7
$\begingroup$

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

$\endgroup$
6
  • $\begingroup$ Does it count as the same shape if you use its mirror image? $\endgroup$ – Tweakimp Jan 4 '18 at 16:16
  • $\begingroup$ See 5 geometric shapes, all touching each other for why five is impossible. $\endgroup$ – Joseph O'Rourke Jan 4 '18 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. $\endgroup$ – Serge Bouwens Jan 4 '18 at 16:42
  • $\begingroup$ Does the shape have to be continuous? $\endgroup$ – corsiKa Jan 4 '18 at 20:41
  • $\begingroup$ @corsiKa, yes. The 4-colour theorem that this problem references doesn’t apply otherwise. $\endgroup$ – Bass Jan 5 '18 at 8:21
9
$\begingroup$

This seems pretty simple

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.