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At Right Angles | Vol. 3, No. 2, July 2014 49
in the classroom
Gautham Dayal
Keywords: Pentomino, puzzle, spatial, congruent, transformation, translation,
rotation, reflection
Pentomino puzzles were invented (or discovered) in the early
1900s by Henry Dudeney, an English inventor of puzzles
(who is unfortunately not as well known as he should be).
They then appeared sporadically in recreational mathematical
magazines in the 1930s and 1940s. Interest in them was revived
when Solomon Golomb wrote about them in the 1950s. They were
popularized by Martin Gardener in his column MATHEMATICAL
GAMES that appeared in the Scientific American as well as in his
books on recreational mathematics. While they are a valuable
educational resource in their avatar as puzzles, they can also be
used effectively to build spatial intuition.
So what are pentominoes?
Take five identical squares. Now place them one at a time so that
(apart from the first square) each square touches at least one of the
squares already placed along a complete edge. The various shapes
that can result are known as pentominoes ('penta' meaning five and
'mino' to suggest that they are related to dominos).
PENTOMINOES
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50 At Right Angles | Vol. 3, No. 2, July 2014
It turns out that it is possible to make twelve such
shapes using five squares. These twelve shapes
make up a set of pentominoes. In order to talk
about the various pieces, it helps to name them.
They are usually denoted by alphabets as in
Figure 1.
Pentominoes are often used in the same way that
tangrams are. So, a pentomino puzzle is a shape
drawn on a sheet of paper. Solving the puzzle
requires one to place the 12 (or sometimes fewer)
pentomino pieces on a plane surface to form
a shape similar to the one given on the sheet.
A Google search gives a large number of sites
devoted to such puzzles; the more useful of these
sites grades the puzzles in order of difficulty.
In this article, we are not as concerned with
difficult pentomino constructions (which is a
natural and worthwhile place to aim to go once
we are familiar with these pieces), as with ways
in which pentominoes can be used in a classroom
with a group of students to develop geometric
intuition. However we give some examples of
these construction puzzles at the end of the article,
including some references.
Finding pentominoes – an activity
A worthwhile question to investigate with a
group of students is that of finding all possible
pentominoes and we sketch here one way to do
this. While this can be done with paper and pencil,
it sometimes helps to have a number of squares
cut out of card (or other suitable material).
One begins with the definition of pentominoes
and what it can mean for two squares to touch
correctly. We observe that there is only one
monomino (the square). We see different ways to
add a square to this monomino and see that all the
possibilities end with the same result (a domino).
To move to trominoes, one sees the various ways
to join a square onto a domino. While there are
many ways of doing this, there are only two
distinct ones (‘I’ and ‘L’; see Figure 2). This is
a good time to use words such as ‘congruent’,
‘rotate’ and ‘reflect’.
Now add a square to the two possible trominoes
to get tetrominoes. At this stage, it may be noticed
that some pairs of shapes are not identical if only
translations and rotations are allowed, but can
be superposed if reflection is permitted (that
is, flipping over). We find that there are five
tetrominoes; see Figure 3.
Figure 1: The twelve pentominoes
Figure 2: From domino to tromino
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At Right Angles | Vol. 3, No. 2, July 2014 51
Add another square to obtain pentominoes.
Students will need to keep track of the different
ways in which a square can be added and the
various congruences which occur to find the full
list of possible pentominoes. Using the symmetries
of each tetromino also makes this enumeration
more efficient.
Getting to know the pentominoes
There are a couple of ways to develop a sense of
familiarity with the pieces.
• One way which works well with younger
students is to give them a set of pentominoes
and let them construct any figure they like
(this strategy also works with other dissec- tion puzzles like the tangram): figures of
animals, houses, vehicles, etc. They could be
given sheets of squared paper to copy out the
silhouette of the figures they have built. They
can also be asked to mark out the positions of
the pieces on their drawings. Often this gives
rise to arrangements that can later be used as
puzzles for other children.
• While it is commonly thought that a pen- tomino puzzle must be made using all the
twelve pieces, this is not necessary. One could
start off with puzzles using just two pieces
and build from there. For examples of such
puzzles, please look at the CIMT website from
which the Space-filling problems in Figure 4
have been taken.
While these are particularly suited to develop
spatial abilities in children, they are also suitable
to train children to think methodically and
develop reasons for eliminating possible options.
Figure 3: The five tetrominoes
Figure 4: Pentomino puzzles from the CIMT website (http://www.cimt.plymouth.ac.uk/)