# The Twelve Apostles

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:

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?

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

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.

• Nice solution. Just realised that the example one follows a similar pattern. Aug 1, 2023 at 7:30
• You can now try another version of this puzzle. Aug 1, 2023 at 7:41

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

• To get everything connected: you can add vertical bridges between the islands in columns 2 and 3.
– daw
Aug 1, 2023 at 7:12
• Ah rigth.... I will try it Aug 1, 2023 at 7:18
• Rules state that all islands "must be connected into a single component". Aug 1, 2023 at 15:38
• @ConnieMnemonic Not at the time the answer was posed, which he states explicitly Aug 1, 2023 at 19:55