Partial Solutions and/or Statistics for Problems in Section A
A1 3 x 5 Rectangles. The other five solutions use the following groups of pieces: {P, U, V}, {N, P, U}, {P, U, Y}, {F, P, U}, and {L, P, V}.
A2 4 x 5 Rectangles. (b) There are 2 solutions, one for the P and the U. (c) One solution uses {L, P, V, Y}. (d) One solution uses {F, L, U, V}.
A3 5 x 5 Squares. (b) Solutions exist for the F, P, T, U, V, W, and Z. If the rectangle does not contain the I, then solutions only exist for the P, U, W, and Z.
A4 Many Rectangles.
Size of Rectangle 
Number of Solutions 
1 x 5 
1 
3 x 5 
7 
2 x 10 
2 
4 x 5 
50 
5 x 5 
107 
3 x 10 
145 
5 x 6 
541 
5 x 7 
1396 
4 x 10 
2085 
5 x 8 
3408 
3 x 15 
201 
5 x 9 
5902 
5 x 10 
6951 
5 x 11 
4103 
A5 Congruent Groups. {F, T}
A7 Duplications. The number of solutions is given in parenthesis after each piece: F (0), I (2), L (2), N (7), P (48), T (1), U (5), V (0), W (4), X (0), Y (2), Z (6). Duplications which have solutions that use its smaller counterpart: I, L, N, P, U, and Z. Duplications which have solutions that do not use its smaller counterpart: N, P, T, U, W, Y, and Z.
A8 10square Simultaneous
Solutions. (a)
The three groups are {N,Y}, {P,Z},
and {F,T}. (b) The three groups are {I,L}, {N,V}, and {T,Y}. (c)
The three groups are {L,N}, {V,Z},
and {P,U}. (d) There are two solutions. The first is {L N}, {W,Z}, and {P,T}. The second is {L,N},
{W,Z}, and {P,Y}.
A9 15square Simultaneous Solutions. (a)
One solution uses {N,V,Z}, {P,W,Y},
and {L,U,X}. (b) One solution uses {T,V,W},
{I,L,P}, and {F,U,Y}. There may be others.
A10 Simultaneous Rectangles. The number
of solutions for each pair of simultaneous rectangles are given in the
table below. Blank spaces indicate
either that either the construction is impossible (in that it would require
more than 12 pieces) or that the number is given elsewhere in the table.

1 x 5 
3 x 5 
2 x 10 
4 x 5 
5 x 5 
3 x 10 
5 x 6 
3 x 5 
7 
5 





2 x 10 
0 
0 
0 




4 x 5 
36 
29 
0 
28 



5 x 5 
35 
25 
0 
60 
12 


3 x 10 
82 
10 
1 
25 
6 
0 

5 x 6 
205 
67 
0 
133 
20 
0 
2 
5 x 7 
398 
84 
0 
22 
1 
 
 
4 x 10 
621 
9 
0 
5 
 
 
 
5 x 8 
775 
29 
0 
0 
 
 
 
3 x 15 
19 
0 
 
 
 
 
 
5 x 9 
780 
1 
 
 
 
 
 
5 x 10 
416 
 
 
 
 
 
 
5 x 11 
112 
 
 
 
 
 
 
A12 A Pentomino Farm. The maximum possible area is 128.