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It is known that one can have 4 shapes in a plane all touching each other, and not 5. You can add requirements to the 4 shape problem:

  • Can you do it with 4 equal triangles? (No)
  • Can you do it with 4 equal rectangles? (No)
  • Can you do it with 4 triangles with equal area? (Yes) (figure 1)
  • Can you do it with 4 equal shapes? (Yes) (figure 2).

My question is: what is the simplest single shape which you can do it with?
(Figure 2 was easy to find, but it looks like it can be improved upon ;-).
(Simpler is undefined, but build from less than 11 squares would fit)
(I could not find a source to this problem. Is there?)

enter image description here

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  • $\begingroup$ Does it count as the same shape if you use its mirror image? $\endgroup$
    – Tweakimp
    Commented Jan 4, 2018 at 16:16
  • $\begingroup$ See 5 geometric shapes, all touching each other for why five is impossible. $\endgroup$ Commented Jan 4, 2018 at 16:38
  • $\begingroup$ @Tweakimp: figure 2 has mirror images. Avoiding this would be nice (just my feeling), but I did not state that requirement. It is a bonus. $\endgroup$ Commented Jan 4, 2018 at 16:42
  • $\begingroup$ Does the shape have to be continuous? $\endgroup$
    – corsiKa
    Commented Jan 4, 2018 at 20:41
  • 1
    $\begingroup$ See also this Mathematic SE post or a solution with mirror-symmetric pieces. $\endgroup$ Commented Mar 3, 2022 at 6:24

1 Answer 1

10
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This seems pretty simple

enter image description here
(each piece has 8 squares and one bend)

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