1a. What is the least number of identical squares that can be placed in a plane such that each shares a side with exactly two others?

1b. What is the least number of identical squares that can be placed in a plane such that each shares a side with at least one other and they form a contiguous region with no rotation or reflection symmetry?

1c. What is the least number of identical squares that can be placed in a plane such that each shares a side with exactly two others and they form a contiguous region with no rotation or reflection symmetry?

2a-c. Same as 1a-c. but replace 'squares' with 'equilateral triangles.'

3a-c. Same as 1a-c. but replace 'squares' with 'regular hexagons.'

I assume that sharing a side means that the sides correspond exactly, and that the shapes therefore share the vertices that correspond with those sides.

1a) 4, arranged as follows

      X X
      X X 

1b) 4, arranged as follows
    XX
    X
    X
1c) The best I could come up with was 14, arranged as follows:

    XXX
    X XX
    XX X
     X X
     XXX 

Posted by Steve Herman on 2013-02-27 20:28:35