been stuck on solving/proving the following puzzle:

You need to make a hole in the wall, so that a 1 meter line can pass it through the hole at all angels, find a shape with minimum surface area that would satisfy the above conditions ?

  • $\begingroup$ A cone maybe, not sure $\endgroup$ – Kushal Bhuyan Nov 11 '15 at 7:20
  • $\begingroup$ Isn't this an unsolved problem? $\endgroup$ – Christopher Carl Heckman Nov 11 '15 at 7:33
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    $\begingroup$ Isn't this the Kakeya needle problem then? en.wikipedia.org/wiki/Kakeya_set $\endgroup$ – MarsOneRover Nov 11 '15 at 7:48
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    $\begingroup$ Doesn't the aforementioned Wikipedia article show that the minimum area (measure) is zero? $\endgroup$ – copper.hat Nov 11 '15 at 7:56
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    $\begingroup$ @copper.hat yes. $\endgroup$ – MarsOneRover Nov 11 '15 at 7:59

Take an equilateral triangle with sides of length $1$, and then use each vertex as the center of a circular arc passing through the other two vertices:

enter image description here

This is at least a contender, with area $3\cdot\frac{\pi}{6}-2\cdot\frac{\sqrt3}{4}$, less than a circle of diameter $1$ or a quarter-circle of radius $1$, which are other convex shapes meeting the description. This shape may be the winner if you require a convex shape, but I have no ideas for proving it.


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