4
$\begingroup$

Previously I posted collections of my Find-the-stars puzzles here, here and here.

This post is different.
You are asked to create a puzzle.


Find-the-stars rules:

A Find-the-stars puzzle is played on a square grid that is initially empty except some squares contain arrows.

The goal of the puzzle is to fill some empty squares with stars so that:

  1. Every row contains exactly one star.
  2. Every column contains exactly one star.
  3. Stars are NOT placed in squares that contain an arrow.
  4. If a square contains a SOLID arrow, THEN exactly one square in the direction of the arrow will contain a star.
  5. If a square contains a DASHED arrow, THEN none of the squares in the direction of the arrow will contain a star.

Clarification: Despite the above rule about SOLID arrows, it is possible that a star is not pointed to by a SOLID arrow. It is also possible that a star is pointed to by more than one SOLID arrow.


Sample 9x9 puzzle:

9x9 Find-the-stars puzzle

Sample solution:

Solution to above puzzle


Main question:

What is the smallest number of arrow clues that must be provided so that a 9x9 Find-the-stars puzzle has a unique solution according to the rules given above.

Puzzle construction constraints:

  1. Each square in the 9x9 grid is either empty or contains a single arrow.
  2. Each arrow must be SOLID or DASHED.
  3. The puzzle you create need not necessarily contain both SOLID and DASHED arrows.
  4. There are 8 possible directions for the arrows: up, down, left, right, diagonally up and to the left, diagonally up and to the right, diagonally down and to the left, and lastly diagonally down and to the right. Note that each of these 8 directions appears at least once in the sample puzzle. You need not necessarily use all 8 directions in the puzzle you create.
  5. Arrows on the edge of the puzzle CAN'T point outwards.
  6. Arrows CAN'T point to other arrows.

Partial answers are welcome (for example, using a square grid smaller than 9x9).

$\endgroup$
4
  • $\begingroup$ What is the consequence of constraint 6? $\endgroup$
    – Sny
    Commented yesterday
  • $\begingroup$ That's not a puzzle construction constraint, just a consequence of the genre. (So I was confused and thought that the puzzle doesn't require one star per row / column.) $\endgroup$
    – Sny
    Commented yesterday
  • $\begingroup$ @Sny I have removed the unnecessary constraint. Let’s delete our comments. $\endgroup$ Commented yesterday
  • $\begingroup$ A small observation to reduce the problem space a bit. rot13(Nal ubevmbagny be iregvpny fbyvq neebj pna or ercynprq ol n qnfurq neebj, naq ivpr irefn, cbvagvat va gur bccbfvgr qverpgvba jvgubhg nssrpgvat nalguvat nobhg gur chmmyr.) $\endgroup$
    – Pranay
    Commented 23 hours ago

3 Answers 3

5
$\begingroup$

I can make a 9×9 puzzle

with six arrows:
enter image description here
Try it here!

How did we get here?

A two-arrow deficit is quite easy to achieve. Here are two small puzzles that achieve it:
enter image description here enter image description here
With very, very few exceptions, an $N\times N$ puzzle with $A$ arrows can be trivially extended to an $N+1\times N+1$ puzzle with $A+1$ arrows by inserting a one-cell arrow somewhere, so I looked for some bigger puzzles with a larger deficit.
I found the following 6x6 grid with three arrows and two solutions:
enter image description here enter image description here
This wasn't enough for a three-arrow deficit, but I did find a way to force three stars into three rows and three columns with three arrows, one of which is outside said rows and columns:
enter image description here
and combining these gives the full 9x9, by using the outside arrow to block one of the solutions.

$\endgroup$
3
  • 1
    $\begingroup$ Puzzle construction constraint #6 is broken (Arrows CAN'T point to other arrows) $\endgroup$ Commented 20 hours ago
  • 1
    $\begingroup$ @ArrowNought I did forget that, but you can trivially modify the puzzle so that constraint 6 is satisfied. $\endgroup$ Commented 19 hours ago
  • 1
    $\begingroup$ I've fixed the puzzle to make it compliant, and added the reasoning that led to it. $\endgroup$ Commented 10 hours ago
3
$\begingroup$

Here is my best result, that I claim, without proof, to be optimal:

trivial puzzle with solution going down a main diagonal

$\endgroup$
3
  • $\begingroup$ This is a nice valid puzzle but not optimal. $\endgroup$ Commented yesterday
  • $\begingroup$ this is not unique. $\endgroup$
    – Oray
    Commented 17 hours ago
  • $\begingroup$ It is unique, but not optimal. $\endgroup$
    – Sny
    Commented 6 hours ago
2
$\begingroup$

Here is my unique solution:

enter image description here

here is another solution:

enter image description here

used my own player to find this, totally trial-and-error so I am not sure this is optimal though.

Please feel free to find with 5 or less arrows by using the game creater below, all rules are implemented (hopefully :P)

create the puzzle here

$\endgroup$
1
  • 2
    $\begingroup$ Nicely made applet! I found another 6 with it: 2,3,UL 3,4,DL 3,7,DR 7,5,UL 7,6,DL 7,7,DR $\endgroup$ Commented 10 hours ago

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.