The 10th Pentacube Contest 1999 and CFF Contest 37.   Torsten Sillke, 1999-04-19
 
  Build the figure "Pentacube Frame on a Board" with all 29 pentacubes.
 
                             Each number says, how
                             many cubes are stacked.
  1 1 1 1 1 1 1 1 1 1 1
  1 1 1 1 1 1 1 1 1 1 1
  1 1 2 2 2 2 2 2 2 1 1
  1 1 2 1 1 1 1 1 2 1 1
  1 1 2 1 1 1 1 1 2 1 1
  1 1 2 1 1 1 1 1 2 1 1
  1 1 2 1 1 1 1 1 2 1 1
  1 1 2 1 1 1 1 1 2 1 1
  1 1 2 2 2 2 2 2 2 1 1
  1 1 1 1 1 1 1 1 1 1 1
  1 1 1 1 1 1 1 1 1 1 1

Dissect the figure in the following way:

  a a a a d d d d d d d
  a a a a d d d d d d d
  a a a a d d d d d d d
  a a a a d d d d d d d
  a a a a d x c c c c c
  a a a a x x x c c c c
  a a a a a x b c c c c
  b b b b b b b c c c c
  b b b b b b b c c c c
  b b b b b b b c c c c
  b b b b b b b c c c c

There are 18683 possible subsets (26701 solutions) for one corner.
There is only one partition and upto a (11,30) swap a unique solution.
- 13 20 21 32 33 41 72
- 12 31 42 61 70 71 80
- 10 22 34 35 40 82 90
- 11 30 36 37 51 60 81

The solution of the four corners
  .  .  .  .  .  . 32
 41 41 41 33 33 32 32
 13 20 33 33 21 32 72
 13 20 20 20 21 72 72
 13 13 13 20 21 21 72
  .  .  .  .  .  .  .
  .  . 41  .  .  .  .
  .  . 41 33 21 32 72
  .  .  .  .  .  .  .
  .  .  .  .  .  .  .

  .  .  .  .  .  . 70
 71 71 71 61 70 70 70
 80 71 61 61 31 70 42
 80 80 80 12 31 31 42
 80 12 12 12 12 31 42
  .  .  .  .  .  .  .
  .  . 71  .  .  .  .
  .  . 61 61 31 42 42
  .  .  .  .  .  .  .
  .  .  .  .  .  .  .

  .  .  .  .  .  . 40
 90 90 82 82 82 40 40
 90 34 34 82 35 40 22
 90 90 34 35 35 40 22
 10 10 10 10 10 22 22
  .  .  .  .  .  .  .
  .  . 34  .  .  .  .
  .  . 34 82 35 35 22
  .  .  .  .  .  .  .
  .  .  .  .  .  .  .

  .  .  .  .  .  . 81
 11 51 51 51 37 37 81
 11 30 51 36 37 81 81
 11 30 30 36 36 60 60
 11 11 30 30 60 60 60
  .  .  .  .  .  .  .
  .  . 51  .  .  .  .
  .  . 36 36 37 37 81
  .  .  .  .  .  .  .
  .  .  .  .  .  .  .

I found this figure in 1992. When I sent it E. K"unzell, he told me
that he had already selected the contest problems for several years
and this figure will become the jubilee contest 10 problem. Then it
seems that there would not be a 10th contest. So I was lucky to find
indeed the problem he selected so long ago. 

Much luck:
Of the following 6 dissections into like corners only the last one
is possible. Luckily I tried it at second.

             1            1 1          1 1 2        1 1 2 1     1 1 2 1 1
1 1 2 1      1 1 2 1      1 1 2 1      1 1 2 1      1 1 2 1     1 1 2 1
1 1 2 1 1    1 1 2 1 1    1 1 2 1 1    1 1 2 1 1    1 1 2 1 1   1 1 2 1
1 1 2 1 1    1 1 2 1 1    1 1 2 1 1    1 1 2 1 1    1 1 2 1     1 1 2 1
1 1 2 2 2    1 1 2 2 2    1 1 2 2 2    1 1 2 2      1 1 2 2     1 1 2 2
1 1 1 1 1    1 1 1 1 1    1 1 1 1      1 1 1 1      1 1 1 1     1 1 1 1
1 1 1 1 1    1 1 1 1      1 1 1 1      1 1 1 1      1 1 1 1     1 1 1 1