.458......6734.........1......4.....7...... 123......4562.....789.....3...129....1.458......673.4...7...........4.........1.. 123......4562.....789.....3...129....1.458......673.4...7............1.......4... 123......4562.....789.....3...129....1.458......673.4......4.....7............1.. 123......4562.....789.....3...129....1.458......673.4.......1....7...........4... 123......4562.....789.....3...129....1.458......673.4......4.........1....7...... 123......4562.....789.....3...129....1.458......673.4.......1.......4.....7......

Box 5 can really only have 10,080 ED settings, and so I've been testing only those.

The only other B5 setting for which I've had any joy is this one, and it seems to be closely related:

Code: Select all: +-------+-------+-------+ | 1 2 3 | . . . | . . . | | 4 5 6 | 3 . . | . . . | | 7 8 9 | . . . | 2 . . | +-------+-------+-------+ | . . . | 1 3 8 | . . . | | . . 1 | 4 6 9 | . . . | | . . . | 5 7 2 | . 4 . | +-------+-------+-------+ | . 7 . | . . . | . . . | | . . . | . . 4 | . . . | | . . . | . . . | . . 1 | +-------+-------+-------+

Once again, all solution counts are 550, and there are 36 different puzzles. Note that the clue values in B6-B9 are the same as above (1,4,4,7).

36 x 550 (case 2): Show

Code: Select all: 123......4563.....789...2.....138.....1469......572.4..7............4...........1 123......4563.....789...2.....138.....1469......572.4..7...............1.....4... 123......4563.....789...2.....138.....1469......572.4......4....7...............1 123......4563.....789...2.....138.....1469......572.4.........1.7............4... 123......4563.....789...2.....138.....1469......572.4......4...........1.7....... 123......4563.....789...2.....138.....1469......572.4.........1.....4....7....... 123......4563.....789...2.....138.....1469......572..4.7............4..........1. 123......4563.....789...2.....138.....1469......572..4.7..............1......4... 123......4563.....789...2.....138.....1469......572..4.....4....7..............1. 123......4563.....789...2.....138.....1469......572..4.......1..7............4... 123......4563.....789...2.....138.....1469......572..4.....4..........1..7....... 123......4563.....789...2.....138.....1469......572..4.......1......4....7....... 123......4563.....789....2....138.....1469......5724...7............4...........1 123......4563.....789....2....138.....1469......5724...7...............1.....4... 123......4563.....789....2....138.....1469......5724.......4....7...............1 123......4563.....789....2....138.....1469......5724..........1.7............4... 123......4563.....789....2....138.....1469......5724.......4...........1.7....... 123......4563.....789....2....138.....1469......5724..........1.....4....7....... 123......4563.....789....2....138.....1469......572..4.7............4.........1.. 123......4563.....789....2....138.....1469......572..4.7.............1.......4... 123......4563.....789....2....138.....1469......572..4.....4....7.............1.. 123......4563.....789....2....138.....1469......572..4......1...7............4... 123......4563.....789....2....138.....1469......572..4.....4.........1...7....... 123......4563.....789....2....138.....1469......572..4......1.......4....7....... 123......4563.....789.....2...138.....1469......5724...7............4..........1. 123......4563.....789.....2...138.....1469......5724...7..............1......4... 123......4563.....789.....2...138.....1469......5724.......4....7..............1. 123......4563.....789.....2...138.....1469......5724.........1..7............4... 123......4563.....789.....2...138.....1469......5724.......4..........1..7....... 123......4563.....789.....2...138.....1469......5724.........1......4....7....... 123......4563.....789.....2...138.....1469......572.4..7............4.........1.. 123......4563.....789.....2...138.....1469......572.4..7.............1.......4... 123......4563.....789.....2...138.....1469......572.4......4....7.............1.. 123......4563.....789.....2...138.....1469......572.4.......1...7............4... 123......4563.....789.....2...138.....1469......572.4......4.........1...7....... 123......4563.....789.....2...138.....1469......572.4.......1.......4....7.......

by **Mathimagics** » Thu Jun 18, 2020 4:55 pm

Same deal, different B5, 36 flavours, all with NS = 550:

Code: Select all: +-------+-------+-------+ | 1 2 3 | . . . | . . . | | 4 5 6 | 2 . . | . . . | | 7 8 9 | . . . | 3 . . | +-------+-------+-------+ | 5 . . | 1 2 7 | . . . | | . . . | 4 5 3 | . . . | | . . . | 6 9 8 | . 1 . | +-------+-------+-------+ | . . 8 | . . . | . . . | | . . . | . . 1 | . . . | | . . . | . . . | . . 2 | +-------+-------+-------+

by **coloin** » Thu Jun 18, 2020 6:32 pm

my top 4

Code: Select all: 264......835.9....971...4...8.723......459.7....186.....9..................2....8 550 sol. 264......835.9....917...4...8.123......459.2....786.....9..............8...2..... 687 sol. 624......835.9....917...4...8.123......459.2....786.....9..............8...2..... 784 sol. 264......385.9....917...4...3.128......459.2....736.....9..................2....3 785 sol.

the 550 is the same as JPF's - which is isomorphic to the 36 others

btw there are 124 ED B1B5 combinations only

Hidden Text: Show

Code: Select all: 123......456......789.........256......137......984.............................. 123......456......789.........162......845......739.............................. 123......456......789.........162......859......734.............................. 123......456......789.........162......958......734.............................. 123......456......789.........162......754......938.............................. 123......456......789.........162......745......938.............................. 123......456......789.........162......758......934.............................. 123......456......789.........162......754......839.............................. 123......456......789.........162......759......834.............................. 123......456......789.........162......758......943.............................. 123......456......789.........164......758......932.............................. 123......456......789.........164......738......952.............................. 123......456......789.........164......752......839.............................. 123......456......789.........164......752......938.............................. 123......456......789.........164......738......259.............................. 123......456......789.........164......735......829.............................. 123......456......789.........164......739......852.............................. 123......456......789.........164......925......738.............................. 123......456......789.........164......258......739.............................. 123......456......789.........164......725......938.............................. 123......456......789.........164......725......839.............................. 123......456......789.........164......738......529.............................. 123......456......789.........168......943......725.............................. 123......456......789.........168......934......752.............................. 123......456......789.........168......749......253.............................. 123......456......789.........168......759......234.............................. 123......456......789.........168......734......925.............................. 123......456......789.........168......954......732.............................. 123......456......789.........168......945......723.............................. 123......456......789.........168......943......752.............................. 123......456......789.........168......735......942.............................. 123......456......789.........168......753......942.............................. 123......456......789.........168......745......932.............................. 123......456......789.........168......754......329.............................. 123......456......789.........168......735......249.............................. 123......456......789.........168......753......924.............................. 123......456......789.........168......743......952.............................. 123......456......789.........168......243......759.............................. 123......456......789.........168......742......935.............................. 123......456......789.........168......724......953.............................. 123......456......789.........168......945......732.............................. 123......456......789.........169......734......258.............................. 123......456......789.........169......734......528.............................. 123......456......789.........169......534......728.............................. 123......456......789.........169......754......823.............................. 123......456......789.........169......748......532.............................. 123......456......789.........169......758......234.............................. 123......456......789.........169......854......723.............................. 123......456......789.........169......745......832.............................. 123......456......789.........169......842......753.............................. 123......456......789.........169......842......735.............................. 123......456......789.........169......824......735.............................. 123......456......789.........169......254......738.............................. 123......456......789.........169......758......324.............................. 123......456......789.........169......754......328.............................. 123......456......789.........169......724......358.............................. 123......456......789.........169......524......738.............................. 123......456......789.........169......758......243.............................. 123......456......789.........169......724......538.............................. 123......456......789.........169......342......758.............................. 123......456......789.........169......352......748.............................. 123......456......789.........169......345......728.............................. 123......456......789.........674......318......259.............................. 123......456......789.........674......358......219.............................. 123......456......789.........674......913......258.............................. 123......456......789.........674......213......859.............................. 123......456......789.........674......291......853.............................. 123......456......789.........674......951......328.............................. 123......456......789.........674......915......328.............................. 123......456......789.........674......921......853.............................. 123......456......789.........674......253......819.............................. 123......456......789.........674......123......958.............................. 123......456......789.........674......315......928.............................. 123......456......789.........679......413......825.............................. 123......456......789.........679......813......425.............................. 123......456......789.........679......214......853.............................. 123......456......789.........679......324......158.............................. 123......456......789.........679......453......128.............................. 123......456......789.........679......815......243.............................. 123......456......789.........679......218......354.............................. 123......456......789.........357......426......981.............................. 123......456......789.........357......296......148.............................. 123......456......789.........352......147......968.............................. 123......456......789.........352......841......967.............................. 123......456......789.........352......186......497.............................. 123......456......789.........352......964......187.............................. 123......456......789.........352......164......897.............................. 123......456......789.........351......987......426.............................. 123......456......789.........351......297......486.............................. 123......456......789.........456......298......371.............................. 123......456......789.........456......219......873.............................. 123......456......789.........451......387......269.............................. 123......456......789.........457......289......361.............................. 123......456......789.........457......689......213.............................. 123......456......789.........457......219......863.............................. 123......456......789.........457......289......163.............................. 123......456......789.........453......816......297.............................. 123......456......789.........453......279......168.............................. 123......456......789.........918......645......327.............................. 123......456......789.........918......362......745.............................. 123......456......789.........917......345......628.............................. 123......456......789.........917......328......645.............................. 123......456......789.........231......546......798.............................. 123......456......789.........854......967......321.............................. 123......456......789.........854......367......921.............................. 123......456......789.........853......421......967.............................. 123......456......789.........835......471......926.............................. 123......456......789.........349......176......258.............................. 123......456......789.........349......168......257.............................. 123......456......789.........349......167......258.............................. 123......456......789.........349......167......285.............................. 123......456......789.........349......852......167.............................. 123......456......789.........348......159......267.............................. 123......456......789.........348......672......915.............................. 123......456......789.........185......296......374.............................. 123......456......789.........182......653......479.............................. 123......456......789.........194......672......853.............................. 123......456......789.........543......129......786.............................. 123......456......789.........123......789......564.............................. 123......456......789.........465......789......123.............................. 123......456......789.........417......568......239.............................. 123......456......789.........456......789......123.............................. 123......456......789.........834......672......159.............................. 123......456......789.........417......528......639..............................

by **JPF** » Thu Jun 18, 2020 10:39 pm

coloin wrote:btw there are 124 ED B1B5 combinations only

I agree and there are 237083 ed-B1B5B9.

If 550 is the lowest number of solutions, I am really lucky because I found it almost by chance starting from:

Code: Select all: +---+---+---+ |123|...|...| |456|...|7..| |897|...|...| +---+---+---+ |...|625|...| |...|914|.2.| |...|783|...| +---+---+---+ |...|...|851| |...|...|342| |...|...|697| +---+---+---+ 23 sol.

JPF

by **Mathimagics** » Fri Jun 19, 2020 8:49 am

Ok, trying again ... :roll:

These 2 have distinct Minlex forms, so presumably one corresponds to JPF's and one must be new:

Code: Select all: +-------+-------+-------+ +-------+-------+-------+ | 1 2 3 | . 8 . | . . . | | 1 2 3 | . . . | . . . | | 4 5 6 | . . . | . . 9 | | 4 5 6 | . . . | . . 3 | | 7 8 9 | . . . | . . . | | 7 8 9 | . 1 . | . . . | +-------+-------+-------+ +-------+-------+-------+ | 8 . . | 1 2 4 | . . . | | . 7 . | 1 2 5 | . . . | | . . . | 6 3 9 | . 2 . | | . . . | 6 9 4 | . 2 . | | . . . | 8 7 5 | . . . | | . . . | 7 8 3 | . . . | +-------+-------+-------+ +-------+-------+-------+ | . . . | . . . | 7 . . | | . . . | . . . | 1 . . | | . . . | . . 2 | . . . | | . . . | . . 2 | . . . | | . 6 . | . . . | . . . | | . . 4 | . . . | . . . | +-------+-------+-------+ +-------+-------+-------+

by **JPF** » Fri Jun 19, 2020 7:23 pm

some improvements:

Code: Select all: 1 2 3 | 8 . . | . . . 4 5 6 | . . . | . . 2 7 8 9 | . . . | . . . -------+-------+------- . . . | 1 2 9 | . . . . . 2 | 3 5 6 | . . . . . . | 4 8 7 | . 1 . -------+-------+------- . . . | . . . | . . . . . . | . . 5 | . . . 3 . . | . . . | . 8 . 251 sol.

JPF

by **JPF** » Fri Jun 19, 2020 11:34 pm

more:

Code: Select all: 1 2 3 | . . . | . . . 4 5 6 | . 3 . | . . . 7 8 9 | . . . | . 1 . -------+-------+------- . . . | 1 2 7 | . . . . . 4 | 9 5 6 | . . 3 . . . | 8 4 3 | . . . -------+-------+------- . . . | . . . | . . . . . . | . . . | . . . . 1 . | 2 . . | . 8 . 154 sol.

and [edit]

Code: Select all: 1 2 3 | . . . | . . . 4 5 6 | . . . | . . . 7 8 9 | . 1 . | 2 . . -------+-------+------- 8 . . | 1 5 7 | . . . . . . | 4 8 3 | . . . . . . | 6 9 2 | . . 3 -------+-------+------- . . . | . . . | 8 . . . . 4 | 3 . . | . . . . . . | . . . | . . . 117 sol.

JPF

by **coloin** » Sun Jun 21, 2020 5:04 pm

That is very surprizing, though for clarity, it has been shown that 5 clues, one in each box B3B6B7B8B9 cant produce valid puzzles...
and an example of 8 cues solving the puzzle [2 clues in B9]

Code: Select all: +---+---+---+ |672|...|..1| |318|...|...| |..5|6..|...| +---+---+---+ |5..|.63|.9.| |...|5.8|...| |...|271|...| +---+---+---+ |...|...|.4.| |...|...|...| |..1|..2|..8| +---+---+---+

I searched 4 patterns , there seems to be a favourite !

Hidden Text: Show

Code: Select all: 362.....1918........53.....4...39.7....2.1......478.......................1..2..8 154 362.....1918........53.....4...39.7....2.1......478.......................1..2..8 154 672.....1318........56.....4...63.9....4.8......271.......................1..2..8 154 672.....1918........56.....4...69.3....4.8......271.......................1..2..8 154 958.....1312........73.....4...69.3....4.8......251.......................1..2..8 154 958.....1312........73.....4...69.3....4.8......251.......................1..2..8 154 978.....1312........53.....4...69.3....2.1......438.......................1..2..8 154 978.....1312........53.....4...69.3....2.1......438.......................1..2..8 154 672.....1318........56.....5...63.9....5.8......271.......................1..2..8 184 958.....1312........73.....4...79.3....4.8......251.......................1..2..8 186 732.....1918........67.....4...79.3....5.8......231.......................1..2..8 222 372.....1918........53.....4...39.7....2.1......648.......................1..2..8 222 938.....1712........67.....4...79.3....5.8......231.......................1..2..8 240 968.....1312........53.....7...39.6....2.1......478.......................1..2..8 240 362.....1918........53.....7...49.3....7.8......231.......................1..2..8 252 362.....1918........53.....4...79.3....2.1......468.......................1..2..8 252 372.....1918........53.....4...69.3....2.1......478.......................1..2..8 252 372.....1918........53.....4...69.3....4.8......231.......................1..2..8 252 958.....1612........76.....4...69.3....4.8......231.......................1..2..8 252 958.....1612........76.....4...69.3....4.8......231.......................1..2..8 252 578.....1312........63.....4...35.9....2.1......478.......................1..2..8 252 958.....1612........76.....4...69.3....2.1......458.......................1..2..8 252 678.....1912........49.....4...96.3....2.1......478.......................1..2..8 258 978.....1312........43.....4...39.5....2.1......478.......................1..2..8 258 978.....1312........43.....4...39.6....2.1......478.......................1..2..8 258 938.....1712........57.....4...79.3....4.8......231.......................1..2..8 278 372.....1918........53.....4...69.7....4.8......271.......................1..2..8 278 362.....1918........53.....4...79.6....2.1......438.......................1..2..8 278 958.....1612........76.....4...69.5....2.1......438.......................1..2..8 278 462.....1718........54.....9...67.3....9.8......231.......................1..2..8 294 352.....1918........63.....4...59.7....2.1......438.......................1..2..8 294 478.....1312........53.....9...74.3....6.8......291.......................1..2..8 294 468.....1312........53.....9...64.3....9.8......231.......................1..2..8 294 468.....1312........53.....9...64.3....2.1......738.......................1..2..8 294 578.....1312........63.....4...75.3....2.1......498.......................1..2..8 294 372.....1918........53.....4...39.7....4.8......271.......................1..2..8 300

but not as good as the latest one !!

by **tarek** » Sun Jun 21, 2020 6:28 pm

I couldn't get below 500 :oops:

Colin's method of starting from a valid puzzle with 8 clues is a good idea for a different approach

tarek

by **coloin** » Sun Jun 21, 2020 6:53 pm

Well, I too couldnt get below the 550 when I searched with +9 clues and three in box 9 [ remove 2]
Searching with 9 clues , 2 boxes having 2 clues [ remove 1 from each] 4 ways was more profitable
There are relatively not many puzzles with +8 clues [and no empty box ]...

by **JPF** » Tue Jun 23, 2020 4:22 pm

With the technique I am using my limit seems to be 117 sol.
I got many 146 and 154 sol. and nothing in between.

Here's an example with 146 sol.

Code: Select all: 123...4..456......789.6....3..271......954......683.5........6.2....7............ 146 sol.

I'll try a last run and stop if no improvement.

coloin wrote:That is very surprizing, though for clarity, it has been shown that 5 clues, one in each box B3B6B7B8B9 cant produce valid puzzles...

Which posts are you refering to?

JPF

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