11
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CrSb0001 has been studying the minimum number of clues to force a unique solution in Hidato for variously sized boards. They came up with a 5x5 Hidato with 5 clues but it was found that it has multiple solutions.

Goal of Hidato:

Fill in a grid with a series of consecutive numbers that connect each other horizontally, vertically, or diagonally.

I was able to find a 5x5 Hidato puzzle with just 4 clues having a unique solution. This was verified with a Picat program (the instance is replaced with a 4x4 one to avoid spoilers; you're free to experiment with the program). Can you find one?

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3 Answers 3

7
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     .  .  .  5  .
     .  .  .  .  .
     .  .  7  .  .
     2  . 19  .  .
     .  .  .  .  .
 

with verified unique solution

    11 10  9  5 25
    12  8  4  6 24
    13  3  7 18 23
     2 14 19 17 22
     1 15 16 20 21
 

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1
  • $\begingroup$ This is solvable by hand. $\endgroup$
    – mathlander
    Mar 9 at 4:47
12
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It can actually be done with 3:

     .  .  .  6  .
     .  .  .  . 13
     .  .  .  .  .
     9  .  .  .  .
     .  .  .  .  .
 

with verified unique solution

     1  2  3  6  5
    22 21  7  4 13
    23  8 20 12 14
     9 24 11 19 15
    25 10 18 17 16
 

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6
  • 2
    $\begingroup$ This is so good I forgive you for blatantly stealing my post layout ;-) $\endgroup$
    – loopy walt
    Mar 8 at 8:52
  • 1
    $\begingroup$ @loopywalt I don't know what you are talking about =-P $\endgroup$ Mar 8 at 8:55
  • $\begingroup$ All 3 numbers 1 lower also seems to work $\endgroup$
    – Retudin
    Mar 8 at 9:54
  • $\begingroup$ @Retudin the program linked in OP finds 3 solutions, then. $\endgroup$ Mar 8 at 10:37
  • 1
    $\begingroup$ Great job! This is solvable by hand but very hard. $\endgroup$
    – mathlander
    Mar 9 at 5:25
1
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     .  .  .  4  .
     .  .  .  .  .
     .  .  6  .  .
     1  . 19  .  .
     .  .  .  .  .
 

Also works as well as its mirror image

     .  4  .  .  .
     .  .  .  .  .
     .  .  6  .  .
     .  . 19  .  1
     .  .  .  .  .
 

Both with unique solutions:

     25   4   8   9  10
     24   5   3   7  11
     23  18   6   2  12
     22  17  19  13   1
     21  20  16  15  14
 

and

     10   9   8   4  25
     11   7   3   5  24
     12   2   6  18  23
      1  13  19  17  22
     14  15  16  20  21
 

and in that logic there should be plenty more in a similar manner where mirror solutions also work. I haven't found any with 3 clues since (I think) you can't force a wall with a gap with only 3 if that makes sense. I'm unsure if the positions (where they are now) are vital to forming the wall with the gap though.

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4
  • 2
    $\begingroup$ I would expect, given the nature of the puzzle, that horizontal and vertical symmetries both always apply, no? $\endgroup$ Mar 7 at 13:40
  • $\begingroup$ ... and rotations too. I would not consider those to be different puzzles/solutions. $\endgroup$ Mar 7 at 13:48
  • $\begingroup$ True, both symmetries apply as well as rotations of solutions since the direction of execution would simply be mirrored/rotated as well. I don't mind taking the mirror "solution" away @JaapScherphuis I just figured people could still have those puzzles as "unique" too. $\endgroup$ Mar 7 at 14:09
  • $\begingroup$ You also can reverse the numbers, i.e. apply n -> (26-n). That would be less obvious. $\endgroup$
    – Florian F
    Mar 7 at 20:40

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