You are asked to create puzzle pieces by joining together identical squares. This needs to be done such that the puzzle pieces can be arranged in a pattern with each piece being in contact with exactly 5 others. In terms of number of starting squares, how low can you go?
Background and further clarification
- Here we saw that 64 identical squares can be joined into puzzle pieces which can subsequently be grouped together in a square such that each piece is in contact with exactly 5 other pieces.
- Here we saw that 42 identical squares can be joined into puzzle pieces which can subsequently be grouped together in a rectangle such that each piece is in contact with exactly 5 other pieces.
- Now we are getting rid of the requirement to group the puzzle pieces into a specific shape: any shape will do!
- We still require two squares being 'joined' or two puzzle pieces being 'in contact' to mean that both share a finite portion of their perimeter.
- Also, the joining of the squares into puzzle pieces as well as the grouping of these pieces, is to be done within a plane and without creating any overlap. In other words, there should be no loss of total area covered: the resulting total puzzle needs to cover an area identical to that of the sum of the squares used.
So far this puzzle has seen zero attempts towards a solution. As long as progress is lacking, every one or two days I will publish a hint.
The first hint:
In contrast to the 64 square and 42 square solutions, the solution sought does deploy single square puzzle pieces.