Smallest integer $k$ so that no Sudoku grid has exactly $k$ solutions

Question

Inspired by this question, consider hints on a Sudoku board. A regular puzzle has a unique solution. It is clear that there are puzzles with 2 or 3 solutions, and therefore, I guess, puzzles with say 4, and 6 solutions.

Now, what is the smallest integer $k$ such that there is no set of Sudoku clues resulting in exactly $k$ solutions?

Answer 1

Consider the process of randomly generating single solution puzzles and randomly dropping one or more clues as shown in the pseudo-code below:

GenerateValidNotNecessarilyMinimalSudoku() {
  clues := {}
  consequences := {}
  for literal in RandomPermutation(Range(729)) {
    if literal not in consequences {
      switch (CountSolutions(clues U {literal}, limit=2)) {
       case 0:
         continue
       case 1:
         return clues U {literal}
       case 2:
         clues := clues U {literal}
         consequences := UnitPropagate(clues)
      }
    }
  }
}

samples = []
for _ in Range(NumSamples) {
  clues := GenerateValidNotNecessarilyMinimalSudoku()
  relaxed := clues - PickRandomElements(clues, NumToDrop)
  samples += [CountSolutions(relaxed, limit=10000)]
}

Interestingly, though maybe not surprisingly, this process gives rise to an approximately lognormal distribution of solution counts. Here's a chart showing the empirical distribution (black) of solution counts from a sample of 15 million random puzzles with 1 dropped clue vs. the fitted lognormal model (green).

And here's the same chart for another 15 million random puzzles, this time dropping 4 clues:

This behavior suggests an absence of the kinds of pattern in solution count probability that might conspire to produce a "low" k. Maybe it also hints at a way to at least guess at the order of magnitude of k. (e.g., choose a useful way to partition the space of possible puzzles, estimate the number of distinct non-equivalent puzzles in each partition, find a way to sample from these puzzles uniformly, fit lognormal models for each partition, given lognormal model and size of sample space, ask in what range does it become likely that some count is zero).

Answer 2

This is just a partial answer, which can perhaps be turned into a community wiki, where examples of partial fillings that allow $n$ complete valid fillings.

This admits $k=2$ solutions: \begin{array}{ccccccccc} . & . & . & . & 5 & 1 & 8 & 7 & 6 \\ . & 7 & . & 6 & . & 9 & . & . & 3 \\ 6 & . & 8 & . & . & 2 & . & 9 & . \\ . & 4 & . & 1 & . & . & . & 8 & . \\ . & . & 1 & . & 3 & . & . & . & 4 \\ 8 & . & . & . & . & 7 & 1 & . & . \\ 2 & . & . & 5 & 6 & . & . & 3 & . \\ . & . & . & . & 1 & . & 9 & . & 8 \\ 3 & . & . & . & . & 4 & . & . & . \\ \end{array}

This allows $k=3$ fillings: \begin{array}{ccccccccc} . & . & . & . & . & 1 & 8 & 7 & 6 \\ . & 7 & . & 6 & . & 9 & . & . & 3 \\ 6 & . & 8 & . & . & 2 & . & 9 & . \\ . & 4 & . & 1 & . & . & . & 8 & . \\ . & . & . & . & 3 & . & . & . & 4 \\ 8 & . & . & 9 & . & 7 & 1 & . & . \\ 2 & . & . & 5 & 6 & . & . & 3 & . \\ . & 6 & . & . & 1 & . & 9 & . & 8 \\ 3 & 8 & 7 & . & . & . & . & . & . \\ \end{array}

If my calculations are correct, this allows $k=5$ complete fillings: \begin{array}{ccccccccc} . & . & . & . & 5 & 1 & 8 & 7 & 6 \\ . & 7 & . & 6 & . & 9 & . & . & 3 \\ 6 & . & 8 & . & . & 2 & . & 9 & . \\ . & 4 & . & 1 & . & . & . & 8 & . \\ . & . & 1 & . & 3 & . & . & . & 4 \\ 8 & . & . & . & . & 7 & 1 & . & . \\ 2 & . & . & 5 & 6 & . & . & 3 & . \\ 5 & . & . & . & 1 & . & . & . & 8 \\ 3 & . & . & . & 9 & . & . & . & . \\ \end{array}

Currently, $k=7$ is still unknown.

This is the mathematica code I use to solve Sudokus. It is just basic recursion, and can most likely be optimized a lot.

(* Returns true iff we can place n at r,c in the filling. *)

IsValidQ[partFil_, n_, {r_, c_}] := Module[{sr, sc},
   (* Check row, col, sub-box for already existing. *)
   Catch[
    If[MemberQ[partFil[[r, All]], n], Throw[False]];
    If[MemberQ[partFil[[All, c]], n], Throw[False]];
    sr = Floor[(r - 1)/3]; sc = Floor[(c - 1)/3];
    If[MemberQ[
      Join @@ partFil[[3 sr + 1 ;; 3 sr + 3, 3 sc + 1 ;; 3 sc + 3]], 
      n], Throw[False]];
    True
    ]
   ];

SolveSudoku[partialFilling_List] := Module[{recSolve, sols = {}},
   recSolve[pFil_List] := Module[{nFil},
     Catch[
      Do[
       If[pFil[[r, c]] == 0,

        (* Try all 9 numbers. *)

        Do[If[IsValidQ[pFil, n, {r, c}],
          recSolve[ReplacePart[pFil, {r, c} -> n]]]
         , {n, 9}];

        (* We tried all 9 numbers here. Now abort. *)

        Throw[False]
        ]
       , {r, 9}, {c, 9}];
      AppendTo[sols, pFil];
      (*Print["NSOLS: ",sols//Length];*)
      True
      ]
     ];

   recSolve[partialFilling];
   sols
   ];

Answer 3

Like it usually happens with these conjectures, if the solution is not k=7 or k=11, it may be really hard to find. Of course, there exists a solution, for there is a maximum (huge) number of solutions that a blank sudoku allows. A property that may help to construct solutions is that for small values, if we have a sudoku that allows a solutions and another that allows b solutions, we can probably find a sudoku that allows a·b solutions. For example:

This allows k=4 fillings: $$ 2,9,6|3,1,8|5,7,4\\5,8,4|9,7,2|6,1,3\\7,1,3|6,4,5|2,8,9\\6,2,?|?,9,7|3,4,1\\9,3,1|4,2,6|8,5,7\\4,7,?|?,3,1|9,2,6\\1,?,7|2,?,3|4,9,8\\8,?,9|7,?,4|1,3,2\\3,4,2|1,8,9|7,6,5 $$

And this allows k=3 fillings: $$ 2,9,6|3,1,8|5,7,4\\5,?,4|9,7,2|6,?,3\\7,?,3|6,4,5|2,?,?\\6,2,5|8,9,7|3,4,1\\9,3,1|4,2,6|8,5,7\\4,7,8|5,3,1|9,2,6\\1,6,7|2,5,3|4,?,?\\8,5,9|7,6,4|1,3,2\\3,4,2|1,8,9|7,6,5 $$

From this, we can easily see that this: $$ 2,9,6|3,1,8|5,7,4\\5,?,4|9,7,2|6,?,3\\7,?,3|6,4,5|2,?,?\\6,2,?|?,9,7|3,4,1\\9,3,1|4,2,6|8,5,7\\4,7,?|?,3,1|9,2,6\\1,?,7|2,?,3|4,?,?\\8,?,9|7,?,4|1,3,2\\3,4,2|1,8,9|7,6,5 $$

allows k = 4·3 = 12 fillings.

Although it is hard to generalize, probably the number you are looking for is prime, because if it was to be a composite number, its factors (smaller than the number) would have to be a number of fillings for some sudoku, and then this non-rigorous multiplication rule would fail.

Answer 4

Here is an example for $k=7$:

\begin{array}{ccccccccc} . & . & . & . & . & . & 7 & . & .\\ 9 & . & 7 & 1 & 2 & 3 & . & 5 & 6\\ 4 & . & 6 & . & 7 & . & . & . & 3\\ 3 & 1 & 2 & 8 & . & 5 & . & 6 & .\\ . & . & . & 3 & . & . & . & 4 & 5\\ 8 & 4 & . & 7 & . & 9 & . & . & .\\ . & . & . & . & 9 & . & 6 & 7 & 8\\ . & . & 8 & . & . & . & 5 & . & .\\ . & . & 4 & . & 8 & . & . & 3 & 1\\ \end{array}

Which resolves to:

\begin{array}{ccccccccc} . & . & . & . & 5 & . & 7 & 8 & 9\\ 9 & 8 & 7 & 1 & 2 & 3 & 4 & 5 & 6\\ 4 & 5 & 6 & 9 & 7 & 8 & 1 & 2 & 3\\ 3 & 1 & 2 & 8 & 4 & 5 & 9 & 6 & 7\\ . & . & 9 & 3 & 1 & 2 & 8 & 4 & 5\\ 8 & 4 & 5 & 7 & 6 & 9 & 3 & 1 & 2\\ . & . & . & . & 9 & . & 6 & 7 & 8\\ . & . & 8 & . & 3 & . & 5 & 9 & 4\\ . & 9 & 4 & . & 8 & . & 2 & 3 & 1\\ \end{array}

Here I extend the prime list of $k$ to the first $142$ primes ($2$ to $821$).

To do so I implemented Knuth's dancing links for Algorithm X, iterated through solves of an empty sudoku, randomly removed between 20 and 53 clues and counted solutions. The first of each prime count was then added to a dictionary. I left it running for about half an hour. Progress had not slowed all that much after the initial burst to around $37$. It discovered another $173$ larger prime solution counts, ranging from $827$ to $17,137$ solutions.

To avoid this post being ridiculously long each is in single line form, and all are in a code block:

  2: 1...5678..8..2345...6..8..331..4.967..7.128.....9....223157469...8.3157..74.89.31
  3: 12...67..78.1234..45.....2.3.2.45..7.673.29...458...1.2.15..6..6...3157.....98.31
  5: ..3.5...9.8......64....9.23.12.....7..73.294.9...6..12231.9.67....2.1594....7.231
  7: ......7..9.7123.564.6.7...33128.5.6....3...4584.7.9.......9.678..8...5....4.8..31
 11: .2...6.8.7.8123456....891233.26..89....3.264...5...3..2.1.....8.7.231.64...8....1
 13: 1.3.5.7...98...4.6.569.81.331.8.5......3.28..8.....3..2315......7..3.5.45.....2..
 17: 1..4.67.99.......6.5..87.2..12....67.693.2.4..4576.3.2.3...4.7...82...9..9.6...3.
 19: ....5.7.98.7..34..4.6......31.....9.9....26.5.4..7931.....6...87..23...45....723.
 23: .2..5..9..7.12..5.456.79.2.31..4...99......456.59....2231...9..7.8.3.5..5.4.9....
 29: 1...5...9..9.....6....8.1233....5....98.126.....89731.2.1.6....9...315.4..4.7.2..
 31: .2.4.689.798.........7.81.........78..9.12.4.64.87....2.15..78..872.1...5.498....
 37: ..3456....9....45..5....123.12.45...9..3....5.......1.2..5...9..8...15..5..9.7.3.
 41: ...45.7.99.812....4..78..2..12.45.9...7312.....59....2........8....31..45..89.2..
 43: ........8897....5.4.6..8...3....5.7....3...45.45.8.3.2.......8797.231.64.64.9.2..
 47: .2..........123.56456.89...3..6.58...7..126....5........1..4.7.79..3........78...
 53: .2.4.6...97.1..4..45...8.23..2..59.......2.45645.7.31....5.4...7.9..1........72..
 59: .......7.97..23456....78.2..........76.3.28...45.693.2...59....6....1....94....3.
 61: ...456.98.87....56...89...3.1.64.....9.3..64...57..3...3.5.......9....645..97...1
 67: .2..5.8......2....45..9..23312.......9..1264..4...9..2..1...9..9..2..5..56..78..1
 71: ..3.....8.8.1234...5...8..3..2...8.696..12.45...8....2..1....8....23.5745749.6.31
 73: .....68.978912.45...68...2.3........86.312945..5...3..2....4........1.945...8..3.
 79: 1.345.......1.34..4..9....3.12.45967.7...284.84..9...22..5...9.9.7.3.............
 83: .23...87..79......45.7.8..3...6......98..2...64598...223..........2.15.4..48.9...
 89: ..345.......1.......6..8......64.8.7...3....564587...223.5.4.6......1..45..9....1
 97: 123.5..8.8.7...4....96...2.....4..6..8...2..........12.315.467.6...31.....4...23.
101: 1...5...896.1.34.7....9...33....58..89..1..........3.22..5..6..6..23157....96823.
103: .2.4.7.68.....345.4....6..331.7..896.........74...8.1..31584...9.....5....4.79.3.
107: 1..4.....79.1..4....6.8.123.1.6......87.....5.4.798....315.......92..5.4.64.7.2..
109: 1..4.679...9..3.......9....31.945.6.768..2.4..456......31..46.9...231..45.4..9...
113: .....67....7..3.5..5.7....3.....5...769...845.4.9...1223...49...9.....7.5746..23.
127: ...459......1.34..4.....1..31.9.5...7..31.....4.678....3.59.87....2..5.4.9..67...
131: .2...7....8.......4...8.12...2.4.......3.2.4..4.89631223.5.4968........45.4.6...1
137: .2.4578.......345.45.....23..2..5........2.4..45..93....1...6899.8.3.5.4.........
139: .234.687.8....3..6..687.1.....64..877.93.2.....5..7..............8..1...56...82..
149: ...45...........5.4...681.3....457...6...2......8...1.....94.8.68723...4....8723.
151: 123..7869..912345.....96.2...27..9....63...............31..4..8..8...5.4.7..6.2..
157: ...4.786...81.3.5.4......2.31..45...8693...4.7.5.96.1....5..69.....3..7........3.
163: .234.6....98.2...6..6...1......4.........2....4....3..2.1.64.78...2.15....49.8.31
167: .2.....8...........5698..2331..4..9....3..74.74..69.12.31...8..6..........4698..1
173: 1.34...796........45..76..3.1..4..67......8...4.769.....1...7.6.96...5.......7.31
179: .2.....8.8691.345745..6.1.3.12..5.6.67......5945..8.1......4.7........9..9.....3.
181: ..34..7......2.4..4.8.7.1.3........7...3.284.84.96.31....5.497....2.15...8.796.31
191: .2.4...9..8.1...5.4........3....5.89...31.64.6459.8...2...64.7.87.2...64..4......
193: ..3..6.....7.2.4...56.9......2....6.....12.....5.78..2231..4.7..7.....8.5.47.92..
197: .....98.78.712..5....76......2.459.........4....8.....2....479.7..23..84..4...2.1
199: ..345...8.97.2..5.4..78..2........87.783.2..5..5..8.....1......78..3.5..5.....23.
211: ...4....66.7.....94......2..1....9...76.1..45845...3....1.9.....6...15...94.8..31
223: 1...5.8.686..2...7....8...3...7......98..274......8...2..57....9.62..57..7..962.1
227: 1...5..69...1.345.4.....12...2.45.......1.....45.9..1.2...7.......23.5.45746...3.
229: .2.....89..9...4...5....123.....56.7...3...4..4...73..231564..897......4.6....23.
233: .2.456.8...........5.7...2.31..45.......12.4.6....9...2.1...97.7.....5.4564...2..
239: ....5.869..8..34.745..9.1.......598.8....2..5...689.....1...6...8.2.1....7...8...
241: ...4...96...12......7...12331.7........312745....983..2.15.....8..2..5.......9...
251: ..34.......61234....96.8.2.3...456.8..8.1.9..9...8....2..86....5.7...8....4......
257: 12.4598.68..12...94.9..6..3..2..5..8.8....945........2...5.....6.8.31.........2..
263: ..........76...45.458....2...284........1284...567..1......4.......3.5845..96.2..
269: ..3.5.69.....2..5.....8.1.331.....86.8......5.4..9.3......7..69...2..57..7486....
271: ...4.6...7.8123..6..6...1...12.4..7....3.....64..............8.9872..5.4564....31
277: 1234..6.....12....4.879...33..8....6.........8..96.....3.5.4.676....15.....6.9.3.
281: .23.58769.6..23..8.......2...28.569.6..3.2.4.84.......2......7...62.1.....4..6.3.
283: 12.45.6...7....45....76......2...98..8....5..6..89..1.....7.....682..794...6...3.
293: 1.34..89..78...4...5...7.233.....7.9..73......45....12.3..6.9....9....64.......3.
307: ....5.8.98.912....4.7..8....1.....9...8..2...6...79.1.2.....97...6.31.8..8.....31
311: .2.4.....6.9.23.58..89..1.3...84...7.673...4......7...2.1..........31.84.8...9...
313: ..3..8.....9...45......9..3..26..789.973..645...9...1........96.8....5..57...62..
317: 12.4.6.......2..5.4........3.254..8.6873....5.4.68......186...9.7......4.6...5...
331: ..3.....8876..........671..3..9.56.77.83.....9....83.2......8766.72..5..5.4...2..
337: .23457.6.9.8.23457..796812....6.5...........56.5.........5....678..31.945.47.62..
347: 12....967....2.......97612.3..8.5.9...9.1...5.........23.....7.7..2.1.....4.9.2.1
349: .23.5.7..97..2.4......79.....264.8....9..26..64..98...2...64..7.9.....64....8..3.
353: ..3..9..7..6....5...9...12....9.5.7.7683....5..56..31.....9.78..872...9..9.7..2..
359: 1.34.....87912.4....69...2.3.....9.....3..64..4....3..2......9..9.2...6.76...923.
367: 12.....98.........4...9..2.31274....9...1.....4..6831..31574..68...3.5........2..
373: .23..6.........4.....8...23.12.45.7.9......4.......31.23.5.47.9.89.....456....2..
379: ..3.5....7.9.....64.6.98.233..9..8...6....9...45..73.......4..96982..5..........1
383: ...4.......6...45.4.8..61...1...9.8..8.....4664...7..22...6.....952..8..8.4...2.1
389: 1.3...7697..1.3.58..86.91....2............8.....9.....2.1..46..967..15.45.47.....
397: 123.......98.......5....12....9...7...63..94...567.3.2.3....7.9..9...5..5.....231
401: .2.....7...6....5.4.......3..25.7....69..2.4.5.7...3..2..68...5.9........8.795231
409: 1.3.5.7............56..9..3.....5...8.....64...5.8.3.2...5.4.7......1564.6.79..31
419: 1........9..12.4..4.....1...126.9..5..531..4964.8..3..2.15.4..7..723.5.45..9.....
421: 12.457.9698.123......6..12......8..9........88..97631.2.1.6.....9.....7.....9..3.
431: 12.......6.7.2.......6.7.2331.9.......6..2..5.457.......159....7.8231...5....6.3.
433: 1.3......7961.....4...7..2..12.49..76...1..49.4..6.............5.9..18.4.6479....
439: 1..458.....6....5....9.61.33.264.........2...64.78..1.2......6...7......89..672..
443: .2...8.79..71.3.58.586.7..33...4.....8..12..5.....63...31.6.............5..8...3.
449: ...4......7...34594.97.6.....2...6.7.....29...4.....1.2..594.6..86....9..9.6.8...
457: ....5.98.9.7...4..4..9..1.....548679.....28....8......23...4....652.....7...652..
461: ....5.9...7.12..5.45...7...3.274.....9.31.74.7...89312.3....8...8.....6.......2..
463: 1.3...6..67.....5.45..7..2.3.......689.31..4..47...3...3........6523..8....7....1
467: 1...........1...7....9..123.12.4...97.5312....46.9..12231.8.5...79...8...........
479: .....6978....2...64..8..12.3.2...78....312..56..78.3...31......9.82..5...........
487: ...45....6...2....4..6...233..9.5..6..6..29.....786.12.31..4..9..523......4.6....
491: 12.4.8......1......58.7612...2.49.6.86..1..4.74.....1.2.1......5.7.......8.5...3.
499: 1...576..6..12.......6..1.3.........796.....5.489...1...1....6..6..3..94.7..6..31
503: ..3...89..9.123....56.9...3..25....8.6.31.....45.........6..98.5....1......9.5.31
509: ......8....7..3.5.4.6.8....3127..69.........8.4..9.3..2....498.8.9...5..5..8.9.3.
521: .234.7....691.34.745796.1.33..5.6..9.7....6......89..2...6..........1....9.......
523: ...4......6.1..4.7..7.69...3.29.5..67....29.594..78.......94...6..2.1..4....86...
541: 12.4598..8..1....9..........1.745....6...27.5....6..1..3.5.4.8...7.315..........1
547: 12.4.867.76912..5.....67123.1......6.8...2.....7.......3...4.6..7..3...4.94...23.
557: 12.4.8.7....12..58....6..2.3.2..5.9......2..5......3...3...4...679231.8.....9.2..
563: 12345........2....4...781...1..4....8...1.....4..8.3...31594.876.....5.45....723.
569: ..3.579.6....2.45...7...12.3.2..5.79..8....4........1....7.....5...3.7.4..4.8.231
571: 123.......7......6..6.7.12..1...........1..4.6..7...122....47...8..31.64.6.9.723.
577: 12.4.6....89......4.69..1..3.2..5...69.3.2......76..1....694...8..........4.7.23.
587: .2....976..6...4.5......1.3.12.4..5.5.7.1..4..49.8531..3.....6.....3.8.....56....
593: ...4.7.86.89....5..57.8.12...2....9.9...12.4....89......1....6989.2.1.......6....
599: .....87..7....3458.......2......6.85...3.26.7.475.931....87459....2..8..8...6....
601: ....59...7...2.........61.33.25..9....73....5...9......31.9.58....2317949.4685..1
607: ..345.....7..23.5.4.6..71.33.2.4.....95.12......569.12..........672............31
613: 1.34....66...2..57....89..3.12.4.7......1........9..1...1..4...8.6.3.5.4.5.9..2..
617: ....56.......23.5.4......2....5..76..693......4.679.......9.6.....231..498.76..3.
619: ...45......9123....5....1...1......76.7.128..5.8.6......1.94.....5.3....9.4..523.
631: 1.3....9..9.1..4..4..89.1...12..9...78..1..4.6.....3.2.3.....6.5682..97..........
641: 1.345....7..1..4.6..6......31.8..9...6.3.....847.........584.97..9......5.4..9.3.
643: 12..5.6.8.6.123.5.....8612.....4..6..7..129....56...122.....7..................31
647: ..3.5.....9...3.....7..6.23.12.49.75..5..2..9....7.31..3....56.....31......56823.
653: ....587.9..9......45.7..12......59.6.9631.84.8.596..1..3................5..6.9...
659: 1....76.9.......7....8...23312......6..3..7........31223.....6.9.....8.4..4.7.231
661: .2.4..6.9.9............9...3.2..8....6931.5..5.8..6.1...1....566.5.3..9......5.31
673: 12.4..7...9..23..8...9.7123.....96...6.......5...7...2.....4....5623..74......2.1
677: 1...59......1....9..9.8.123.....8.7.69......8........2.3.8..7.5..5.3.89498...5.3.
683: ..3..9.....8.2.459..96..1.33.2.....69.....5..5.....3..2.1...8..87..3.6946....8..1
691: .2...9....86.....9....76..3.1...8....75..2..86.8.95.12.3...4..........84894.6.231
701: ...45....79.12.4.....89.123..2.........3...............315..8.7..7231..4.547892..
709: .23.5....9..1.3.574579..1..31....6......1.84..4.76..1....6.4......2.176..........
719: .....9..7......4..4....81......46....8..12..6.4.9..3....17.4.8..9..31.7..74.95..1
727: 1.34569..7891...6...67....3...9.75....5..........6.3.2..16.48.7.78........4......
733: 1.345.6.7....2....45.....2..1..469.57..3..84.84.....1.23....5..5.............5.3.
739: 12.........91.34...5.9....3....4.8.998.3...45.4.7.9....3.89.......2.1..4.9..6....
743: ..3...7.9.6.1.3...4..7..1..31.8......9.3..84....9..3...3...49......3..7457468....
751: ..345.7..7......5.4..76.1..31..4....87.........587.3...3...4..7597...8....4...2..
757: ...4.6.98.8...3....5........1..4.8....8.1.6......7......18...6.5.7231.8.8.4.67231
761: 12.4....9......4..457..9..3...........5.12648....7.31...17.4..65.92...8...4......
769: .23...7.687612.4...5...6.2..125....7.....25.85....73.2.31...8...8......4......231
773: 1...........12.....5....1........7.8......645.....9..223189.576.652.1.84..4..6...
787: ..3.5...88....3.57......1..3.2..9........2...9.8675.1.23...4..668.23..9.........1
797: ...45.798.98.........7981.........6.867.....5....8.3...3.8..579..92......8..7..3.
809: .....87......2....4...7.123.1.84...696....84.84576.....3...4.........584.....7.3.
811: 1....86.9.........4..7.....3..5...8.8..31..45547.893....1....97..5.....4.84..52..
821: ..3....696...23.......96.2...2..7..69....25..54..69....3.98...57..23.8.....6.....

Which resolve to:

  2: 123456789789123456456798123312845967..7312845845967312231574698..8231574574689231
  3: 12.4567..78.12345.45.78912.3129458678673129459458673122315746..698231574574698231
  5: .23.5..89.89.23.56456.89.23312945867867312945945.6.312231594678678231594594.7.231
  7: ....5.789987123456456978123312845967..9312845845769312....9.678..8.3.594.94.8.231
 11: 123..6789798123456...7891233126..897987312645..59783122.1.6.9.8879231564...89.2.1
 13: 12345679879812345645697812331.8.5...9..3.28..8.....3..231584...679231584584...231
 17: 1.34.67.99.7...4.6.5.987123312.4.967769312.4.845769312.3..94.7..782...9..9.67..3.
 19: ....5.7898.7..34..4.67....3312.4.897978312645645879312....6..787..23...45....723.
 23: 123.5..9.87912..5.456.79.2.31..4...998.....456459....2231...9..798.3.5..56479.2..
 29: 123.56..98.9123.56....8.123312645...798312645...8973122.1.6....9..2315.4..4.782.1
 31: 123456897798123...4567981..31.64..78879312.4.64.87....2.15.47899872.1...5.4987...
 37: 123456....9.123456.5....123.12.45...9..31...5..5....1.2..5.4.9..892.15..5..9.7.3.
 41: ...45.78997812....45.789.2..12645.97.97312....45978..2....6...8.8..31..45..89.2..
 43: ........8897.23.5.4.6..8...3....5.7...93.2.45.45.8.3.2......9879782315645648972..
 47: 123456789987123456456789...3..6.58.787.3126.5..58.7....31.64.78798.31.64....78...
 53: .2.456...97.12345.45.798.23..2.459...97.12.456458793122..5.4...7.92.15.45.49.72..
 59: .......78978123456.5..78.23...8.....76.3.28..845769312...59....6...31...594.8..31
 61: .2.456798987...456.5.897123.1.64.....9.3..64...57..31..3.56.....79...5645..97..31
 67: .2..5.8......2.45.45.89.123312...7...9.3126456457893122.1...9..9..2.15..564978231
 71: ..3.....8689123457.57..8..3..2...8.6968.12.45..586...2..1...689896231574574986231
 73: .....68.9789123456.568...233....5...8673129459.5..83..2....4...6.8..1.945...8..3.
 79: 1.345....7..1.34..4..9....3312845967.793.284.84..9...22..5...9.9.7.3.........9...
 83: 123...87.879......4567.8..3.126......98..2...64598...2231......9872.15.45648.9...
 89: ..345.......1.......6..8..3312645897...312645645879..2231584769.....1..45.49....1
 97: 12345..8686712.4....96..12..12.45.6..86.12.....5....12231594678678231.....4...231
101: 1...5.9.896.1.34.7....9.1.33....589689.31.745.....93122..5..68.6..23157....96823.
103: .234.7.68.....345.4....6..331.74.896.......4.74...8.1..31584...9.....584..4.79.3.
107: 12.4.....79.1..4..456987123.126......873...45.45798.122315.....8792..5.456487.2.1
109: 1.34.679...9..3.......9....3129458677683129459456......31..46.9.9.2315.45.4.69...
113: ....567.9..7..3.56.5.7....3.....5.97769...845.459.7.1223.57496..9....5745746..23.
127: ...459......1.34..4..7.61..31.945...7..312.4..4.678....3.594876...231594594867...
131: .2...7....8.......45..8.12.3.274.......3.2.4.745896312231574968........45.4.68.31
137: .2.4578......2345.45.....23..2.45..8.....2.45.45..93.22315746899.823.5.45.4...23.
139: .234.687.8....3..64.687.1.....64..877.93.2...6.5..7........4..8..8..1...5647.82..
149: ...45..7..7.12.45.4..768123....4576.76..12.4....876.1....594687687231594...687231
151: 123457869..9123457...896.2...27.59.6..63........6......31.74698..8.31574.7..682..
157: ...4.786...81.3.5.4......2.312745986869312.4.745896.1....5..69.....3..7........3.
163: .234.6....98.2...64.6...123....4.........2....4....3..2.1.64978...2.1564..49.8231
167: .2.....8...........56987.2331.74..9..6.3..74.745869312.31...8..6.........74698.31
173: 1.34..67967....4..45..76..3.1..4..67.67...8...4.769....31...7.6796...5....46.7.31
179: 123..7.8.869123457457869123312..5.6.678....459456.8.12.....4.7........9..9.....3.
181: ..34..7......234..4.867.1.3....4.6.7...3.284.84.96731....5.497....2.15.4584796231
191: .2.4...9.98.1..45.4........312645789798312645645978...2...64.7.87.2...64.64......
193: .23..6.....7.2.4.6.56.97.2...2....6...8.12.....5.78..2231..46796792315845847692..
197: ...4598.78.7123.5....768.....2.459.........4....8.....2....479.7..23..84..4...2.1
199: ..345.7.8897.2..5.45678..2........87.783.2..5..5.78..2..1...87.78..3.5..5..8.723.
211: 1..4....66.71..4.94.....123.1....96.976.1..45845...31...1.9.....6...159..94.8..31
223: 1...5.8.686..2...7...68...3...7.5....98..2745...9.8..22..57....9.62..57..7.896231
227: 12.45..69...12345.45.96.12..12.45.......1..45.45.9..1.23.574......23.57457468..3.
229: .2..5.789..9...456.5....123.....56.7...3..84..4...73..231564978978...564.6....231
233: .2.456.8...........5.7...2.31..45.......12.4.64..79...2.15.497.7.....5.4564..72..
239: ....5.869..81234574578961.......59868....2..5...689.....1...6.8.8.2.1....7...8...
241: 12.4..896...12.457..798.1233127........312745...6983122.15.....8..2.15.....8.92.1
251: .234.......61234....96.8.2.3.29456.8..8.1.9..9.5.8....2.186....5.72..8....45.....
257: 1234598768..123459459..6123..2..5..8.8...2945........22..5.....6.823159.......2..
263: ...4......76...45.458.9..2...284........1284.84567..12.....4.......3.5845.496.2..
269: .23.5.69.....2..5.....8.12331..4..86.8.....45.4..98312....7..69...2..57..7486.231
271: ...4.6...7.8123..64.6...1..312.4..7.87.3.....64............4.8.987231564564...231
277: 123458679...123...458796..33..8....6...3.....8..96.3...3.584.676..2315.....679.3.
281: .23.58769.6..23..8.......2...284569.6..3.284.84.......2......76..62.1.8...4..6.3.
283: 12.45.67..76...45....76..2...2...98..8....54.6..89..1.....7..6..682..794...6...3.
293: 1.34..897.78...45..5...7.233.....7.9.973......45...312.3..6.97...9....64.......3.
307: 1.3.5.8.98.912....4.7..81...1.....9...8.12...64..79.1.231...97...6.31.8..84....31
311: .2.4.8...6.9.23458..89..123...84...7.673...4......7...2.1.84.......31.84.8...9...
313: ..3..89....9...45......9..3..26..789.973..645...9..312.......96986...57.57..9623.
317: 12.4.6.9.7...2.45.4........31254..8.6873...45.4.68......1864..9.7......4.64..5...
331: ..3...768876.........8671..3..9.56877.83.....9....83.2......8766.72..5..5.4...2..
337: 123457.6.968123457457968123...6.5.........6.56.5.........5....67862315945.47.62..
347: 12....9679..12.......97612.31.8.5.9...9.12..5.....9.1.231....7979.2.1.....4.972.1
349: .23.5.7..97..2.4......79.....264.8....9..264.64..98...2...64.87.9.....64....8..3.
353: ..3..9.67..6....59..9...123...9.567.7683..9459.56..31.....9.78..872..59..9.7..23.
359: 1234....98791234564569..1233.....9..9..3..64.64..9.3..23....59.59.23..6.76...923.
367: 12..5..98....2....4...9..2.31274.86.9...1.....4..6831.2315749868...3.5......8.2..
373: 1234.68..8.....4.....8..1233126459789......4.......31.2315.47.9789.....4564...2.1
379: ..3.56.98789....56456.98.233..9..8..86....9..945..73.2.....4..96982..5....4..9..1
383: ...4..6....6...45.4.8..61...12649.8..8.....4664...7.122...6.....952..86.864...2.1
389: 123...7697961234584586791....2........9...8....59.....2.1..46..967..15.45.47.....
397: 123....9..98....5..5....12.3..9...7...63..94.9.567.312.3....7.9..9...5.45.....231
401: .2..5..7...6....5.45..7...3..25.7....69..25475.7...3..2..68..95.95......684795231
409: 1.3.5.7.........5..56..9..3.....5..78.....645..5.8.3.2...564.7......1564564798231
419: 1..4.....9..12.4..4..7..1..312649..57853126496498..3..2.15.4..78.723.5.45.49.....
421: 123457896986123..7...689123.....8..9..9.....88..97631.2.1.6.98..9.....7.....9..3.
431: 1234.....6.7.2....4.96.7.233129......76..2..5.457.....231594...768231...594876.3.
433: 123......7961.....4...7..2..12.49..76...1..4994..6......1......5.9..18.4.6479....
439: 1..458.....6....5....9.61.33.264.........2...64.78..122......6..67......89..672..
443: .23458.79..71.3.58.586.7..33...4.....8.312..5....863...31.6........3....5..87..3.
449: ...4.9..667...34594.97.6.....2...6.7.6...29...4..6..1.2..594.6..86....94.946.8...
457: ...45.98.9.7...4..4..9..1.....548679.....28....8......23...4...8652.....7...652..
461: .2..5.9...7.12..5.45...7.2.31274.....9.31274.7...89312.3....8...8.....6.......2..
463: 123...6..67.....5.45..7..2.312.....689631..4.547...31..31....6..65231.8..847...31
467: 1.....9..9..1...7....9..123.12.4..59795312....46.9..12231.8.5...79...8........2..
479: .....6978....2.4564..8..1233.2...78....3126.56..78.3.2.31......9.82..5...........
487: ...45....6...2....4..69..233..9.5..6..63.29..549786312.31.74..9.6523......4.6..3.
491: 12.4.8....7.1......58.7612.312.49.6.86..1..4.74.....1.2.1......5.7.......8.5...3.
499: 1...576..6..12.......6..1.3.1.......796.1...5.4897..1...1....6..6..31.94.7..6..31
503: ..3.5689..9.123....56.9...3..25....8.6.31.....45........16..9855.9..1...6..9.5.31
509: ..3...8...87..3.5.4.698...33127.869..95.....8.48.9.3..2.1..498.8.9...5..5.48.9.3.
521: 123457...8691234574579681233.25.6..997....6......89..2...6..........1....9.......
523: ...4.7....6.1.34.74.7869...3.29.5..67.63.29.5945678.......94...6..231..4..4.86...
541: 12.4598..8..1....9......1...1.745....6..127.5....6..1..315.4.8...7.315..........1
547: 123458679769123458...967123.1......6.8...2.....7.....2.3...4.6..7..3...4.94...23.
557: 12.458.7....12..58....6..2.3.2..5.9....3.2..5......3.223...4...679231584....9.23.
563: 12345........2....4...781..31..4....8...1.....4..8.3..2315946876...3.5.45.4.6723.
569: .23.57986....2.457.57...1233.2..5.79..8....4...5....1.23.7.....58..3.7.4..458.231
571: 123.......7.1....6..6.7.12..1...........1..4.6..7...1223...47...8.231.64.6.9.723.
577: 12.4.6...789......4.69.71..3.28.5...69.3.2...5487693122..694...8........9.4.7823.
587: .23.5.976..6...4.5.5....1.3312.4..58587.1..49649.8531..3....56..65.3.89....56..3.
593: ..3457986.8912..5..57.8.12...2....9.9.8.12.4....89......1....698962.1.......6....
599: .....87..7....3458.....712....746985...312647647589312...87459....23.8..8..96.2..
601: ..3.59...7...23..9..9..61.33.25..9...973....5...9.....231.9.586...231794974685..1
607: ..345.....7..23.5.456..7123312748...695312...7..569312.3........6723...........31
613: 1.34....66...2..57...6891.3.12.4.7......1........9..1...1..4...8.6.3.5.4.5.9..2.1
617: ....56.......23.5.4...87.2....54.76976931.....4.679......89.6.....231..498.76..3.
619: 1..45......9123....5....1...1..4...76.7.128..5.8.6..12..1.94.....5.31...9.4..5231
631: 1.3....9.89.1..4..4..89.1..312..9...785312649649...312.3.....6.5682..97..7.......
641: 1.3456...7.81..4.64.6..8...31.8..9..96.3.....847.......31584.97..9......5.4..9.3.
643: 123.5.678.6.123459...786123.12.4..67.7..129....56...122.....7..................31
647: ..3.5.....9...3.5...7..6.23.12.49.75..5..2..9....75312.3....56.....31......56823.
653: ...4587.9..9......45.79612......5976.9631.8458.596..1..3................5..6.9...
659: 1....76.9.......7....8...233127....66..3..7........31223....9679.....854..4.7.231
661: .2.45.6.9.9......5.5...9...3125.8.6..6931.5..5.8..6.1..31....566.5.3..9......5.31
673: 1234..7...9.123..8...9.7123.....96...6...2...5...7...2.....4....56231.74......2.1
677: 1...59..7...1....9..9.8.123.....8.7.69......8........2.3.89.7.5..5.3.89498...523.
683: ..3..9.6..68.2.459..96..1.33.2...9.69.....5..5.....3..2.1...8.5875.3.6946.4..8..1
691: .2...9....86.....9.5..76..3.1...8....75..2..8648.95.12.3...4....6.....84894567231
701: 12.45....79.12.4.....897123..2.........3..............231564897987231..4654789231
709: 123.57..69..1.3.574579.61..31....67....31.84..4.76..1....6.4......2.176..........
719: ...4.9..7...1.34..4....81......46....8..12.46.4.9873....17.4.8..9..31.74.74.95..1
727: 1234569787891..465..67....3..29.75....5...7....7.6.3.2..16.48.7.78........4.7....
733: 12345.6.7....2..5.45.....2.3128469757.53..84.84.5...1.23....5..5.............5.3.
739: 123........91.34...5.9....3.1..4.8.998.31..45.4.789....3.89.......231..4.9..6....
743: 1.34..7.9.6.1.34..4..7..1..31.84..9..9.3..84..4.9..3...31.749......31.745746892..
751: ..345.7..7......5.45.76.1..31..4..7887........4587.3...3...4..7597...8....4..72..
757: ...4.6.98.8...3....5..89....1..4.8....8.1.6......78.....1894.6.567231.8.894567231
761: 12.4....9......4..457..9..3.1..4......5.12648.4..7.31...17.4..65.92...84..4......
769: 123...7868761234...5...6123.125....7.....25.85.8..73.2.31...8...8...1..4......231
773: 1...........12.....5....1........798......645.....9312231894576765231984..4..6231
787: ..3.5...88....3.57......1..3.2..9......3.2..9948675.1.23...4..668.23..9.........1
797: ...45.798798....5....79812.......86.867...9.5....8.3...3.8..579.792...8..8..7.23.
809: ...4.87......2.4.84...7.123.1.84...696....84.84576..1..3...4.........584..4..7.3.
811: 1..4.86.9......4..4..7..1..31.5...8.8..31..45547.893....18..597..5...8.4.84..52..
821: .23..8.696...23.5.....96.2.3.2..7..69....25..54..69...23.98.6757..23.8.....67....