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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 that are the first 12 odd primes: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41? Bonus: what is the smallest rectangle (by area) that can fit such an arrangement?

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1 Answer 1

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These islands can be arranged in one line: numbers in brackets are the islands, number between dashes are the number of bridges between islands.

 [3] -3- [5] -2- [7] -5- [11] -6- [13] -7- [17] -10- [23] -13- [37] -24- [41] -17- [29] -12- [31] -19- [19]
 

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  • $\begingroup$ Great work, well done! $\endgroup$ Aug 1, 2023 at 12:09
  • $\begingroup$ @DmitryKamenetsky is this the optimal area? $\endgroup$ Aug 1, 2023 at 12:14
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    $\begingroup$ @newQOpenWid yes: it is the minimal number of cells: 12 islands + 11 bridges (which are necessary in order that all the islands are connected) $\endgroup$
    – daw
    Aug 1, 2023 at 12:53

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