## Prime number of solutions for multiple solutions sudokus

Everything about Sudoku that doesn't fit in one of the other sections
In fact your 29-permutable set is not unique, neither your 11-permutable; and so your disproof is invalid.
What?!?! Go and re-read your own conjecture. "Given a p(n) solutions-sudoku ..." -- I gave you a 29-permutable set. It didn't have a 23-permutable subset. So your conjecture is false.

You are right, it should be :" There exists at least a p(n) solutions-sudoku... to n max"

define a set of clues to be n-FPPS if it is p(n)-permutable and it contains at least one p(k)-permutable subset for each k=1,...,n-1

we seek the maximum n such that there exists at least one n-FPPS set of clues

Yes.We seek *all the successive* n-FPPS from n=4 (1,2,3 are already known) to maximum n-FPPS found ,for each n (n=4,5,6,7,8,..,max n). In S-1 or S-2 or S-3,... (Just use S-1, if this suffices).

Though i was thinking too in a n-FPPS that would contain at least one k-FPPS for each k=1,...,n-1. But this would be probably too difficult or might be impossible.

Posts: 23
Joined: 22 May 2006

Here is a p(10)=29 solutions sudoku, such that if we add 1 or more clues, we transform it in a p(k) solutions sudoku for each 1<=k<10.

Code: Select all
` *-----------* |...|...|...| |9.2|..4|.8.| |.5.|92.|.6.| |---+---+---| |...|.7.|...| |...|..8|..7| |..7|...|.53| |---+---+---| |7..|...|6.8| |.4.|5.9|...| |8..|3..|...| *-----------* `

Remove it and add 6 to r6c1->19 solutions

Posts: 23
Joined: 22 May 2006

Define S+1 to be the set of distinct prime solutions when adding just 1 clue to the precedent 29 solutions sudoku.

S+1={2,3,5,7,11,13,17,19,23} is a full prime set, because it contains all primes p(k) for each 1<=k<10.

In increasing order the operations to get S+1 are {4@r1c8,1@r1c7,5@r1c7,5@r1c9,9@r1c7,4@r1c1,5@r1c5,6@r5c4,2@r4c1}

The p(10)=29 Fully Prime Permutable Set (10-FPPS) is:

Code: Select all
`  *-----------* |4.3|.5.|912| |...|.1.|..5| |1..|...|..4| |---+---+---| |2.4|1..|.9.| |..9|6..|42.| |6..|4.2|1..| |---+---+---| |...|2.1|...| |3.6|...|2.1| |...|...|549| *-----------* `

With 31 cells and 7 variables : (1,2,3,4,5,6,9)

Could anybody find a (minimal number of cells) 10-FPPS ???

For bigger primes S+2 or S+3 might be used.

A curious, or may be not so curious : 29 is a prime of the form 4*k+1 (1 mod 4). These primes are always sum of 2 squares in just one way (They never are a solution to the Diophantine equation a^2+b^2=c^2+d^2 (3 of (a,b,c,d) distint at least). I could solve this equation after a long work, with a free variable. And used it to find n for all n^2+1 primes, substracting sets, but not seiving.
Unfortunately, i am a self-taught programmer, and amateur, so my program was sligthly slowlier than a traditional sieving.

So 29=2^2+5^2.
And in cell r8c7
1@r8c7--->5^2 solutions
2@r8c7--->2^2 solutions

and 2 and 5 are prime.

Next 4*k+1 primes sum of the squares of 2 primes are
53=2^2+7^2
173=2^2+13^2
293=2^2+17^2
Guess if this will happen again.

Posts: 23
Joined: 22 May 2006

This is the p(9)=23 permutable set
Code: Select all
`*-----------*  |4.3|.5.|912|  |...|.1.|..5|  |1..|...|..4|  |---+---+---|  |..4|1..|.9.|  |..9|6..|42.|  |6..|4.2|1..|  |---+---+---|  |...|2.1|...|  |3.6|...|2.1|  |...|...|549|  *-----------* 30 cells`

The p(8)=19 permutable set

Code: Select all
`*-----------*  |4..|.5.|912|  |...|.1.|..5|  |1..|...|..4|  |---+---+---|  |..4|1..|.9.|  |..9|...|42.|  |...|4.2|1..|  |---+---+---|  |...|2.1|...|  |...|...|2.1|  |...|...|549|  *-----------* 25 cells`

The p(7)=17 permutable set
Code: Select all
`*-----------*  |4..|.5.|912|  |...|.1.|..5|  |1..|...|..4|  |---+---+---|  |..4|1..|.9.|  |..9|...|42.|  |...|4.2|1..|  |---+---+---|  |...|2.1|...|  |...|...|2.1|  |...|...|5.9|  *-----------* 24 cells`

The p(6)=13 set
Code: Select all
`*-----------*  |...|.5.|912|  |...|.1.|..5|  |...|...|...|  |---+---+---|  |..4|1..|.9.|  |..9|...|42.|  |...|4.2|1..|  |---+---+---|  |...|2.1|...|  |...|...|2.1|  |...|...|5.9|  *-----------* 21 cells`

The p(5)=11 set
Code: Select all
`*-----------*  |...|.5.|912|  |...|.1.|..5|  |...|...|...|  |---+---+---|  |..4|1..|.9.|  |..9|...|42.|  |...|4.2|1..|  |---+---+---|  |...|2.1|...|  |...|...|2.1|  |...|...|...|  *-----------* 19 cells`

The p(4)=7 set
Code: Select all
`*-----------*  |...|...|912|  |...|...|...|  |...|...|...|  |---+---+---|  |..4|1..|.9.|  |..9|...|42.|  |...|4.2|1..|  |---+---+---|  |...|2.1|...|  |...|...|2.1|  |...|...|...|  *-----------* 16 cells`

The p(3)=5 set
Code: Select all
`*-----------*  |...|...|912|  |...|...|...|  |...|...|...|  |---+---+---|  |..4|1..|.9.|  |..9|...|4..|  |...|4.2|1..|  |---+---+---|  |...|2.1|...|  |...|...|2.1|  |...|...|...|  *-----------* 15 cells`

The p(2)=3 set
Code: Select all
`*-----------*  |...|...|91.|  |...|...|...|  |...|...|...|  |---+---+---|  |..4|1..|.9.|  |..9|...|4..|  |...|4.2|1..|  |---+---+---|  |...|2.1|...|  |...|...|...|  |...|...|...|  *-----------* 12 cells`

And the p(1)=2 set

Code: Select all
`*-----------*  |...|...|...|  |...|...|...|  |...|...|...|  |---+---+---|  |...|...|...|  |...|...|...|  |...|1.2|...|  |---+---+---|  |...|2.1|...|  |...|...|...|  |...|...|...|  *-----------* 4 cells`

Posts: 23
Joined: 22 May 2006

And here is a p(12)=37 solutions sudoku with 27 clues.

Call x the number of clues.

We define a n- Fully Prime Solutions Sudoku+1 (n-FPSS+1), to be a Sudoku with p(n) solutions (p(n)=n-th prime), such that adding *just 1* clue in an appropriate cell,*and there is never more than x+1 clues*,we obtain all p(k) solutions fot 1<=k<n.

In a n-FPSS+q there should never be more than x+q clues.

This is a 12-FPSS+1

Code: Select all
` *-----------* |...|.9.|...| |8.2|..4|.9.| |.5.|82.|.64| |---+---+---| |...|.7.|...| |...|..9|..7| |..7|...|.53| |---+---+---| |7..|9.2|6.8| |.4.|5.8|...| |9..|3..|5..| *-----------*  `

When we add a clue, we must remove it, before adding a new one.

2@r8c1->31 solutions
4@r1c3->29
3@r3c1->23
9@r4c2->19
6@r1c1->17
6@r5c2->13
2@r1c8->11
6@r1c2->7
1@r5c1->5
2@r1c7->3
3@r1c1->2

This new definition comes from the fact that it seems that there is not
here a unique permutable set for all the p(k) solutions.
In fact I could not get a 1 solution in any cell.

May be we should include p(0)=1, in the n-FPSS+1 (n-FPPS+q)

Red Ed, I hope this new statement is more suitable, and you can begin to work on it.
I am very interested to know how far we can go up to n max with q as low as possible.

We have here a kind of topology (under Sudoku-constraints). of primes!!

Posts: 23
Joined: 22 May 2006

And here is a beautiful 25-FPSS+1 found by Red Ed:

Code: Select all
` *-----------* |...|395|2..| |.5.|...|.3.| |3..|4.2|157| |---+---+---| |516|...|.4.| |4..|...|...| |...|..4|7..| |---+---+---| |...|8.7|...| |9..|.53|87.| |7..|.4.|..1| *-----------*  `

Thanks, Red Ed.

Was just wondering if it contained also a k-FPSS+1 for each 1<=k<25.

Probably not, but too long to be checked by hand (and the help of Simple Sudoku program).

Difficult general problem, indeed, if we wish to find a constructive method
to build n-FPSS+q for n with 3 digits and q<=2.

Posts: 23
Joined: 22 May 2006

This 10-FPSS+1 which has been already posted:
Code: Select all
` *-----------* |...|...|...| |9.2|..4|.8.| |.5.|92.|.6.| |---+---+---| |...|.7.|...| |...|..8|..7| |..7|...|.53| |---+---+---| |7..|...|6.8| |.4.|5.9|...| |8..|3..|...| *-----------*  `

contains a k-FPSS+1 for each 1<=k<10, because the permutable set:
Code: Select all
` *-----------* |4.3|.5.|912| |...|.1.|..5| |1..|...|..4| |---+---+---| |2.4|1..|.9.| |..9|6..|42.| |6..|4.2|1..| |---+---+---| |...|2.1|...| |3.6|...|2.1| |...|...|549| *-----------* `

is a common structure (with less elements each time) for all the odd prime solutions (the p(1)=2 solution is the only exception).
Also, this permutable set is contained in the p(0)=1 solution of the 10-FPSS+1.

All the odd primes solutions can be "constructed" this way:

1+2*t+r

3->(t,r)=(1,0)
5->(2,0)
7->(3,0)
11->(5,0)
13->(6,0)
17->(8,0)
19->(9,0)
23->(9,4)

Posts: 23
Joined: 22 May 2006

YAC (Yet Another Conjecture):

Every prime >3, is a sum of at most 3 primes, and all the previous *odd* primes appear at least once.

This is: For all n>=3, p(n)=q+r+p(k) (q,r are primes, and one of them can be 0), and for all p(k) 2<=k<n.

(p(n)=n-th prime).

Example:

p(27)=103=

2+101
3+3+97
3+11+89
7+13+83
5+19+79
13+17+73
3+29+71
13+23+67
11+31+61
7+37+59
7+43+53
13+43+47
31+31+41

13 steps required to form p(k) for each 2<=k<n

Of course, it would be impossible to find a 27-FPSS+1 in which 12 cells only, with 3 variables in each of them, + 1 cell with 2 variables,would suffice.

If you do not believe this conjecture, then, find a counterexample.

Posts: 23
Joined: 22 May 2006

Adding something to the previous post:

1,2,2,3,3,4,4,4,5,5,6,6,7,7,7,8,8,9,10,10,11,11,11,12,13

Minimal number of steps needed to represent a prime >=5, as a sum of at most 3 primes, such that all the previous odd primes are represented

Would be nice if somebody extended this sequence.

Posts: 23
Joined: 22 May 2006

Every prime >3, is a sum of at most 3 primes, and all the previous *odd* primes appear at least once.
We won't be able to disprove that. Suppose we have a counterexample to your conjecture: some p>3 and smaller odd q (both primes) such that p cannot be written as q plus two other primes. Then p-q is an even number not equal to the sum of two primes, contradicting Goldbach's conjecture.

I think you should move your prime number ponderings to a maths forum, as this has nothing to do with Sudoku!
Red Ed

Posts: 633
Joined: 06 June 2005

Red Ed:

Goldbach conjecture only tell us that every even number is a sum of 2 primes.We are certain this conjecture is true, but there seems to be no way to prove it.

I said long ago that Poincaré conjecture would be easier to prove than Goldbach.

Nevertheless, GC does not claim that all the distinct primes are represented, which I do:

Every prime contains additively all the previos primes, with not more than 3 summands!!

We are talking here of multiple prime-solutions Sudokus.
And of n-FPSS+q (Sudokus with all the prime solutions up to p(n), when adding not more than q clues).And we certainly have at least 2 variables per cell,in the permutable sets, to find all the previous primes of p(n) for a constructed for this purpose p(n)-solutions Sudoku.I claim that the constructive structure of these n-FPPS+q would be a fertile matter, though hard.

You should be aware of that, since you found a 25-FPSS+1, but computer
methods, tend to be brute-force ones.Rather, we should find a constructive algorithm, but that´s much harder...
We are therefore absolutelly ON-TOPIC.

Posts: 23
Joined: 22 May 2006

I'm going to make just a couple more comments on this thread and then walk away from it, since I am starting to believe that this is all some sort of weird wind-up.

you wrote:Nevertheless, GC does not claim that all the distinct primes are represented
Yes it does, implicitly. Let SC = sopadeajo's claim; GC = Goldbach conjecture. I explained previously that not(SC) => not(GC). So GC => SC. You're not claiming anything new.

shortly afterwards, you wrote:We are talking here of multiple prime-solutions Sudokus. [...] I claim that the constructive structure of these n-FPPS+q would be a fertile matter, though hard.
Ha ha ha, for a moment there I thought you were suggesting that Sudoku research might contribute something of value to the study of prime numbers. But of course you don't mean that, since then you wouldn't be so much "on topic" as "on" something much funkier ...

Good luck. I've a feeling you'll need it.
Red Ed

Posts: 633
Joined: 06 June 2005

Without really bothering to read most of the thread so far, it is nevertheless clearly true, as Red Ed points out, that GC implies YAC (=SC). Just to explain again, suppose we want to represent (say) 389 as a sum of three primes, in which one of them is (say) 5. We simply have to represent 389-5=384 as the sum of two primes, e.g., 384=191+193, using GC. And then 389=5+(191+193).
frazer

Posts: 46
Joined: 06 June 2005

Red Ed wrote:
I explained previously that

You are certainly very nice, and explain yourself pretty well.
Congratulations .

Posts: 23
Joined: 22 May 2006

In the category "Research That Serves No Purpose",
here is a list of puzzles with a prime number of solutions for all primes upto 449:

Code: Select all
`#    2 solutions.39...12....9.7...8..4.1..6.42...79...........91...54.5..1.9..3...8.5....14...87...2.8...6...9.6.85.....7.2.1...78...7.4...2..6...94........2.6....3.5.42..5.6...8#    3 solutions..3.....6...98..2.9426..7..45...6............1.9.5.47.....25.4.6...785............2.839..5.674........6..2...1.9....8........7...3.24913......1.4....6.72.71......#    5 solutions......27..7....5.41.2..8....91.43..28...6.....47.21..97.4..9....1....9.6......34..24.65..95..2...861..9......93.2....8..7...5.6......1........32.6..547..93....5..#    7 solutions1.54.96.3..6...2..2.......5.28...94...........1..5..7.....8......42.18.....596.......81..51...............4.3.4.6.17...9..2..65..8.......2.34......1.5.97.......5.#    11 solutions81..674..42........7.84..........53....2.1....48..........78.4........91..249..8728.6..5.......53.6....7.18.1........37.9.8.61........9.98.6....6.12.......4..9.38#    13 solutions..8....9..3.4..5......58..2.59.3...4..7...8..3...2.76.2..54......1..2.4..8....6...6..........6135......5.7.9.89...156.........613...24.5.7.6......1895..........3.#    17 solutions..............2734713.....9.....53.2..8.1.9..3.64.....2.....5976573..............9...82......6.952.31......6..4....81.........19....6..6......75.452.6......85...9#    19 solutions..145.98....86.7..4...79.6.71.......................37.4.59...8..8.37....63.841..42.....768....21...3..54.8..4.......78.....54.......6..7.32..1...98....736.....25#    23 solutions43...75.6.27...9.1.....5..3.4........8..6..4........9.3..9.....1.6...73.2.47...156327.5.9.....3167........8.4..9.3...............1.4..6.1........2341.....9.3.2168#    29 solutions25..1.........71...9.....3...28.13...35...86...13.47...2.....4...94.........2..85.........52.3......34...9.2.1.4.3..57.6...3.13..7.9.8.1.2...86......8.23.........#    31 solutions...7...3..3.....9719...6845..32.....6.......1.....72..5794...1834.....6..6...5...87....16.............6.4..5...14..3.31.7.2.59.8..95...2..4.6.............98....72#    37 solutions13.8.......2...........638......9.5748.....1269.7......419...........5.......1.98.....142.......3....926317..3.52..4...........5..37.8..687952....1.......456.....#    41 solutions..96...53....397.6......8..3...2...74.......25...6...9..4......9.751....12...43..5.........47.65...2...1...78.2...3..43.8.6.25..6...7.96...4...1...52.67.........2#    43 solutions..74.23....5.....78..7.9..4.14..........9..........96.6..3.4..27.....8....39.86.....2.......836..2.6...57.3...2..56...57...21...14..7...1.94...5.3..719.......6...#    47 solutions.......543.8...9..9526....8.8.4.6...............3.8.1.8....1647..5...3.242..........57..2.....8..9663..92.....3......54.....71......9.....71..6818..6.....7..23...#    53 solutions..172.....3.1..9768.......5.2..1......69.24......7..5.5.......1682..1.3.....576....8.............7...9..716.3.61.4.9..17...65..9.6.58.3.719..4...3.............5..#    59 solutions.76....83.2...9..4.1.....9..5.39.1..2.......9..9.12.7..6.....5.1..2...4.48....26.79...2...16.759.8.............1..742.........342..5.............3.524.18...8...35#    61 solutions2...8...94...6..18.....1..7...6.25......3......78.4...7..3.....51..7...28...9...4......3...8...51......795..2...9..3.3.......6.5..2...8..245......78...9...1......#    67 solutions...52....64.........9....7.5..31.86.8.1.5.7.4.34.78..1.8....4.........83....86...16.9...533.7..69..9.....71..8...........4...........3..73.....5..93..8.224...8.69#    71 solutions.....2856...6.1397........4......1.8.14...96.3.9......4........1385.4...5921......8....937167........3......3..46...2.26...18.9...28..3......6........329859....7.#    73 solutions...5...........2...2.847.3.8.32...57..6...3..47...36.1.5.176.4...7...........8....93.5.....7.8.1..34..9..2.6....1.8.............7.8....1.6..8..59..5.6.4.....4.36.#    79 solutions786..3................798.2.5.4.67...9..1..2...73.2.9.4.583................9..513..2.5........82.1.8.......2.7.9....6.264.158.5....8.3.4.......1.9.26........1.9..#    83 solutions.......1.......7......27546.526.8...61.....75...5.496.72913......3.......6.............96442..........53...7.9..57...5.4...3.8...41..9.7...83..........19146......#    89 solutions.31.9.75.56.....317.......6....3.....87...54.32.....89............973....4..2..1.62..4...............92.783.3.5...68.1..3..........4.925...8.....839..............#    97 solutions37...4...2........84...257.53..81......2.3......74..19.638...57........3...4...62...........17.68...23.4.51.5..1.2..8.9.....2............8...2...79...15.61.2.3.87#   101 solutions...5.3....186.793.....8.....59...87...........34...15.....6.....917.436....3.5........126.........8...29........2.9.59.375....75....8...14.3.59.2.9..47.........8.#   103 solutions6...4...9...3.6....93.2.67..6.....1.2.8...9.3.1.....2..89.6.73....9.7...7...5...69..8.....7.8....3......6..9..136..72.3..4......621..94.....9..88.5....6.3..6.....#   107 solutions4..3............6...9865...3.74..6....4.7.9....2..87.4...6924...4............3..1.............4....8.47.91.2.7.....8...29.37..413...295..51328...47...62..........#   109 solutions..217.6.....9..8...46.23.71..83..2.....2..1.8....81.67..5..27.4.7....3..............4..52.4..5...9.9..7.....3.2....5..8..5..3..6....4.8.....4..1.9...5..7.43..9...#   113 solutions..18......582..36426..3....47.........2..8.......2469..2...9.7..4...21...3............2...3..74.6.22...6.5..5836.....1.......3.....3856..1.2...49.2.75..8...9.....#   127 solutions...6..4..1..9..6......82..5.5..3...4..1.4.8.7.9..2...1....17..82..4..7.....2..1..1.....4....61.8.....79...356.8......4.9.....17.5........17...86..46.9...9.....7..#   131 solutions7.2.439................5.7232......7.47...23.6......9826.4................371.8.4.47.196..1.......9...6.......91..3.....8.79...23..........62....58.....2.6.4.1...#   137 solutions...2.8....18...25...67.38......9....457...938....7......54.23...34...69....9.1........6..7...7.9.34....8......2.......8....4695.....81..........85.23.7...21..4..5#   139 solutions...23...9.....92.7......6319...78.5.6..9......3.6......24........61.....395.....8..87139....7...6...1.586.4...........7..3..6.2..8.9..41.......3.8.....2.73.....95#   149 solutions.7.56............8.9.8.4..5...419....5....2.18......34.8615..........15....6.....3.7...2.5.........9...4...61.62.74.9....6....8.51.46.76...1...8.........7.2...1.4#   151 solutions.3.65..84..78......5......77.......6.6....97.4.......8.1......9..31......4.26..31..1.....9.38....7.5....7..4...2.8.5.1......3....4.1.8.6....4..3.24....6...3.....1#   157 solutions..........65947.1......1.6..461.2.8..5.....2..2.7.514..7.6......1.23869..........2...3...7..62.15.....856................1....81.....9474.....56..35.49....9...3..#   163 solutions...2.7...8...3...5..21546..1.......9...421...6.4...7.1.........59.....12...3.9...5.24.......9......64...5...3..25.......1....6..7..9513.....6.35.....41.2....3294.#   167 solutions.237691..61.......8...4..7.3....65..7.6......5..4.....9..2....5..8....27......31..8...4.7....5...2....7.1.94.45.3......8.....3.37.9.......8.2.39...1...8..1...9.6.#   173 solutions...291....6.....2...1...3..7..8.2..64.......32..3.4..1..4...6...9.....1....926....46.3.9........5.87..6....2...............2..25.3.8.74...9.67.....5...4.46.2....1#   179 solutions4...1...9..........9..5..7.8.24.19.5..52.38..9.35.86.2.1..8..2..........5...2...4......4.............2.4..31.....3176..6.125.....86...31..32...5..95.......31.46.8#   181 solutions..26....84.7...36.8.9....7.9.5......1.6.2............4....73456.............9673.....2..969.....2....2.8...5..52..98....6..1.3..34..62...8.4...91.....8......9..61#   191 solutions7.......3..6.1.......3.4.8...36.72...9.....5...54.28...6.1.5.......4.9..8.......58..5.4..6....2....64.....25...1.5....89...64....6.9...43.....17....5....2..4.8..9#   193 solutions87...........2.........1596....4..654.57.69.336..5....5864.........3...........84.5..1..3..........9.3...6.8..84.57..4..7.6..5..51.29..2.9...3.4..........3..7..8.#   197 solutions3......1.5..2....86..3.....8..6.....496........1..436.7..84........3......2.95643........5...1...7.6.....389..5.41..............2.97...2....5.3....61489....8..4..#   199 solutions91.......862........5...9..4..3..16.5..79..4.68.1.5......5..........7294........31.62..............3.......17..5...12....82.6.....3.945.....6.2....37569...71.4...#   211 solutions7.....3.....3278...........3.76.........83..1.64.....26.....12..932..7.6.......98....4.....9.82.5...2.95..1..7...9....3....245.82...79..574.......178235..........#   223 solutions....371.9..9625.7.............7.89.4456.9..8..9....3..............2714...3..........9138.55.7..6...............3.....824...5.3.3....6...687.51.22............28...#   227 solutions........1.9.53..4..3.24.8.98.........79..438...36.9.........47..85.1...6..............81.4.4..21.....29....53.4.57.126...34.7....2..8.......62.........8.....82...#   229 solutions..13..86.7......9.34..8.....6..92...............63..4.....7..86.3......4.54..87..9....3....45.1.6...2.79.4.........7....16..2.21.3............676..5.....3.7.24...#   233 solutions.9.....1.8.6.4.2.3.4..2..6.....7.....748.312.....9.....6..1..5.1.3.8.9.2.8.....7........325......8.1.67..........5..3.543.971.7..4..........72.1.1......998.......#   239 solutions34.29.6.......3...........4.7..8..2......5..9.59.....829.6.....5.69.7..2.17.....6.5..4...8.21..5.4...718.....7.5...1.2...6.....1.2...8...582.....92..6.5..6..5...2#   241 solutions.4.....2..........3..642..9...4.1....1389524.8.4...3.572.....31..5...9......5......249....4...27.1...9.1.....54....7..98.6.321........5....714.8....58..........9.#   251 solutions..........7165...9..69375.2....793...........8.92.57..6.479....7......64.........32.4.9...............61.4..........4958..3......75.9.81728..6.3.9.32.............#   257 solutions..83645...9.2.1.3...1...8...2.....5...........8.....1...3...4...1.5.2.8...29461...6.2.1....547...8.....3...46..............9..8.......57..15.2........19..32..48..#   263 solutions..25....9.9..6.57884............8...2.......3...9............67378.5..9.4....91.......8.9....25.7.8...43..2...4.....7.6..7.51..7....36..4...1....3872.............#   269 solutions31......6.62....48...2...5.....5.4.....968.....3.7.....4...9...73....69.9......128..3.4..1..........1..5..9...6.2.7....3.9.1..9.71463.25.9...6.8..1...4..4.......7#   271 solutions41..8..299.......6...7.3.....1...9..2...7...5..6...1.....4.5...3.......754..3..18....8.4.5.2756..9.............7.384.........9.35.2....749......8.347.............#   277 solutions.....937.......659....5.4.....61...5..35.71..6...8....935.7....78........1.9....7..3...2..7.4...8.1..93416...7623948.4.......2.............9....267...194.........#   281 solutions4...97...............5..34656...8..4..2.7.6..7..6...52374..2...............81...7....8....6.......8894...325...369....4..1..3....742...257...9649.......3....2....#   283 solutions.6...1....4..2.7..98...54..3...46......8...4.8...39...51...73...3..1.9...7...3...1.5..47....49....6...............2........41..7..516.........575..2..8.498...6..2#   293 solutions.29......1..5.....5...21....5.97.8....24.89....8.15.2....28...6.....7..8......15.....65..9.2...71....9132.................645.3514.87...14.........6...219........#   307 solutions2971.....4.3......85.6...927.9.81.3....4........9..247.....98....48.6.....6..4.........48....569......4815...8...46.1..9..7....1...23.8...7258.....398.........95.#   311 solutions.4..75...8..3........6.2....91..87.47.......64.61..58....9.1........6..9...75..1....8...71.....6..24.31.....7.54...1..8.........9..78.6.4.7.19....7.6.........37..#   317 solutions....2...5....934....4.65.3.......69.931...258.26.......8.14.5....925....4...3.......6.3.....72.48...36.7.49.7...4...8.........2..7.8..4............4.2.....83.16..#   331 solutions....8...5.54....2.6.2...84.......43..4.6....2......98.4.5...16..76....5.....5...959...4...1.8..6.45.432.......1...8........51236..........49.2...5..1.....1..2....#   337 solutions.327.............3..1...5...........2..6374.5.46.....958.1......673.5.8.......7..4......39....19..6..7...2.....48..6..8.632.9..3..95.....1...6..8..36....64......3#   347 solutions9.7......612..85......7..9........632..5.9..113........6..8......91..687......3.92......8..697.....75826........2.....1.5......4..............61..3.96......37489.#   349 solutions6312......5....3.19.......4....72........52....961..4...2..3..........9.7.4..6..5.3..82...........8......4..27.......4.69.1.2.5...73..66..................543976..#   353 solutions....9....42.....87....5....15.9.3.46..2...8..37.4.8.95....4....61.....72....3......71..5.2.......94513...76...25...................34...38...64576.......4.5..69..#   359 solutions........33..............87......72455.1.63..97..8....6..9...53......9....4.7361.8.2........1.......9..27..451.8.495..6..1....94.......7...8..................97613#   367 solutions......42.....9.6.7....7..95...82.....987.156.....49...73..5....9.4.6.....51.......6...3..22........4...8.31..1....23.5.......7.7....65.1...3.58.8.........5...8..9#   373 solutions.....2.95.8.415......63.....67.......91...65.84...123.....86...1...24...6.............8.6.........2...927......14..7..23.8.14................6..3...5.89.867..2..4#   383 solutions..........2.9....7...4583..59..1..2....6.2.....678....64...15.........7........9............89.15....3.4.1..9.4.5.2.13..4.2..55.1.9.4.3..9.2.8....25.47...........#   389 solutions93..875..1.52.9.7.......3......6...9....4..1......12.........5..7.954.8.2.......436..................2..8.9.251..94...386.......482...........26.1....3.5..356...1#   397 solutions.3....96....1.48................5.96.4..........36.....518.7....8..5.41...42.1.3...........629.4.....817...5........6721.9..8...3.5...........2.........42.5.487..#   401 solutions.....67...8.......6.571....84.3...2....69..1...1..8......5.........79...95...34..........5.........7265.3..........484.....17..73.8.62..6.8......41..6.....8.37...#   409 solutions.64..2....8.3.....529..64.......954..........9....8.3...351...74..6......7.8.............72..5...3.75.91.........5.9..2.6....8.752.3....68.......4.8693...........#   419 solutions..8....3....95....1.7...95..126.9..85..2............76....268............853..4.7..............6....1285....693.....2.2.1..........47...8..7341.16........37..8.2.#   421 solutions27.6...8.....59..33.5.1.92.....6.37191..7.....4...........91....................66.........49.....3..3518..4.......9...1......2......58.......1..3.7.4....6....73.#   431 solutions79.....53.8..247....4........7..832.869.4.....3.5.....54.9......................8.2..4..13.6..7328..4...95...1....6...72.........8.1...3..5..........2.75.........#   433 solutions79.............4.9...18.....................59.8....364..35...7..571....86..941...95...8.1................54....8479....59...247....1..95.....63...2.....6273.....#   439 solutions........57.........3...9.7...1..3....4..9...1.2...8.4...38.....2173.6.584...7.61.4.1.9268.96...4.....81..4.5........2832....14................2....75.......4.8..3#   443 solutions....1.35......28............2..3..9..1..946.84..8.753.3.12..7..2...46.8............52...3998..4.....731......64.........97...625.....8.83.6.......1.93............#   449 solutions.19..42....7....5...5..3.181.....................9..4.5...47.........59..435.682..7.....3....12.8........915.49....7.........282.5.....6.5.................1..2596`

Ruud