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You are given an empty square grid. Each cell can be an island with a positive integer $n$ or a bridge connecting islands. The following rules apply:

  • Each island with a number $n$ must have exactly $n$ bridges attached to it
  • Bridges must be either horizontal or vertical with one or more cells in length
  • Bridges cannot intersect, but they can meet at an island
  • All islands must be connected into a single component

For example, here is a valid arrangement for islands with numbers 1 to 4:

enter image description here

Can you find an arrangement for islands with numbers 1 to 12? Bonus: what is the smallest rectangle (by area) that can fit such an arrangement?

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    $\begingroup$ If you are wondering, the title is referring to this: en.wikipedia.org/wiki/The_Twelve_Apostles_(Victoria) $\endgroup$ Aug 1, 2023 at 6:54
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    $\begingroup$ Number of bridges between islands is unlimited? Islands are not allowed to touch? All islands need to be connected? $\endgroup$
    – daw
    Aug 1, 2023 at 7:04
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    $\begingroup$ ah yes all islands should be connected... will add it in $\endgroup$ Aug 1, 2023 at 7:26

2 Answers 2

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Islands can be arranged in one line: putting islands with odd numbers on one end, and islands with even numbers on the other end: numbers in brackets are islands, numbers between dashes are bridges.

 [ 1] -1- [ 3] -2- [ 5] -3- [ 7] -4- [ 9] -5- [11]
                                               -6-
 [ 2] -2- [ 4] -2- [ 6] -4- [ 8] -4- [10] -6- [12]
 

This arrangement needs $3*11=33$ cells. Putting all islands on one long line
needs $23$ cells.

This is the minimum: we need $12$ cells for all the islands, if all islands are to be connected, we need at least $11$ bridges.

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    $\begingroup$ Nice solution. Just realised that the example one follows a similar pattern. $\endgroup$ Aug 1, 2023 at 7:30
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    $\begingroup$ You can now try another version of this puzzle. $\endgroup$ Aug 1, 2023 at 7:41
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In case not all the islands need to be connected (I have a feeling they might, but the rules do not say this at the time of writing this) a simple solution can be

Split them in 4 groups of 3 in such a way that the islands in the group are connected, but the groups are not connected.
1 - 12 - 11
4 - 10 - 6
3 - 8 - 5
2 - 9 - 7

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  • $\begingroup$ To get everything connected: you can add vertical bridges between the islands in columns 2 and 3. $\endgroup$
    – daw
    Aug 1, 2023 at 7:12
  • $\begingroup$ Ah rigth.... I will try it $\endgroup$
    – Marius
    Aug 1, 2023 at 7:18
  • $\begingroup$ Rules state that all islands "must be connected into a single component". $\endgroup$ Aug 1, 2023 at 15:38
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    $\begingroup$ @ConnieMnemonic Not at the time the answer was posed, which he states explicitly $\endgroup$
    – No Name
    Aug 1, 2023 at 19:55

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