S{{{{]]]g
Sq{{q]ggg
Sq{{q]ggg
Sqqqq]]g?
SSSI ]???
SIIIIIII?
55555!???
5++55!!!!
++++++!!!
 
Balloon Balance problems, by Bob Kraus.
****************************************************
Puzzle #1:
Weights: -5,-4,-3,-2,-1,+1,+2,+3,+4,+5

             O-O-|-|-|-|-|-|-|-O
                 |
               O-|-|-|-|-O
                   |
             |-|-O-|-|-|-|-O
             |
   O-|-|-O-|-|-O

****************************************************
Puzzle #2:
Weights: -5,-4,-3,-2,-1,+1,+2,+3,+4,+5

     O-|-|-|-|-|-|-|-|-O
               |
             |-|-O-O-|-|-|-O
             |
             |-|-O-|-O
               |
   O-|-|-|-|-O-|-|-|-|-O

****************************************************
Puzzle #3:
Weights: -5,-4,-3,-2,-1,+1,+2,+3,+4,+5

   O-|-|-|-|-|-O-|-|-|-O
             |
             |-|-|-|-|-O-|-|-O
                         |
                       O-|-|-|-|-O
                           |
                   O-|-|-|-|-|-|-|-O-|-|-|-O

****************************************************
Puzzle #4:
Weights: -5,-4,-3,-2,-1,+1,+2,+3,+4,+5

     O-|-|-O-|-|-|-O
             |
     O-|-|-|-|-|-O
           |
       O-|-|-|-|-|-|-|-|-|-O
                 |
       O-|-|-|-|-|-|-|-O-O

****************************************************
Puzzle #5:
Weights: -5,-4,-3,-2,-1,+1,+2,+3,+4,+5

   O-|-O-|-|-O-|-|-O
           |
           |-|-|-|-|-|-O
             |
           |-|-|-|-|-|-|-|-|-|-|-|-O
           |
   O-|-|-|-|-|-|-O-O-O

****************************************************
Puzzle #6:
Weights: -6,-5,-4,-3,-2,-1,+1,+2,+3,+4,+5,+6

         O-|-|-|-O-|-|-|-|-O
                   |
         O-|-|-O-|-|-|-|-O
                 |
   O-|-|-|-O-|-|-|-|-|-O
             |
     O-|-|-|-|-O-O

****************************************************
Puzzle #7:
Weights: -6,-5,-4,-3,-2,-1,+1,+2,+3,+4,+5,+6

     O-O-|-|-|-|-|-O
           |
   O-|-|-O-|-|-|-|-|-|-|-O
               |
       O-|-O-|-|-|-|-|-|-|-|-|-O
         |
   O-|-|-|-O-|-|-O

****************************************************
Puzzle #8:
Weights: -6,-5,-4,-3,-2,-1,+1,+2,+3,+4,+5,+6

    O-|-O-|-|-|-O
              |
        O-O-|-|-|-O
            |
      O-O-|-|-O-|-|-|
                    |
      O-|-|-|-|-|-O-|-|-|-O

****************************************************
Puzzle #9:
Weights: -6,-5,-4,-3,-2,-1,+1,+2,+3,+4,+5,+6

    O-|-|-|-|-|-O-|-|-|-O
                  |
        O-O-|-|-|-|-|-O
              |
  O-|-O-|-|-|-|-|-O-|
                    |
            O-|-|-O-|-|-|-|-|-O

****************************************************
Puzzle #10:
Weights: -6,-5,-4,-3,-2,-1,+1,+2,+3,+4,+5,+6

   O-O-|-|-|-|-|-O
         |
     O-|-|-|-|-|-O-O
           |
     O-|-|-|-O-|-|-|-|-|-O
       |
   O-|-|-O-|-O

****************************************************
michael reid
andrew is correct; there are only six rectifiable heptominoes:
the straight one, and those five listed on my page.  in fact,
the same goes for all sizes up to and including 9.  i.e. the only
rectifiables are those listed, and rectangular polyominoes.

at size 10, things get more interesting.  there are 3 "open" cases:

         *
 **     **
***     ***     ***
*****   ****   *******

but some may have been settled by others.  the second was supposedly
eliminated (i.e. proved to be non-rectifiable) by ingo wrede.
i was never convinced by his proof (partially because it's in german),
but helmut postl told me that the proof "can be made to work".
in other words, there are a lot of details to check carefully,
and presumably he's done that.  i'm also told that solomon golomb
has shown that the third is non-rectifiable, but i haven't seen the
proof.

if anyone can prove that these are non-rectifiable, i'd really like
to see the proof.  or, if you can prove that any are rectifiable,
i'd like to see that proof, too!

for size 11, there are two open cases:

*        *
*****   ****
*****   ******

and i think the first is likely to be rectifiable.  it won't have a
small rectangle though; according to my computer program, any rectangle
will require at least 544 pieces!  i don't know about the second,
but i've checked all rectangles up to some fairly big size.

for size 12, there are seven open cases:

                            **      *
  *        **       ***    ***     **      * ***      **
 *** *     ****    ****    ***    ****     * *     **  * *
*******   ******   *****   ****   *****   ******    ******

the first one has an easy half-strip tiling.  the last two are
probably not rectifiable, but i didn't find an easy proof,
maybe i'll try again.

for size 13, there is one open case:

**
*****
******

if anyone can settle any of these, please let me know!

mike