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This is an extension of Nilster's great puzzle: The Greenhouse Problem

The task is the same, but this time sprinklers cover only a 3x3 square around them. For completeness, here is the full set of rules:

  • The greenhouse floor is made of tiles arranged in a 9x11 rectangle. The door is on the center of the 11-length edge.
  • Each plant takes up 1 tile.
  • Each plant needs to be watered by a sprinkler. Each sprinkler waters a 3x3 square around it.
  • A plant and a sprinkler cannot be placed on the same tile.
  • I must be able to reach every plant orthogonally. I cannot walk on plants or sprinklers. I can only move orthogonally.
  • Asymmetry is allowed. The fewer sprinklers, the better, but there's no limit.

What is the most number plants possible in this greenhouse?

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4 Answers 4

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Here's a symmetric solution with

50 plants:

\begin{matrix} &S &P &P &S &P &. &P &S &P &P &S\\ &P &. &. &P &P &. &P &P &. &. &P\\ &P &. &P &S &P &. &P &S &P &. &P\\ &S &. &P &P &P &. &P &P &P &. &S\\ &P &. &. &. &. &. &. &. &. &. &P\\ &S &. &P &P &P &. &P &P &P &. &S\\ &P &. &P &S &P &. &P &S &P &. &P\\ &P &. &. &P &P &. &P &P & . &. &P\\ &S &P &P &S &P &. &P &S &P &P &S\\\end{matrix}

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  • $\begingroup$ Excellent work Rob! $\endgroup$
    – Dmitry Kamenetsky
    Commented Feb 2, 2021 at 22:07
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I could place

49 plants

in this way (which is also more symmetric than the previous one):

S P . P S P S P . P S
P P . P P . P P . P P
. . . P P . P P . . .
P P . P S . S P . P P
S P . P P . P P . P S
P P . . . . . . . P P
. . . P P . P P . . .
P P . P S . S P . P P
S P . P P . P P . P S

Another one with the same count:

S P . P S . S P . P S
P P . P P P P P . P P
. . . . . . . . . . .
P P . P S P P P . P P
S P . P P P S P . P S
P P . . . . P P . P P
. . . P P P S P . . .
P P . P S P P P . P P
S P . P P . . . . P S

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  • $\begingroup$ This is a great start! $\endgroup$
    – Dmitry Kamenetsky
    Commented Feb 2, 2021 at 6:09
  • $\begingroup$ I tried various other patterns and non-patterns, but tweaking them all converged to the existing ones. I'm suspecting the current number is optimal. $\endgroup$
    – Bubbler
    Commented Feb 2, 2021 at 7:48
  • $\begingroup$ I can confirm that this is not optimal. Better solutions exist. $\endgroup$
    – Dmitry Kamenetsky
    Commented Feb 2, 2021 at 10:56
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Same P-count as @Bubbler but I managed to squeeze out one S:

 
    P S P . P S P . P S P
    . P P . P P P . P P .
    . . . . . . . . . . .
    P P P P P . P P P P P
    S . P S P . P S P . S
    P . P P P . P P P . P
    P . . . . . . . . . P
    S . P P P . P P P . S
    P . P S P . P S P . P

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My solver found some more solutions with the optimal number of plants:

50 plants 

 S P P S P P S P P S P
 P . . . . . . . . . .
 P . P P P P P P . P P
 S . P S P P S P . P S
 P . P P . . P P . P P
 S . P S P . P P . . .
 P . P P P . P S P P P
 P . . . . . P P P P S
 S P P S P . . . . . P

and

 S P . P S P P S P P S
 P P . P P . . . . . P
 . . . P S P P P P . P
 P P . P P P P S P . S
 S P . . . . . P P . P
 P P . P P P P S P . P
 . . . P S P P P P . S
 P P . P P . . . . . P
 S P . . . . P S P P S

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