Polyomino question

[Jack Brennen]

> For n = 5, the best I can do is 9:
>   O
> OOOOO
> OOO
> I don't know if this is unique; I would be very surprised if 8 is possible.

It's not unique.  See also the somewhat similar:

     O
  OOOOO
   OOO

Eight is not possible.  First you need to have five in a row:

  OOOOO

Then consider the "X" pentomino:

   O
  OOO
   O


In order to overlay this onto the
five-in-a-row without exceeding a total area of 8, you have three
basic possibilities (ignoring reflections and rotations):

   O        O         O
  OOOOOO   OOOOO    OOOOO
   O        O         O

Now consider the "W" pentomino:

    O
    OO
     OO

No matter how this is oriented with respect to any of the three
basic possibilities, it can only overlap two squares of the central
row and at most one square not on the central row -- meaning that
two of its five squares must be added to the enclosing polyomino.
This pushes the minimum area to 9, and since that has already been
shown to be possible, the answer for n = 5 is shown to be exactly 9.


   Jack