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Pentominoes
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Pentominoes

Pentominoes are the most diverse puzzles I have ever had the pleasure to use.  They can be used as a kids puzzle or to baffle the most accomplished mathematicians.  Pentominoes provide intriguing puzzles, interesting patterns, exciting games and they are effective teaching tools in the classroom because:

  1. The activities nurture a nonanxious attitude towards mathematics and science.
  2. They promote an atmosphere of cooperation.
  3. They support the development of the problem-solving process.
  4. They are an excellent source of spatial-ability skill exercises.
  5. They can be used to introduce students to elementary number theory.
  6. Practicing these activities can give students positive perceptions about the world of mathematics and science.
  7. Pentominoes furnishes the means for initiating discussions about many mathematical concepts.
  8. They serve as concrete representations that help ease students'understanding of abstract notions.

Kids learn best from hands on activities that they enjoy.  For more information about the learning process read Princeton University’s holistic approach to learning, the 70/20/10 theory.

Pentominoes are shapes that use five square blocks joined together with at least one common side.  We sell two types of sets, Pentominoes 3D and Pentominoes Plus.  There are twelve unique shapes that are mathematically possible using this criterion.  These shapes make up a set of pentominoes by using the following two rules:

  1. If one shape can be rotated to look like another, the two shapes are not considered to be different.
  2. If one shape can be flipped to look like another, the two shapes are not considered to be different.

Each of the twelve pentominoes shapes in the set are named T, U, V, W, X, Y, Z, F ,L ,I ,P , and N.  As a mnemonic device, you can remember these as the end of the alphabet (TUVWXYZ) and the word FILiPiNO. 

The labeling of the N and Z as shown is often reversed.  Since there are twelve distinct pentomino shapes with each covering five squares, their total area is sixty squares.  If you count reflections and rotations, there are 63 shapes: 8 possibilities of rotation and reflection for the asymmetrical pentominoes F, L, N, P, and Y; 4 possibilities for the mirror-or point-symmetrical pentominoes T, U, V, W, and Z; 2 possibilities for the I and 1 for the completely symmetrical X.

History

The first pentomino problem was written by the great English inventor of puzzles, Henry Ernest Dudeny.  It was published in 1907 in the Canterbury Puzzles.  However, the discovery that there are twelve distinctive patterns that can be formed by five connected stones on a Go board is quite ancient.  Go is an old Chinese game which later spread to Japan.  It is played by placing black and white stone markers on a board, and an ancient master of that game is attributed with making the observation of the twelve distinct patterns.  Later under the heading of "dissection problems," extensive literature on the subject of pentominoes appeared in the 1930s and 1940s in the Fairy Chess Review, a British puzzle journal.  In 1953 SolomonW. Golomb gave a speech at the Harvard Mathematics Club in which he authored the name pentomioes.  About a year later this talk was published in American Mathematical Monthly.  In the May 1957 issue of ScientificAmerican,  Martin Gardner reprinted some of this material, thus bringing pentominoes to the attention of public.  These publications generated excitement.  In1965, Solomon W. Golomb's book Polyominoes appeared.  Martin Gardner included several chapters on pentominoes in his books one being Mathematical Puzzles and Diversions.  Many people found pentominoes addictive, most recovered, others (including me) did not.  In 2004 Chasing Vermeer, an art history mystery by Blue Balliett, featured pentominoes in an ingenious plot twist.  This was a Harry Potter styled mystery in which pentominoes 3D was used to solve the mystery.  This revived interest in pentominoes with the main focus being the Pentominoes 3D set.  This addition to the puzzles diversity brings the book into a situational perspective that involves the reader.  Therefore the puzzle complements the experience of the book while the book brings the puzzle to life.  In 2005 its sequel, The Wright 3 became available.

Pentominoes Puzzles

There are several ways to place all twelve different pentominoes on an 8 x 8 board, with four squares always left over. By artistically specifying the positions ofthe four extra squares, many interesting patterns can be created.  Another evident possibility is to require that the four extra squares form a 2 x 2 area (a square tetromino) in a specified position on the board. This placement leads to a remarkable theorem, which can be proved by using only three constructions: Wherever on the checkerboard a square tetromino is placed, the rest of theboard can be covered with the twelve pentominoes.

Another problem is to determine the least number of pentominoes which will span the checkerboard.  In other words, some of the pentominoes are placed on the board in such a way that none of the rest can be added.  Although there are several different configurations for solving this puzzle, the minimum number is five pentominoes.

Other patterns include forming the configurations:

  1. 6 x 10 rectangle that can be made 2339 different ways.
  2. 5 x 12 rectangle that can be made 1010 different ways.
  3. 4 x 15 rectangle that can be made 368 different ways. 
  4. 3 x 20 rectangle that can be made 2 different ways.

Each of the above problems creates rectangles using all twelve pentominoes.  The 3 x20 is the most difficult to derive.  There is only one unique solution, except for the possibility of rotating the central portion by 180 degrees.  I should note that the other rectangles are far from easy.  Smaller rectangles such as 3 x 5, 4 x 5, 5 x 6, etc. can also be made in many different ways.  There are also numerous ways to build cross or bridge constructions.

R.M. Robinson, professor of mathematics at the University of California at Berkley, has suggested another intriguing construction with pentominoes, which he calls the "triplication problem."  Given a pentomino, use nine of the other pentominoes to construct a scale model that is three times as high and three times as wide as the given piece.  Constructions of all twelve of the pentominoes are possible.  When you purchase from us, we include solutions for these configurations.

Pentominoes in the Classroom

Not only do pentominoes have a wide appeal among mathematical-recreations fans, they also serve as "enrichment" material in school mathematics programs.  There are at least five good reasons to incorporate pentominoes in the classroom:

  1. Pentominoes nurture a non-anxious attitude toward mathematics and science.
  2. They  promote an atmosphere of cooperation.
  3. They support development of the problem-solving process.
  4. They provide spatial-ability skill exercises.
  5. They introduce students to elementary number theory.

Through various studies, it has been discovered that students with anxieties toward mathematics do not perform as well as non-anxious students on achievement tests.  The informal geometry that is ingrained in pentomino discovery is an excellent non-threatening activity with which to teach students.  Pentominoes can also be used to examine the concepts of congruence, similarity, transformations (flips,turns, and slides), tessellations (tilting), perimeter, area, and volume in a relaxed atmosphere.

The value of cooperation can also be demonstrated through the use of pentominoes.  The process of working together in small groups on pentomino problems promotes a desire among students to become involved in more formal cooperative learning activities in the future.

Pentomino activities provide good practice using the four-step problem solving process which includes:

  1. Understanding the problem.
  2. Devising a plan.
  3. Carrying out a plan.
  4. Checking the work.

Skill in representing, transforming, generating, and recalling symbolic, nonlinguistic information is referred to as spatial ability.  Mental rotation, one type of spatial ability, requires students to rotate a two-or three-dimensional figure rapidly and accurately.  Another type of spatial ability, (spatial visualization) demands students demonstrate an ability which involves complicated, multi-step manipulations of spatially presented information.  Spatial-ability skills have been found to correlate with mathematics and science achievement, and the ultimate goal of  practicing spatial-ability skills is for students to be able to mentally manipulate figures.  For additional  information see Pentominoes and Art.

For those of you interested in learning creative uses for pentominoes in the classroom, you might try Polyominoes : a guide to puzzles and problems in tiling by George E. Martin. Publisher, [Washington, D.C.] : Mathematical Association of America, c1991.

Number Theory

Provoking questions can be asked about the family of polyominoes.  Pentominoes are the most popular of the polyominoes because twelve pieces is a number small enough to be manageable yet large enough to provide diversity.  The unique set of biominoes (shapes consisting of two squares joined together by at least one common side) can easily be discovered; there is only one unique biomino (Dominoes).  The unique set of triominoes (shapes consisting of three squares joined together by at least one common side) can be quickly found; there are only two triominoes.  Identifying the unique set of quadominoes (ortetrominoes) (shapes consisting of four squares joined together by at least one side) is also quite simple; there are five quadominoes.  Even so, discovering the number of unique polyominoes is truly an exercise in analyzing number patterns. Although generating the number of unique polyominoes, uni through hex (1-6), is relatively simple, one should not be misled. Quite frequently, novice number theorists prematurely conclude that since a pattern holds for three or more cases, it will hold for an infinite number of cases; this is not true for the family of polyominoes.  Even though computers have been programmed to semi-concretely generate the shapes of polyominoes, at this time, computers still are not able to be programmed to symbolically predict the number of unique polyominoes using a formula.  The world of mathematics is anxiously awaiting its discovery.  For those of you that may be wondering, tetris was inspired by pentomino puzzles, although it uses four-block tetrominoes.

Pentominoes Games

Besides being an intriguing puzzle, the placement of pentominoes on a checkerboard also makes it an exciting competitive game of skill.  Played by one, two, or three players, the object of the game is to be the last player to place a pentomino piece on the checkerboard.  Players take turns choosing a piece and placing it on the board.  The pieces must not overlap or extend beyond the boundary of the board, but they do not have to be adjacent.  The game will last at least five, and at most twelve, moves, can never result in a draw, has more openings than chess, and will fascinate players of all ages.  Although it is difficult to advise what strategy should be followed, there are two valuable strategic principles.  First, try to move in such a way that there will be room for only an even number of pieces.  (This applies only when there are two players.)  Second, if a player is not able to analyze the situation, he should do something to complicate the placement so that the next player will have even more difficulty analyzing it than he did.

Conclusion

Pentominoes provide intriguing puzzles, interesting patterns, and exciting games. They can also be used as effective teaching tools in the classroom. Pentomino activities nurture a non-anxious attitude towards mathematics and science, promote an atmosphere of cooperation, support the development of the problem-solving process, supply spatial-ability skill exercises, and introduce students to elementary number theory.  Practicing such activities can give students positive perceptions about the world of mathematics and science.  Furthermore, pentominoes furnish the means for initiating discussions about many mathematical concepts and serve as concrete representations that help ease students' understanding of abstract notions.

BIBLIOGRAPHY

Bhat, Rashmi & Fletcher Audrey, Pentominoes. November 20, 1995

Golomb,Solomon W., Pentominoes. Princeton University Press, Princeton, New Jersey, 1994. Pages 6-10, 148.

Onslow,Barry, Pentominoes Revisited. American Teacher, May 1990, v37, n9. Pages 5-10.

Tracy,Dyanne M., Five Good Reasons to Use Pentominoes. School Science andMathematics, December 1990, v90, n8. Pages 665-674.

 

Pentominoes 3D

Pentominoes and Art

Congruent Shapes

http://home.scarlet.be/~demeod/indexe.html

http://www.uen.org/Lessonplan/preview.cgi?LPid=6109

http://www.theory.csc.uvic.ca/~cos/inf/misc/PentInfo.html

http://www.mathematische-basteleien.de/pentominos.htm