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 Touching polygons (Posted on 2013-02-27)
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.'

 See The Solution Submitted by Jer Rating: 4.0000 (1 votes)

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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    X1c) 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

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