PENTOMINOS Infrequently Asked Questions (iFAQ)

Question:

What are pentominos?


Answer:

Pentominos are a special case of the class of shapes called polyominos. A
polyomino of order n is a shape which can be made from n squares, stuck
together with edges coinciding and no overlapping. A polyomino which is the
same as another polyomino except that it is rotated or reflected is counted
as the same.

Pentominos are polyominos of order 5. There are 12 pentominos:
    ﾚｿ                    ﾚｿ ﾚｿ ﾚｿ
 ﾚｿ ｳｳﾚｿﾚｿﾚｿ  ﾚｿ  ﾚﾄｿ ﾚﾄﾄｿｳｳ ｳｳ ｳｳ ﾚﾄｿ ﾚﾄｿ
ﾚﾙﾀｿｳｳｳﾀﾙｳｳｳ  ｳﾀｿ ﾀｿｳ ﾀｿﾚﾙｳｳ ｳﾀｿｳﾀｿﾀｿﾀｿｳ ｳ
ﾀｿﾚﾙｳｳﾀﾄﾄﾙｳﾀﾄｿﾀｿﾀｿ ｳﾀｿ ｳｳ ｳﾀｿｳﾚﾙﾀｿｳ ｳﾚﾙｳﾚﾙ
 ﾀﾙ ｳｳ    ﾀﾄﾄﾙ ﾀﾄﾙ ﾀﾄﾙ ﾀﾙ ﾀﾄﾙﾀﾙ  ﾀﾙ ﾀﾙ ﾀﾙ
    ﾀﾙ
Also known by their letters:
  X  I  U   V   W   Z   T  L  Y  S   F  P

These can be arranged in many ways, as a jigsaw puzzle. As each of the 12
pentominos has 5 squares, there are 60 (=5x12) squares, which can be arranged
as a rectangle of 10x6 squares, 12x5 squares, 15x4 squares or 20x3 squares.
Note that 30x2 squares is not suitable because some of the pieces are 3 squares
high however they are rotated.

Note that the 5 tetrominos (4th order polyominos) have no rectangular
solutions, and are rather boring (although there is an interesting computer
game based on the 7 tetroids, which are the same as tetrominos except that
reflections are counted as different). There are 35 hexominos, which makes for
extremely long and complicated puzzles. Hence 5th order is the optimum size for
polyomino puzzles.


Question:

Apart from the rectangular solutions already described, what other pentomino
puzzles are there?


Answer:

(I am not certain that solutions to the following puzzles exist)

By using n of the 12 pieces (12>n>2) you can make a 5xn rectangle. Using 9
pieces you can make a 3x15 rectangle. Using 8 pieces you can make a 10x4
rectangle. Using 6 pieces you can make a 10x3 rectangle.

Using nine pieces, you can make "scale models" of each of the pentominos three
times as large as the originals. Using only four pieces, ten of the pentominos
can be made at twice the original size.

If the pentominos are made three dimensional, i.e. made out of 1x1x1 cubes
instead of 1x1 squares, there are 3 cuboid puzzles: 3x4x5, 2x5x6, 2x3x10.

If you use 10 pieces, you can make a 2x5x5 cuboid. If you use 9 pieces, you can
make a 3x3x5 cuboid. If you use 8 pieces, you can make a 2x4x5 cuboid. If you
use 6 pieces, you can make a 2x3x5 cuboid.

Using 8 pieces, you can make double size scale models of your 3D pentominos.

If you are extremely clever, or happen to belong to a higher space, you may be
able to arrange the pentominos into a 2x2x3x5 hypercuboid.

The only other solutions (in 2D) involve making stretched versions of the scale
models of the pentominos, 2xscale models of decominos (10th order polyominos)
and pentadecominos (15th order polyominos), stretched versions of the decomino
scale models, and using n pieces, 5n-ominos (this is the most general class of
puzzle). However, because these puzzles have so many variations I don't think
that they are very interesting.

You could also try playing pentominoes on other surfaces, such as a torus,
Mobius strip or Klein bottle.


Question:

How many rectangular solutions are there which use all the pieces?


Answer:

 10x6: 2339 solutions
 12x5: 1010 solutions
 14x4:  368 solutions
 20x3:    2 solutions
Total: 3719 solutions

(not counting rotations and reflections)


Question:

Which shapes appear most often between the prongs of the ﾚｿﾚｿ shape in the
solutions of 10x6 pentominos?                            ｳﾀﾙｳ
                                                         ﾀﾄﾄﾙ

Answer:

         10x6           12x5          15x4          20x3         Total
 ﾚｿ
ﾚﾙﾀｿ 1226 (52.42%)  462 (45.74%)  210 (57.07%)   2 (100.00%)  1900 (51.09%)
ﾀｿﾚｴ
ﾚﾁﾙｳ  345 (14.75%)  138 (13.66%)   93 (25.27%)   0 (  0.00%)   576 (15.49%)
ﾀｿﾚｴ
ﾚﾁﾙﾀｿ 221 ( 9.45%)   65 ( 6.44%)   13 ( 3.53%)   0 (  0.00%)   299 ( 8.04%)
ﾀﾄﾂﾄｴ
ﾚﾄﾙﾚｴ 194 ( 8.29%)   78 ( 7.72%)   12 ( 3.26%)   0 (  0.00%)   284 ( 7.64%)
ﾀﾂﾄﾙｳ 113 ( 4.83%)   74 ( 7.33%)    2 ( 0.54%)   0 (  0.00%)   189 ( 5.08%)
 ｳﾚﾄｴ
 ﾃﾙﾚﾙ  66 ( 2.82%)   66 ( 6.53%)    6 ( 1.63%)   0 (  0.00%)   138 ( 3.71%)
 ｳﾚｴ
ﾚﾁﾙﾃｿ  70 ( 2.99%)   44 ( 4.36%)   23 ( 6.25%)   0 (  0.00%)   137 ( 3.68%)
ﾀﾄｿｳｳ  57 ( 2.44%)   45 ( 4.46%)    0 ( 0.00%)   0 (  0.00%)   102 ( 2.74%)
 ﾚﾁﾙｳ
ﾚﾁﾂﾄﾙ
ｳ ﾀﾂｿ  40 ( 1.71%)   29 ( 2.87%)    9 ( 2.45%)   0 (  0.00%)    78 ( 2.10%)
ﾃﾄﾄﾙｳ   7 ( 0.30%)    9 ( 0.89%)    0 ( 0.00%)   0 (  0.00%)    16 ( 0.43%)
ﾃﾄﾄﾄﾁｿ
ﾀﾄﾄﾄﾄﾙ  0 ( 0.00%)    0 ( 0.00%)    0 ( 0.00%)   0 (  0.00%)     0 ( 0.00%)


Question:

How many solutions are there in which the ﾚﾄﾄﾄﾄｿ piece does not touch any of
                                          ﾀﾄﾄﾄﾄﾙ
the sides?


Answer:

For 10x6 solutions, there are 11:
ﾚﾄﾂﾂﾄﾄﾂﾄﾂﾄｿﾚﾄﾂﾂﾄﾂﾄﾄﾂﾄｿﾚﾄﾂﾄﾂﾂﾄﾄﾂﾄｿﾚﾄﾂﾂﾄﾄﾂﾄﾂﾄｿﾚﾄﾂﾂﾄﾂﾄﾄﾂﾄｿﾚﾄﾂﾄﾂﾂﾄﾄﾂﾄｿ
ｳﾚﾙﾀｿﾚﾙﾚﾙﾚｴｳﾚﾙﾀｿﾀｿﾚﾙﾚｴｳﾚﾙﾚﾙﾀｿﾚﾙﾚｴｳﾚﾙﾀｿﾚﾙﾚｴ ｳｳﾚﾙﾀｿﾀｿﾚｴ ｳｳﾚﾙﾚﾙﾀｿﾚｴ ｳ
ｳﾀｿﾚｴﾃｿｳﾚﾙｳｳﾀｿﾚｴﾚｴｳﾚﾙｳｳﾀｿﾃｿﾚｴｳﾚﾙｳｳﾀｿﾚｴﾃｿｳﾀｿｳｳﾀｿﾚｴﾚｴｳﾀｿｳｳﾀｿﾃｿﾚｴｳﾀｿｳ
ﾃﾂﾁﾙﾃﾙﾀﾁｴ ｳﾃﾂﾁﾙﾃﾙﾀﾁｴ ｳﾃﾂﾁﾙﾃﾙﾀﾁｴ ｳﾃﾂﾁﾙﾃﾙﾀﾁｿﾀｴﾃﾂﾁﾙﾃﾙﾀﾁｿﾀｴﾃﾂﾁﾙﾃﾙﾀﾁｿﾀｴ
ｳｳﾚﾄﾅﾄﾄﾄﾁﾂｴｳｳﾚﾄﾅﾄﾄﾄﾁﾂｴｳｳﾚﾄﾅﾄﾄﾄﾁﾂｴｳｳﾚﾄﾅﾄﾄﾄﾁﾂｴｳｳﾚﾄﾅﾄﾄﾄﾁﾂｴｳｳﾚﾄﾅﾄﾄﾄﾁﾂｴ
ｳﾀﾁｿﾀﾄﾂﾄﾄﾙｳｳﾀﾁｿﾀﾄﾂﾄﾄﾙｳｳﾀﾁｿﾀﾄﾂﾄﾄﾙｳｳﾀﾁｿﾀﾄﾂﾄﾄﾙｳｳﾀﾁｿﾀﾄﾂﾄﾄﾙｳｳﾀﾁｿﾀﾄﾂﾄﾄﾙｳ
ﾀﾄﾄﾁﾄﾄﾁﾄﾄﾄﾙﾀﾄﾄﾁﾄﾄﾁﾄﾄﾄﾙﾀﾄﾄﾁﾄﾄﾁﾄﾄﾄﾙﾀﾄﾄﾁﾄﾄﾁﾄﾄﾄﾙﾀﾄﾄﾁﾄﾄﾁﾄﾄﾄﾙﾀﾄﾄﾁﾄﾄﾁﾄﾄﾄﾙ
ﾚﾄﾄﾂﾄﾄﾂﾄﾄﾂｿﾚﾄﾄﾂﾄﾄﾂﾄﾄﾂｿﾚﾄﾄﾂﾄﾄﾂﾄﾄﾂｿﾚﾄﾄﾂﾄﾄﾂﾄﾄﾂｿ
ｳﾚｿﾃｿ ﾃｿﾚﾙｳｳﾚｿﾃｿ ﾃｿﾚﾙｳｳﾚｿﾃｿ ﾃｿﾚﾙｳｳﾚｿﾃｿ ﾃｿﾚﾙｳ
ﾃﾙﾀｴﾀﾂﾙｳｳﾚｴﾃﾙﾀｴﾀﾂﾙｳﾃｿｳﾃﾙﾃﾙﾀﾂﾙｳｳﾚｴﾃﾙﾃﾙﾀﾂﾙｳﾃｿｳ
ﾃｿﾚﾙﾚﾙﾚﾁｴｳｳﾃｿﾚﾙﾚﾙﾚﾁｴｳｳﾃｿﾀｿﾚﾙﾚﾁｴｳｳﾃｿﾀｿﾚﾙﾚﾁｴｳｳ
ｳﾃﾁﾄﾁﾄﾅｿﾃﾙｳｳﾃﾁﾄﾁﾄﾅｿｳﾀｴｳﾃﾄﾁﾁﾄﾅｿﾃﾙｳｳﾃﾄﾁﾁﾄﾅｿｳﾀｴ
ｳﾀﾄﾂﾄﾄﾙｳﾀｿｳｳﾀﾄﾂﾄﾄﾙｳﾀｿｳｳﾀﾄﾂﾄﾄﾙｳﾀｿｳｳﾀﾄﾂﾄﾄﾙｳﾀｿｳ
ﾀﾄﾄﾁﾄﾄﾄﾁﾄﾁﾙﾀﾄﾄﾁﾄﾄﾄﾁﾄﾁﾙﾀﾄﾄﾁﾄﾄﾄﾁﾄﾁﾙﾀﾄﾄﾁﾄﾄﾄﾁﾄﾁﾙ
ﾚﾄﾄﾂﾂﾄﾄﾄﾂﾂｿ
ｳﾚｿｳﾀｿﾚﾄﾙｳｳ
ﾃﾙﾀﾅｿﾀｴﾚﾄﾙｳ
ﾃｿﾚﾙﾃﾄﾁﾁﾄﾂｴ
ｳﾀﾅｿﾀﾂﾄﾂﾄﾙｳ
ｳ ｳﾀﾄﾁｿﾀﾄｿｳ
ﾀﾄﾁﾄﾄﾄﾁﾄﾄﾁﾙ

There is one 12x5 solution:
ﾚﾄﾂﾂﾄﾄﾂﾄﾂﾂﾄﾄｿ
ｳﾚﾙﾀｿﾚﾙﾚｴﾀﾄｿｳ
ｳﾀｿﾚｴｳﾚﾙﾀﾄｿｳｳ
ﾃﾄﾅﾙﾃﾁﾅﾄﾄﾄﾁﾅｴ
ｳ ﾀｿﾀｿﾀﾄﾂﾄﾄﾙｳ
ﾀﾄﾄﾁﾄﾁﾄﾄﾁﾄﾄﾄﾙ

There are no 15x4 solutions or 20x3 solutions with this property.


Question:

What is an isomorphism set, and how many isomorphism sets of each size are
there?


Answer:

For the purposes of this document, two solutions are isomorphic if a subset of
the pieces can be rotated or reflected to give a different solution. For
example, the only two solutions to 20x3 pentominos are isomorphic, because one
can be transformed to the other by rotating a central 7-piece section (LSFTWYZ)
(or, alternatively, switching the end pieces UXPIL and V over):

 ﾚﾄﾂﾂﾄﾄﾄﾄﾂﾄﾄﾂﾂﾂﾂﾄﾄﾄﾂﾂｿ ﾚﾄﾂﾂﾄﾄﾄﾄﾂﾂﾄﾂﾄﾄﾂﾂﾄﾄﾄﾂｿ
 ｳﾚﾙﾀﾂﾄﾂﾂﾙﾚﾄﾙｳｳﾀｿﾚﾄﾙｳｳ ｳﾚﾙﾀﾂﾄﾂﾄﾙﾃｿﾀｿﾚﾙﾀﾂﾄｿｳｳ
 ｳﾀｿﾚﾙ ｳﾀﾄﾁｿﾚﾙﾀｿﾀｴﾚﾄﾙｳ ｳﾀｿﾚﾙ ｳﾚﾄﾙﾀｿｳｳﾚﾄﾙﾚﾁﾙｳ
 ﾀﾄﾁﾁﾄﾄﾁﾄﾄﾄﾁﾁﾄﾄﾁﾄﾁﾁﾄﾄﾙ ﾀﾄﾁﾁﾄﾄﾁﾁﾄﾄﾄﾁﾁﾁﾁﾄﾄﾁﾄﾄﾙ

Again, for the purposes of this document, the isomorphism set of a given
solution is the set of solutions which can be made by rotating and reflecting
subsets of pieces, including the original solution. Note that a solution which
takes several rotations and/or reflections to transform it to the original
solutions is also included. Hence, isomorphism sets are discrete - the
isomorphism set does not depend on which of the solutions in the set you start
with.

In the solutions of 10x6 pentominos, isomorphism sets of sizes 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 24, 30 and 44 (ismorphism sets of sizes
which are not powers of 2 occur because the subsets can be overlapped.) The
isomorphism sets break down as follows (the table shows numbers of sets, not
numbers of solutions):

Sizeｳ10x6 ｳ12x5 ｳ15x4 ｳ20x3 ｳTotal
ﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄ
  1 ｳ 578 ｳ 183 ｳ  71 ｳ   0 ｳ 832
  2 ｳ 327 ｳ 119 ｳ  37 ｳ   1 ｳ 484
  3 ｳ  55 ｳ  32 ｳ  12 ｳ   0 ｳ  99
  4 ｳ  87 ｳ  30 ｳ  12 ｳ   0 ｳ 129
  5 ｳ  11 ｳ   6 ｳ   0 ｳ   0 ｳ  17
  6 ｳ  27 ｳ  13 ｳ  12 ｳ   0 ｳ  52
  7 ｳ   3 ｳ   5 ｳ   1 ｳ   0 ｳ   9
  8 ｳ  12 ｳ   8 ｳ   2 ｳ   0 ｳ  22
  9 ｳ   1 ｳ   3 ｳ   0 ｳ   0 ｳ   4
 10 ｳ   3 ｳ   4 ｳ   0 ｳ   0 ｳ   7
 11 ｳ   2 ｳ   1 ｳ   0 ｳ   0 ｳ   3
 12 ｳ   3 ｳ   1 ｳ   1 ｳ   0 ｳ   5
 13 ｳ   0 ｳ   1 ｳ   0 ｳ   0 ｳ   1
 14 ｳ   1 ｳ   0 ｳ   0 ｳ   0 ｳ   1
 15 ｳ   1 ｳ   1 ｳ   0 ｳ   0 ｳ   2
 18 ｳ   2 ｳ   0 ｳ   0 ｳ   0 ｳ   2
 20 ｳ   0 ｳ   1 ｳ   0 ｳ   0 ｳ   1
 24 ｳ   1 ｳ   0 ｳ   0 ｳ   0 ｳ   1
 28 ｳ   0 ｳ   1 ｳ   0 ｳ   0 ｳ   1
 30 ｳ   1 ｳ   0 ｳ   0 ｳ   0 ｳ   1
 32 ｳ   0 ｳ   0 ｳ   1 ｳ   0 ｳ   1
 44 ｳ   1 ｳ   0 ｳ   0 ｳ   0 ｳ   1
ﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄ
Tot ｳ1116 ｳ 409 ｳ 149 ｳ   1 ｳ1675


Question:

In 10x6 pentominos, are there any solutions with diagonal lines?


Answer:

Of course, because pentominos are made of squares, true diagonal lines are
impossible, but you can get fairly close with some of the isomorphisms of this
solution (shown here split in two for clarity):

ﾚﾄﾄﾂﾂﾂﾄﾄﾄﾄｿ
ｳﾚｿｳｳﾀﾄﾂﾄﾂﾙ
ﾃﾙﾀｴｳﾚﾄﾙﾚﾙ ﾚｿ
ﾃｿﾚｴﾀﾅﾄﾂﾙ ﾚﾙｳ
ｳﾃﾙﾀﾂﾙﾚﾙ ﾚﾁｿｳ
ｳﾀﾄｿｳﾚﾙ ﾚﾁｿｳｳ
ﾀﾄﾄﾁﾁﾙ ﾚﾙ ｳﾀｴ
       ﾀﾄﾄﾁﾄﾙ


Question:

What is a 4-point, and how many solutions are there with each number of
4-points?


Answer:

For the purposes of this document, a 4-point is a point where 4 different
pentominos meet, for example the pieces T, S, L and W in the answer to the last
question (shown as ﾅ in this file). The number of 4-points per solution is a
number between 0 and 4 inclusive for 10x6 and 12x5 pentominos. The distribution
is as follows:

4-points ｳ  10x6  ｳ  12x5  ｳ  15x4  ｳ  20x3  ｳ  Total
ﾄﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄ
    0    ｳ   563  ｳ   325  ｳ   238  ｳ     2  ｳ  1128
    1    ｳ  1102  ｳ   487  ｳ   117  ｳ     0  ｳ  1706
    2    ｳ   547  ｳ   180  ｳ    13  ｳ     0  ｳ   740
    3    ｳ   118  ｳ    18  ｳ     0  ｳ     0  ｳ   136
    4    ｳ     9  ｳ     0  ｳ     0  ｳ     0  ｳ     9
ﾄﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄﾄ
Average  ｳ   1.11 ｳ   0.89 ｳ   0.39 ｳ   0.00 ｳ   0.98


Question:

How can pentominos be extended?


Answer:

As mentioned above, there are 35 hexominos. This table gives numbers of
polyominos:

Order ｳ Number
ﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄ
  1   ｳ   1
  2   ｳ   1
  3   ｳ   2
  4   ｳ   5
  5   ｳ  12
  6   ｳ  35

You might want to try using all the polyominos between two orders, for example
the set of polyominos between orders 2 and 5 (inclusive) has 20 pieces with
a total of 88 squares (making a nice 8x11 rectangle).

The class of shapes made by sticking cubes together face-to-face are called
polycubes. The 3D pentominos are a subset of the 5th order polycubes or
pentacubes. The following table gives the numbers of pentacubes:

Order ｳ Number
ﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄ
   1  ｳ   1
   2  ｳ   1
   3  ｳ   2
   4  ｳ   7
   5  ｳ  23

Because 5x23 does not decompose into a cuboid (both 5 and 23 being prime) it
may be better to count mirror-image reflections as different. If this is done
the following table is obtained. These are chiral polycubes:

Order ｳ Number
ﾄﾄﾄﾄﾄﾄﾅﾄﾄﾄﾄﾄﾄﾄ
   1  ｳ   1
   2  ｳ   1
   3  ｳ   2
   4  ｳ   8
   5  ｳ  30

Returning to two dimensions for a moment, we are not limited to squares. We
could use rectangles of the same size and shape (rotational symmetry order 2),
triangles (rotational symmetry order 3) or hexagons (rotational symmetry order
6). There are many shapes and sizes of solution to these puzzles. I am not
sure if shapes other than cubes can be used in 3 dimensions.

Of course, there is no need to stick to two or three dimensions, shapes of
four or even more dimensions could be used, however this could cause
visualisation problems.