Polyomino Mazes

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For the purposes of these puzzles:

  • a maze is defined as a branching path with no loops, no corner-wise connections and all connections should be a single unit width.

  • an elegant maze covers the area such that no spaces are more than a single unit in width.

Problem 1: What is the largest area in which an elegant maze can be built with a set of 12 pentominoes?

Hover over an image to see a solution.

   
  17 x 7 solution (not the largest possible solution)  

 

Problem2: What is the longest “correct” path which can be formed in an elegant maze, ignoring dead ends?

   
  path length = 37 (not the longest possible solution)  

 

In a set of mono-through-pentominoes, only the P pentomino and square tetromino contain a 2x2 block.

Problem 3: Using such a set, what is the longest “correct” path which can be formed in an elegant maze, starting at the "P" pentomino and ending at the square tetromino?

   
  path length = 37 (not the longest possible solution)  

Seven of the pieces contain squares which cannot form part of the path. These pieces are shown below with one possible longest path highlighted. So the maximum possible path length for mazes is the total number of squares minus these eliminations. For the pentominoes this gives 60 - 6 = 54, and for the mono-through-pentominoes it is 89 - 8 = 81.

Maximal solutions exist for both puzzles.

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