Partial Solutions and/or Statistics for Problems in Section A

A-1 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}.

A-2 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}.

A-3 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.

A-4 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

A-5 Congruent Groups.  {F, T}

A-7 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.

A-8 10-square 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}.

A-9 15-square 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.

A-10 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 - - - - - -

A-12 A Pentomino Farm.  The maximum possible area is 128.